Steerable Neural ODEs on Homogeneous Spaces

# Steerable Neural ODEs on Homogeneous Spaces
Source: [https://arxiv.org/html/2605.11133](https://arxiv.org/html/2605.11133)
###### Abstract

We introducesteerable neural ordinary differential equationson homogeneous spacesM=G/HM=G/H\. These models constitute a novel geometric extension of manifold neural ordinary differential equations \(NODEs\) that transport associated feature vectors transforming under the local symmetry groupHH\. We interpret features as sections of associated vector bundles overMM, and describe their evolution as parallel transport\. This results in a coupled system of ODEs consisting of a flow equation onMMand a steering equation acting on features\. We show that steerable NODEs areGG\-equivariant whenever the vector field generating the flow and the connection governing parallel transport are bothGG\-invariant\. Furthermore, we demonstrate how steerable NODEs incorporate existing NODE models and continuous normalizing flows on Lie groups\. Our framework provides the geometric foundation for learning continuous\-time equivariant dynamics of general vector\-valued features on homogeneous spaces\.

Emma Andersdotteremma\.andersdotter@umu\.se Department of Mathematics and Mathematical Statistics Umeå University Umeå SE\-901 87, Sweden

Daniel Perssondaniel\.persson@chalmers\.se

Department of Mathematical Sciences Chalmers University of Technology and Gothenburg University Gothenburg SE\-412 96, Sweden

Fredrik Ohlssonfredrik\.ohlsson@umu\.se Department of Mathematics and Mathematical Statistics Umeå University Umeå SE\-901 87, Sweden

###### Contents

1. [1Introduction](https://arxiv.org/html/2605.11133#S1)1. [1\.1Background and motivation](https://arxiv.org/html/2605.11133#S1.SS1) 2. [1\.2Summary of results](https://arxiv.org/html/2605.11133#S1.SS2) 3. [1\.3Organisation of the paper](https://arxiv.org/html/2605.11133#S1.SS3)
2. [2Mathematical preliminaries](https://arxiv.org/html/2605.11133#S2)1. [2\.1Fibre bundles](https://arxiv.org/html/2605.11133#S2.SS1) 2. [2\.2Connections and parallel transport](https://arxiv.org/html/2605.11133#S2.SS2) 3. [2\.3Homogeneous spaces](https://arxiv.org/html/2605.11133#S2.SS3)
3. [3Steerable neural ODEs on homogeneous spaces](https://arxiv.org/html/2605.11133#S3)1. [3\.1Motivation and general setup](https://arxiv.org/html/2605.11133#S3.SS1) 2. [3\.2Feature fields and induced representations](https://arxiv.org/html/2605.11133#S3.SS2) 3. [3\.3Steerable NODEs through parallel transport](https://arxiv.org/html/2605.11133#S3.SS3)
4. [4Equivariance of steerable NODEs](https://arxiv.org/html/2605.11133#S4)1. [4\.1Equivariance via horizontal flows](https://arxiv.org/html/2605.11133#S4.SS1) 2. [4\.2Equivariance in the local formulation](https://arxiv.org/html/2605.11133#S4.SS2) 3. [4\.3Invariant connections and Wang’s theorem](https://arxiv.org/html/2605.11133#S4.SS3)
5. [5Relation to existing NODE models](https://arxiv.org/html/2605.11133#S5)1. [5\.1Parallel transport and the push\-forward](https://arxiv.org/html/2605.11133#S5.SS1) 2. [5\.2Continuous normalizing flows onG/HG/H](https://arxiv.org/html/2605.11133#S5.SS2)1. [5\.2\.1Continuous normalizing flows on a Lie groupGG](https://arxiv.org/html/2605.11133#S5.SS2.SSS1) 2. [5\.2\.2Connection with NODEs onG/HG/H](https://arxiv.org/html/2605.11133#S5.SS2.SSS2) 3. [5\.3Example: Steerable NODEs onS2S^\{2\}](https://arxiv.org/html/2605.11133#S5.SS3)
6. [6Conclusions](https://arxiv.org/html/2605.11133#S6)
7. [References](https://arxiv.org/html/2605.11133#bib)
8. [AProof ofDefinition˜3\.3](https://arxiv.org/html/2605.11133#A1)
9. [BProof of Wang’s theorem forG→G/HG\\to G/H](https://arxiv.org/html/2605.11133#A2)

## 1Introduction

### 1\.1Background and motivation

Neural ordinary differential equations \(NODEs\), introduced by Chen et al\.\[[7](https://arxiv.org/html/2605.11133#bib.bib13)\], are a class of machine learning models in which data is transformed continuously in time according to a learnable vector field, rather than through a finite sequence of discrete layers as in traditional neural networks\. Formally, a NODE defines a dynamical system whose flow map transports input data to output data by integrating an ordinary differential equation\. This continuous formulation endows NODEs with several attractive properties, including reversibility, parameter sharing across depth, and a natural interpretation as diffeomorphisms of the data space\. In addition, NODEs provide a principled way of modelling continuous\-time dynamics and are particularly well suited for problems involving irregular sampling, adaptive computation, and dynamical systems\.

One of the most influential applications of NODEs is in generative modelling, where they give rise to continuous normalizing flows \(CNFs\)\[[7](https://arxiv.org/html/2605.11133#bib.bib13)\]\. In this setting, a NODE defines a diffeomorphic change of variables, and probability densities are transformed via the induced push\-forward along the flow\. CNFs combine the expressivity of deep generative models with exact likelihood evaluation and invertibility, and have been further developed in a number of geometric directions, including group manifolds and quotient spaces\. This includes models for generating molecular and protein structures\[[34](https://arxiv.org/html/2605.11133#bib.bib75),[5](https://arxiv.org/html/2605.11133#bib.bib9),[25](https://arxiv.org/html/2605.11133#bib.bib49),[14](https://arxiv.org/html/2605.11133#bib.bib32),[11](https://arxiv.org/html/2605.11133#bib.bib24)\], and applications to lattice gauge theories and physics\-inspired models\[[15](https://arxiv.org/html/2605.11133#bib.bib30),[18](https://arxiv.org/html/2605.11133#bib.bib80)\]\.

Many modern machine learning problems involve data that are inherently non\-Euclidean and exhibit symmetries, such as data defined on manifolds, graphs, or quotient spaces\. Geometric deep learning aims to incorporate such structures directly into model architectures which respect the underlying geometry and symmetry constraints\. From this perspective, neural networks operating onℝn\\mathbb\{R\}^\{n\}are a special case, and more general constructions are required to handle data living on curved spaces or transforming under group actions\. Lie groups and homogeneous spaces play a central role in this setting, as they provide a natural language for describing continuous symmetries encountered in physics, robotics, and machine learning\.

A natural setting for NODEs in geometric deep learning is to consider ODEs on a smooth manifoldMM, rather thanℝn\\mathbb\{R\}^\{n\}, to accommodate non\-Euclidean data\[[12](https://arxiv.org/html/2605.11133#bib.bib28),[28](https://arxiv.org/html/2605.11133#bib.bib53),[29](https://arxiv.org/html/2605.11133#bib.bib55)\]\. The ODE that governs the NODE model is then given by the Cauchy problem

dd​t​Φp​\(t\)=ϕΦp​\(t\),Φp​\(0\)=p,\\frac\{d\}\{dt\}\\Phi\_\{p\}\(t\)=\\phi\_\{\\Phi\_\{p\}\(t\)\}\\,,\\quad\\Phi\_\{p\}\(0\)=p\\,,\(1\.1\)whereppis a point inMM,Φp:ℝ→M\\Phi\_\{p\}:\\mathbb\{R\}\\to Mis the unique solution curve, andϕ:M→T​M\\phi:M\\to TMis a learnable vector field, e\.g\., parametrised by a neural network\. The NODE can be interpreted as the diffeomorphismψ:M→M\\psi:M\\to Mdefined byψ:p↦Φp​\(1\)\\psi:p\\mapsto\\Phi\_\{p\}\(1\), mapping the inputsp∈Mp\\in Mto the outputsΦp​\(1\)∈M\\Phi\_\{p\}\(1\)\\in Mof the model\. The situation where the data exhibits global symmetries under some groupGGrequires the construction of equivariant NODEs, which were obtained in\[[21](https://arxiv.org/html/2605.11133#bib.bib44)\]for the Euclidean setting and in\[[19](https://arxiv.org/html/2605.11133#bib.bib38)\]for Riemannian manifoldsMM\. In our previous work\[[1](https://arxiv.org/html/2605.11133#bib.bib3)\], we studied equivariant NODEs on manifolds and established conditions under which invariance of the vector field implies equivariance of the resulting flow\. In particular, we showed that equivariant manifold NODEs could be further generalised and parametrised in terms of the differential invariants of the action ofGG\. We also proved they are universal approximators whenMMis connected\.

Incorporating data that also exhibits non\-trivial transformation properties in the presence of local symmetries requires the introduction of feature fields onMM\. The construction of such feature fields has been studied extensively in the context of group\-equivariant convolutional neural networks \(CNNs\)\[[9](https://arxiv.org/html/2605.11133#bib.bib21),[8](https://arxiv.org/html/2605.11133#bib.bib16),[16](https://arxiv.org/html/2605.11133#bib.bib33)\]\. In this setting, features are not treated as scalar\-valued functions, but as objects transforming in representations of the local symmetry groupHH\. Seminal work by Cohen and collaborators\[[9](https://arxiv.org/html/2605.11133#bib.bib21)\]showed that group\-equivariant CNNs can be formulated in terms of induced representations and, more generally, sections of associated vector bundles over homogeneous spaces\. From this viewpoint, convolution operators arise as equivariant maps between spaces of sections, or equivalently, intertwiners between representations ofHH, and steerability is a direct consequence of the representation theory inherent in the associated bundle construction\. This bundle\-theoretic perspective has proven powerful in the discrete\-time setting through the connection between convolutions and equivariant feed\-forward layers consistent with the global symmetry\[[23](https://arxiv.org/html/2605.11133#bib.bib46),[2](https://arxiv.org/html/2605.11133#bib.bib5)\], and more recently in the context of equivariant transformers and their generalisations\[[31](https://arxiv.org/html/2605.11133#bib.bib57)\]\.

However, an analogous framework incorporating local symmetry in continuous\-time models based on neural ODEs is still missing\. In particular, while NODEs and CNFs describe actions on scalars and scalar densities, there is no general formulation describing flow\-based models acting on general vector\- or tensor\-valued feature fields in a geometrically consistent way that respects the symmetries of the problem\.

### 1\.2Summary of results

The goal of this paper is to fill the aforementioned gap by introducing*steerable neural ODEs*, a geometric extension of NODEs that simultaneously transports points on a manifold and associated feature fields\. Our construction is based on the observation that, in the presence of symmetries, feature fields are naturally described as sections of vector bundles rather than as ordinary functions\. In particular, when the underlying manifold is a homogeneous spaceM=G/HM=G/H, feature vectors transform in representations of the stabiliser subgroupHH, and the natural geometric setting is that of associated vector bundles overG/HG/H\. Within this framework, the evolution of feature fields along NODE trajectories inMMis governed by a connection on the corresponding principal bundle\. Equivariance of the NODE on the base manifold under the global groupGGinduces equivariance of the feature transport, provided that the connection is compatible with the group action\. This leads to a notion of an induced action on feature fields, respecting all symmetries of the problem and generalising the push\-forward of densities in continuous normalizing flows\.

More specifically, we consider neural ODEs on the homogeneous spaceM=G/HM=G/H, whereGGis a Lie groupGGandHHa closed Lie subgroup\. The Lie groupGGdefines a principal bundleG→G/HG\\to G/Hwith base manifoldG/HG/Hand fibreHH\. Feature fields are mapsf:G/H→Vf:G/H\\to V, i\.e\., sections of the associated vector bundleG×ρVG\\times\_\{\\rho\}Vwith base manifoldG/HG/Hand fibreVV, whereVVis a vector space equipped with a representationρ\\rhoof the stabiliser subgroupHH\. A principal connectionω∈Ω1​\(G,𝔥\)\\omega\\in\\Omega^\{1\}\(G,\\mathfrak\{h\}\)is a differential 1\-form onGG, taking values in the Lie algebra𝔥\\mathfrak\{h\}ofHH, which defines a notion of horizontal tangent vectors in the principal bundle\. The solution curve of the NODE can be horizontally lifted to the principal bundle and used to define a notion of horizontality inherited by the associated bundle\. The action on a feature fieldffis then obtained asparallel transportalongΦp​\(t\)\\Phi\_\{p\}\(t\)by requiring that the corresponding section ofG×ρVG\\times\_\{\\rho\}VoverΦp​\(t\)\\Phi\_\{p\}\(t\)is everywhere horizontal\. The steerable NODE is defined by imposing horizontality tosteerthe feature vector as it is transported alongΦp​\(t\)\\Phi\_\{p\}\(t\)\. This amounts to extendingψ\\psito the mapψ:p↦Φp​\(1\)\\psi:p\\mapsto\\Phi\_\{p\}\(1\)to the mapΨ:M×V→M×V\\Psi:M\\times V\\to M\\times Vdefined by\(p,f\(p\)\)↦\(Φp\(1\),f\(Φp\(1\)\)\(p,f\(p\)\)\\mapsto\(\\Phi\_\{p\}\(1\),f\(\\Phi\_\{p\}\(1\)\)wheref​\(Φp​\(1\)\)f\(\\Phi\_\{p\}\(1\)\)is obtained through parallel transport\. The geometric construction of the steerable NODEs is illustrated in[Figure˜1](https://arxiv.org/html/2605.11133#S1.F1)

![Refer to caption](https://arxiv.org/html/2605.11133v1/x1.png)\(a\)Neural ODE
![Refer to caption](https://arxiv.org/html/2605.11133v1/x2.png)\(b\)Steerable neural ODE

Figure 1:In a neural ODE on a homogeneous spaceM=G/HM=G/H\([fig\.˜1\(a\)](https://arxiv.org/html/2605.11133#S1.F1.sf1)\), pointsp∈Mp\\in Mare transported to outputsΦp​\(1\)\\Phi\_\{p\}\(1\)along the flow governed by a vector fieldϕ\\phi\. In a steerable NODE \([fig\.˜1\(b\)](https://arxiv.org/html/2605.11133#S1.F1.sf2)\), this framework is extended to include features that are associated to the pointspp, defined by a feature fieldf:M→Vf:M\\to V, where the vector spaceVVcarries a representation ofHH\. The transformation of the features along the NODE trajectory is governed by parallel transport with respect to a principal connection onGG\. A more detailed illustration of a steerable NODE can be seen in[fig\.˜2](https://arxiv.org/html/2605.11133#S3.F2)\.In the steerable NODEs, we treat both the vector fieldϕ\\phiand the connectionω\\omegaas learnable quantities\. Training the model amounts to learning both the flow inM=G/HM=G/Hand the bundle geometry required to induce the observed transformation of points inMMand feature vectors inVVattached to those points\. In contrast to existing discrete\-time models, which probe representation\-theoretic aspects of the associated vector bundles, the continuous\-time steerable NODEs also probe the geometric structure of the bundlesGGandG×ρVG\\times\_\{\\rho\}Vvia the connectionω\\omega\. Consequently, training the steerable NODE not only results in a diffeomorphism approximating the observed transformations of feature vectors, but also provides insight and information regarding the geometry of the underlying problem\. In this way, our construction introduces a novel perspective on NODEs in geometric deep learning, incorporating learning of intrinsically geometric structures and broadening the scope to arbitrary feature vectors on homogeneous spaces\.

The main contributions we describe in this paper are as follows:

- •We introduce steerable NODEs on homogeneous spacesM=G/HM=G/Has parallel transport in associated vector bundlesG×ρVG\\times\_\{\\rho\}V\. In[Definition˜3\.7](https://arxiv.org/html/2605.11133#S3.Thmtheorem7), we provide a concrete description of the model as a system of non\-linear ODEs; the first describes a flow inMM, and the second describes the parallel transport steering the feature vector as it is transported along the flow \(see[fig\.˜1](https://arxiv.org/html/2605.11133#S1.F1)\)\.
- •We show that the steerable NODE is equivariant with respect to the global action ofGGwheneverϕ\\phiandω\\omegaare bothGG\-invariant \([Theorem˜4\.6](https://arxiv.org/html/2605.11133#S4.Thmtheorem6)\)\. In[section˜4\.3](https://arxiv.org/html/2605.11133#S4.SS3), we also provide concrete parametrisations of the spaces of invariant vector fields and connections using our previous result from\[[1](https://arxiv.org/html/2605.11133#bib.bib3)\]and Wang’s theorem \([Section˜4\.3](https://arxiv.org/html/2605.11133#S4.SS3)\)\.
- •In[section˜5](https://arxiv.org/html/2605.11133#S5), we show that existing NODEs on Lie groups are encompassed by steerable NODEs in the special case whereHHis trivial, and how steerable NODEs extend the modelling capabilities of current flow\-based machine learning models\. In particular, we demonstrate that steerable NODEs encompass certain continuous normalizing flows for lattice gauge theories studied recently\.

### 1\.3Organisation of the paper

In[section˜2](https://arxiv.org/html/2605.11133#S2), we review the necessary differential geometric background, including fibre bundles, principal connections, parallel transport, and homogeneous spaces\. In particular, we recall the construction of associated vector bundles and the description of sections in terms of Mackey functions and induced representations\. We then introduce steerable neural ODEs on homogeneous spaces in[section˜3](https://arxiv.org/html/2605.11133#S3)\. We first reinterpret feature fields as sections of associated bundles and then construct the induced evolution of features via horizontal lifts and parallel transport\. This leads to a precise definition of steerable NODEs as coupled flows on base manifolds and associated bundles\. In[section˜4](https://arxiv.org/html/2605.11133#S4)we prove that invariance of the base vector field together with invariance of the connection implies equivariance of the lifted flow and of the induced feature transport\. This establishes sufficient geometric conditions for steerable NODEs to respect global symmetries\. Furthermore, we apply Wang’s theorem\[[33](https://arxiv.org/html/2605.11133#bib.bib71)\]to classifyGG\-invariant principal connections onG→G/HG\\to G/H\. Together with previous results on the classification of invariant base vector fields using differential invariants, this provides an explicit parametrisation of all admissible equivariant steerable NODE models onG/HG/H\. Finally, in[section˜5](https://arxiv.org/html/2605.11133#S5), we discuss the relation of our framework to existing NODE models, including continuous normalizing flows and previously studied equivariant NODE constructions\. We conclude in[section˜6](https://arxiv.org/html/2605.11133#S6)with directions for future work\.

## 2Mathematical preliminaries

The construction of steerable NODEs presented in the following sections relies on concepts from differential geometry, specifically the notions of fibre bundles and homogeneous spaces\. In this section, we recall the relevant definitions and properties of these geometric constructs, assuming throughout that the manifolds and maps considered are all smooth\. Proofs and detailed derivations are omitted for brevity, but can be readily found in standard texts on differential geometry such as\[[20](https://arxiv.org/html/2605.11133#bib.bib42),[30](https://arxiv.org/html/2605.11133#bib.bib56),[24](https://arxiv.org/html/2605.11133#bib.bib48)\]\. The reader who is already familiar with these topics can skip to[section˜3](https://arxiv.org/html/2605.11133#S3)\.

### 2\.1Fibre bundles

Intuitively, a fibre bundle is a base manifoldMMwith copies of a fibre manifoldFFattached to each pointp∈Mp\\in Mand glued together in a possibly twisted manner\. The canonical examples are provided by the cylinderS1×\[0,1\]S^\{1\}\\times\[0,1\]and the Möbius strip, which have the same local structure – both are locally homeomorphic toℝ2\\mathbb\{R\}^\{2\}– but are globally different due to the twist in the fibre\[0,1\]\[0,1\]over the base manifoldS1S^\{1\}in the Möbius strip\. The definition of a fibre bundle formalises this notion of a space that is locally, but not globally, a product space\.

###### Definition 2\.1\(Fibre bundle\)\.

Abundle of manifoldsis a triple\(E,π,M\)\(E,\\pi,M\), denotedE​→𝜋​ME\\overset\{\\pi\}\{\\to\}M, whereEEandMMare manifolds referred to as thetotal spaceand thebase space, respectively, andπ:E→M\\pi:E\\to Mis a surjective map called theprojection\. ThefibreEpE\_\{p\}at a pointp∈Mp\\in Mis the pre\-image of the projection,Ep=π−1​\(p\)E\_\{p\}=\\pi^\{\-1\}\(p\)\. Afibre bundlewith typical \(or canonical\) fibreFFis a bundleE​→𝜋​ME\\overset\{\\pi\}\{\\to\}Mtogether with a local trivialization\{Ui,ϕi\}\\\{U\_\{i\},\\phi\_\{i\}\\\}, where\{Ui\}\\\{U\_\{i\}\\\}is an open cover ofMMandϕi:π−1​\(Ui\)→Ui×F\\phi\_\{i\}:\\pi^\{\-1\}\(U\_\{i\}\)\\to U\_\{i\}\\times Fis a diffeomorphism such thatproj1∘ϕi=π\\text\{proj\}\_\{1\}\\circ\\phi\_\{i\}=\\pionπ−1​\(Ui\)\\pi^\{\-1\}\(U\_\{i\}\)\.

A section of a fibre bundle is a mapσ:M→E\\sigma:M\\to Eassigning to each pointppinMMa point in the corresponding fibreEpE\_\{p\}\. In general, a fibre bundle admits no global sections due to topological obstructions\. However, local sections obtained by restricting the domain ofσ\\sigmato the open setsUiU\_\{i\}inMMalways exist\.

###### Definition 2\.2\(Section of a bundle\)\.

Asectionof a fibre bundleE​→𝜋​ME\\overset\{\\pi\}\{\\to\}Mis a mapσ:M→E\\sigma:M\\to Esuch thatπ∘σ​\(p\)=p\\pi\\circ\\sigma\(p\)=pfor allp∈Mp\\in M\.

We are particularly interested in fibre bundles admitting a Lie group of local symmetries acting on the total space\. Throughout, we will use the notationLgL\_\{g\}andRgR\_\{g\}to refer to the left and right actions, respectively, of a group elementg∈Gg\\in G\. For the situations where the action is on points on a manifold, we will sometimes use juxtaposition for brevity\.

###### Definition 2\.3\(Principal bundle\)\.

AprincipalHH\-bundleis a fibre bundleE​→𝜋​ME\\overset\{\\pi\}\{\\to\}Mtogether with a Lie groupHHand a free right action ofHHonEE,E×H→EE\\times H\\to E, which is fibre\-preserving \(meaning thatπ​\(u​h\)=π​\(u\)\\pi\(uh\)=\\pi\(u\)for anyu∈Eu\\in Eandh∈Hh\\in H\) and transitive on each fibre\.

The fibres of a principalHH\-bundle are theHH\-orbits, all of which are diffeomorphic \(but not isomorphic\) to the groupHH\. Given a principalHH\-bundleEEand a representationρ\\rhoof the symmetry groupHH, it is possible to define a vector bundle which inherits the topology and local symmetry ofEE\.

###### Definition 2\.4\(Associated vector bundle\)\.

LetE​→𝜋​ME\\overset\{\\pi\}\{\\to\}Mbe a principalHH\-bundle, letVVbe a vector space, and letρ:H→GL​\(V\)\\rho:H\\to\\mathrm\{GL\}\(V\)be a representation ofHH\. Let∼\\simbe the equivalence relation onE×VE\\times Vdefined by\(u,v\)∼\(u​h,ρ​\(h−1\)​v\)\(u,v\)\\sim\(uh,\\rho\(h^\{\-1\}\)\\,v\)for anyh∈Hh\\in H, and denote the corresponding equivalence classes by\[u,v\]=\{\(u​h,ρ​\(h−1\)​v\)\|h∈H\}\[u,v\]=\\\{\(uh,\\rho\(h^\{\-1\}\)v\)\\,\|\\,h\\in H\\\}\. Theassociated vector bundleis then defined as the space

E×ρV:=\(E×V\)/∼,E\\times\_\{\\rho\}V:=\(E\\times V\)/\\sim\\,,\(2\.1\)equipped with the projection operatorπρ:E×ρV→M\\pi\_\{\\rho\}:E\\times\_\{\\rho\}V\\to Mdefined byπρ​\(\[u,v\]\)=π​\(u\)\\pi\_\{\\rho\}\(\[u,v\]\)=\\pi\(u\)\.

By construction, the projection is independent of the representative of the equivalence class, sinceπρ​\(\[u​h,ρ​\(h−1\)​v\]\)=π​\(u​h\)=π​\(u\)=πρ​\(\[u,v\]\)\\pi\_\{\\rho\}\(\[uh,\\rho\(h^\{\-1\}\)v\]\)=\\pi\(uh\)=\\pi\(u\)=\\pi\_\{\\rho\}\(\[u,v\]\), and the fibre ofE×ρVE\\times\_\{\\rho\}Vis diffeomorphic toVV\.

A section of the associated vector bundle can be expressed in terms of a sectionσ\\sigmaofEEand a vector fieldf:M→Vf:M\\to VonMMass​\(p\)=\[σ​\(p\),f​\(p\)\]s\(p\)=\\left\[\\sigma\(p\),f\(p\)\\right\]\. Under a change of the principal section of the formσ′​\(p\)=σ​\(p\)​h\\sigma^\{\\prime\}\(p\)=\\sigma\(p\)hfor someh∈Hh\\in H, the vectorf​\(p\)f\(p\)transforms according to the representationρ\\rhoof the local symmetry groupHH,\[σ′​\(p\),f​\(p\)\]=\[σ​\(p\)​h,f​\(p\)\]=\[σ​\(p\),ρ​\(h\)​f​\(p\)\]\[\\sigma^\{\\prime\}\(p\),f\(p\)\]=\[\\sigma\(p\)h,f\(p\)\]=\[\\sigma\(p\),\\rho\(h\)f\(p\)\]\.

Fixing the sectionσ\\sigmaamounts to choosing a reference inVV\. This can be made manifest through the canonical local trivialisationφ\\varphidefined as follows\.

###### Definition 2\.5\(Canonical local trivialization\)\.

Thecanonical local trivialisationφ:M×V→E×ρV\\varphi:M\\times V\\to E\\times\_\{\\rho\}Vrelative to the sectionσ\\sigmaof the associated vector bundle is given by

φ​\(p,v\)=\[σ​\(p\),v\],\\varphi\(p,v\)=\[\\sigma\(p\),v\],\(2\.2\)wherep∈Mp\\in Mandv∈Vv\\in V\.

The canonical local trivialisationφ\\varphiis a \(local\) diffeomorphism by construction\.

### 2\.2Connections and parallel transport

In order to relate geometric information defined at different points onMM, we need a construction relating the fibres of a bundle over different points in the base manifoldMM\. For principal bundles, this is provided by a connection, which defines a notion of parallel transport of a point in the total space of the bundle along a curveMM\. This notion is then inherited by the associated vector bundles\.

In order to define parallel transport rigorously for a principalHH\-bundleE​→𝜋​ME\\overset\{\\pi\}\{\\rightarrow\}M, we first need to introduce precise notions of vertical and horizontal directions in the total spaceEE\. The vertical subspaceVu​E⊂Tu​EV\_\{u\}E\\subset T\_\{u\}Eis the component of the tangent spaceTu​ET\_\{u\}Eatu∈Eu\\in Ealong the fibreEpE\_\{p\}overp=π​\(u\)p=\\pi\(u\),

VuE:=ker\(πu\)∗=\{X∈TuE\|\(πu\)∗\(X\)=0\}\.V\_\{u\}E:=\\ker\\,\(\\pi\_\{u\}\)\_\{\*\}=\\\{X\\in T\_\{u\}E\\,\|\\,\(\\pi\_\{u\}\)\_\{\*\}\(X\)=0\\\}\\,\.\(2\.3\)The vertical subspace can be explicitly constructed using the right action ofHH\. LetA∈𝔥A\\in\\mathfrak\{h\}be an element of the Lie algebra𝔥\\mathfrak\{h\}ofHHand define a curveΓu\\Gamma\_\{u\}onEEthroughuuby

Γu​\(t\)=u​exp⁡\(t​A\)\.\\Gamma\_\{u\}\(t\)=u\\exp\{\(tA\)\}\\,\.\(2\.4\)The curve is contained in the fibre atpp,Γu⊂Ep\\Gamma\_\{u\}\\subset E\_\{p\}, sinceπ​\(u​exp⁡\(t​A\)\)=π​\(u\)=p\\pi\(u\\exp\{\(tA\)\}\)=\\pi\(u\)=p, and its tangent defines thefundamental vector fieldA\#A^\{\\\#\}generated byAA,

A\#​\(u\)=dd​t​u​exp⁡\(t​A\)\|t=0=Γ˙u​\(0\),A^\{\\\#\}\(u\)=\\left\.\\frac\{d\}\{dt\}u\\exp\{\(tA\)\}\\right\|\_\{t=0\}=\\dot\{\\Gamma\}\_\{u\}\(0\)\\,,\(2\.5\)whereΓ˙\\dot\{\\Gamma\}denotes the derivative with respect tott\. Sinceπ∘Γp\\pi\\circ\\Gamma\_\{p\}is constant with respect tott, we have

π∗​\(A\#​\(u\)\)=dd​t​π​\(Γu​\(t\)\)\|t=0=0\\pi\_\{\*\}\(A^\{\\\#\}\(u\)\)=\\left\.\\frac\{d\}\{dt\}\\pi\(\\Gamma\_\{u\}\(t\)\)\\right\|\_\{t=0\}=0\(2\.6\)for anyu∈Eu\\in E\. Consequently,A\#​\(u\)∈Vu​EA^\{\\\#\}\(u\)\\in V\_\{u\}EandA\#A^\{\\\#\}is a vertical vector field onEE\. In fact, the fundamental vector fields provide an isomorphism of vector spaces𝔥≅Vu​E\\mathfrak\{h\}\\cong V\_\{u\}E\.

The horizontal subspaceHu​E⊂Tu​EH\_\{u\}E\\subset T\_\{u\}Eis the complement of the vertical subspaceVu​EV\_\{u\}E\. However, unlike the vertical subspace, it is not uniquely determined by the bundle topology alone but requires additional geometric structure in the form of a connection that describes how vectors in the tangent spaceTu​ET\_\{u\}Eare projected onto the vertical subspace\.

###### Definition 2\.6\(Principal connection\)\.

A \(Ehresmann\)connection1\-formω∈Ω1​\(E,𝔥\)\\omega\\in\\Omega^\{1\}\(E,\\mathfrak\{h\}\)on the principalHH\-bundleEEis a projection onto the vertical componentVu​EV\_\{u\}Esatisfying the following requirements:

- \(i\)ω​\(A\#​\(u\)\)=A\\omega\(A^\{\\\#\}\(u\)\)=A, for allu∈Eu\\in EandA∈𝔥A\\in\\mathfrak\{h\}\.
- \(ii\)Rh∗​ω=A​dh−1​ωR\_\{h\}^\{\*\}\\omega=Ad\_\{h^\{\-1\}\}\\omega, for allh∈Hh\\in H\.

Given a connection 1\-formω\\omega, the horizontal subspaceHu​EH\_\{u\}E, for anyu∈Eu\\in E, is defined by

Hu​E:=ker⁡ωu=\{X∈Tu​E\|ωu​\(X\)=0\}\.H\_\{u\}E:=\\ker\\omega\_\{u\}=\\\{X\\in T\_\{u\}E\\,\|\\,\\omega\_\{u\}\(X\)=0\\\}\\,\.\(2\.7\)It follows that the connection defines a separation of the tangent spaceTu​E=Vu​E⊕Hu​ET\_\{u\}E=V\_\{u\}E\\oplus H\_\{u\}Esuch that any smooth vector fieldXXonEEis decomposed into smooth componentsX=XH\+XVX=X^\{H\}\+X^\{V\}, whereXuH∈Hu​EX^\{H\}\_\{u\}\\in H\_\{u\}EandXuV∈Vu​EX^\{V\}\_\{u\}\\in V\_\{u\}E\. Moreover, the separation is equivariant with respect to the right action ofHH,Hu​h​E=Rh⁣∗​Hu​EH\_\{uh\}E=R\_\{h\*\}H\_\{u\}Efor allu∈Eu\\in Eandh∈Hh\\in H\.

The connectionω\\omegaprovides the desired notion of transporting geometric information along a curve in the base manifoldMM\. Starting with the principal bundle, a curveγ\\gammainMMcan be lifted toEEin a way that respects the decomposition ofTu​ET\_\{u\}Einduced byω\\omegaby requiring that its tangent vector remains inHu​EH\_\{u\}Eat each point\.

###### Definition 2\.7\(Horizontal lift\)\.

Ahorizontal liftof a curveγ:I→M\\gamma:I\\to Mis a curveγ~:I→E\\tilde\{\\gamma\}:I\\to Esuch thatπ∘γ~=γ\\pi\\circ\\tilde\{\\gamma\}=\\gammaandγ~˙​\(t\)∈Hγ~​\(t\)​E\\dot\{\\tilde\{\\gamma\}\}\(t\)\\in H\_\{\\tilde\{\\gamma\}\(t\)\}Efor allt∈It\\in I\.

The horizontal lift can be expressed in terms of a local sectionσ\\sigmaasγ~​\(t\)=σ​\(γ​\(t\)\)​h​\(t\)\\tilde\{\\gamma\}\(t\)=\\sigma\(\\gamma\(t\)\)h\(t\), whereh:ℝ→Hh:\\mathbb\{R\}\\to H\. In terms of the connection, the condition defining the horizontal liftγ~\\tilde\{\\gamma\}isω​\(γ~˙​\(t\)\)=0\\omega\(\\dot\{\\tilde\{\\gamma\}\}\(t\)\)=0, or equivalently,

dd​t​h​\(t\)=−\(Rh\)∗​\[ω​\(σ∗​\(dd​t​γ​\(t\)\)\)\]\.\\frac\{d\}\{dt\}h\(t\)=\-\(R\_\{h\}\)\_\{\*\}\\left\[\\omega\\left\(\\sigma\_\{\*\}\\left\(\\frac\{d\}\{dt\}\\gamma\(t\)\\right\)\\right\)\\right\]\\,\.\(2\.8\)Note that the induced right action\(Rh\)∗\(R\_\{h\}\)\_\{\*\}is acting on the vector in𝔥\\mathfrak\{h\}obtained by evaluatingω​\(σ∗​γ˙​\(t\)\)\\omega\(\\sigma\_\{\*\}\\dot\{\\gamma\}\(t\)\)\. This is an ODE forh​\(t\)h\(t\), and standard ODE theory guarantees the existence and uniqueness of a horizontal lift given an initial conditionγ~​\(0\)∈π−1​\(γ​\(0\)\)\\tilde\{\\gamma\}\(0\)\\in\\pi^\{\-1\}\(\\gamma\(0\)\)\. Since the horizontal subspaces are equivariant,\(Rh\)∗​Hu​E=Hu​h​E\(R\_\{h\}\)\_\{\*\}H\_\{u\}E=H\_\{uh\}E,HHacts transitively on the right on the horizontal lifts ofγ\\gammaby shifting the initial condition\. The pointγ~​\(t\)\\tilde\{\\gamma\}\(t\)is called theparallel transportofγ~​\(0\)\\tilde\{\\gamma\}\(0\)alongγ\\gamma\.

The horizontal lift of curves can be extended to a flowΦ:ℝ×M→M\\Phi:\\mathbb\{R\}\\times M\\to Mgenerated by a vector fieldϕ:M→T​M\\phi:M\\to TM,

dd​t​Φ​\(t,p\)=ϕΦ​\(t,p\),Φ​\(0,p\)=p\.\\frac\{d\}\{dt\}\\Phi\(t,p\)=\\phi\_\{\\Phi\(t,p\)\}\\,,\\quad\\Phi\(0,p\)=p\\,\.\(2\.9\)Given a fixed connectionω\\omega, there is a unique lift ofϕ\\phito a vector fieldϕ~:E→T​E\\tilde\{\\phi\}:E\\to TEwhich is horizontal,ω​\(ϕ~\)=0\\omega\(\\tilde\{\\phi\}\)=0,HH\-invariant,\(Rh\)∗​ϕ~=ϕ~\(R\_\{h\}\)\_\{\*\}\\tilde\{\\phi\}=\\tilde\{\\phi\}for allh∈Hh\\in H, and satisfies the defining propertyπ∗​\(ϕ~\)=ϕ\\pi\_\{\*\}\(\\tilde\{\\phi\}\)=\\phiof a lift\[[4](https://arxiv.org/html/2605.11133#bib.bib7)\]\. The flowΦ~:ℝ×E→E\\tilde\{\\Phi\}:\\mathbb\{R\}\\times E\\to Egenerated byϕ~\\tilde\{\\phi\}gives the horizontal lifts of the integral curves ofϕ\\phiand is equivariant underHHby construction,Φ~​\(t,u\)​h=Φ~​\(t,u​h\)\\tilde\{\\Phi\}\(t,u\)h=\\tilde\{\\Phi\}\(t,uh\)for allh∈Hh\\in H\.

Parallel transport in the principal bundle induces a notion of parallel transport in the associated vector bundle in the following way\. A section ofE×ρVE\\times\_\{\\rho\}Valong a curveγ:ℝ→M\\gamma:\\mathbb\{R\}\\to Mcan always be expressed in terms of the horizontal lift as

s\(γ\(t\)\)=\[\(γ~\(t\),η\(γ\(t\)\)\],s\(\\gamma\(t\)\)=\[\(\\tilde\{\\gamma\}\(t\),\\eta\(\\gamma\(t\)\)\],\(2\.10\)whereγ~\\tilde\{\\gamma\}is the horizontal lift ofγ\\gammaandη​\(γ​\(t\)\)∈V\\eta\(\\gamma\(t\)\)\\in Vis the fibre component\. Thecovariant derivativeof the sectionssalongγ\\gammadescribes how the value of the section changes relative to parallel transport defined by the connection\.

###### Definition 2\.8\.

Letϕ:M→T​M\\phi:M\\to TMbe a vector field andγ:ℝ→M\\gamma:\\mathbb\{R\}\\to Ma curve withγ˙​\(t\)=ϕ\\dot\{\\gamma\}\(t\)=\\phi\. Lets​\(γ​\(t\)\)=\[γ~​\(t\),η​\(γ​\(t\)\)\]s\(\\gamma\(t\)\)=\[\\tilde\{\\gamma\}\(t\),\\eta\(\\gamma\(t\)\)\]be a section of the associated bundleE×ρVE\\times\_\{\\rho\}V\. Then thecovariant derivativeofssalongγ​\(t\)\\gamma\(t\)is given by

\(∇ϕs\)​\(γ​\(t\)\):=\[\(γ~​\(t\),dd​t​η​\(γ​\(t\)\)\)\]\.\(\\nabla\_\{\\phi\}s\)\(\\gamma\(t\)\):=\\left\[\\left\(\\tilde\{\\gamma\}\(t\),\\frac\{d\}\{dt\}\\eta\(\\gamma\(t\)\)\\right\)\\right\]\.\(2\.11\)

We note that the definition of the covariant derivative is independent of the sectionσ\\sigmaused to express the horizontal lift ofγ​\(t\)\\gamma\(t\)\. Furthermore, since all possible horizontal lifts ofγ​\(t\)\\gamma\(t\)are related by the transitive right action ofHH, the covariant derivative in \([2\.11](https://arxiv.org/html/2605.11133#S2.E11)\) is, in fact, also independent of the choice of lift or equivalently, the initial conditionγ~​\(0\)\\tilde\{\\gamma\}\(0\)\. The sectionssundergoesparallel transportalongγ\\gammaif\(∇ϕs\)​\(γ​\(t\)\)=\[\(γ~​\(t\),0\)\]\(\\nabla\_\{\\phi\}s\)\(\\gamma\(t\)\)=\\left\[\\left\(\\tilde\{\\gamma\}\(t\),0\\right\)\\right\]for alltt\.

### 2\.3Homogeneous spaces

In the presence of a global Lie groupGGof symmetries acting transitively on a manifoldMMon the left,G×M→MG\\times M\\to M, the manifold can be described as thehomogeneous spaceM=G/HM=G/H, whereHHis the stabiliser Lie subgroup of the action ofGG\. Such spaces carry a natural bundle structure, allowing for the machinery of the previous principal and associated bundles overMMto be applied\.

###### Example 2\.9\.

Consider the manifoldM=G/HM=G/H, whereGGis a Lie group andHHa closed Lie subgroup\. Letπ:G→G/H\\pi:G\\to G/Hbe the map defined byπ​\(g\)=g​H\\pi\(g\)=gHfor eachg∈Gg\\in G, that is, the map sending each elementg∈Gg\\in Gto its corresponding coset\. ThenG​→𝜋​G/HG\\overset\{\\pi\}\{\\to\}G/His a fibre bundle with typical fibreHH; in fact, it is an example of a principalHH\-bundle sinceπ​\(g​h\)=g​H​h=g​H=π​\(g\)\\pi\(gh\)=gHh=gH=\\pi\(g\)andHHacts transitively onπ−1​\(g​H\)\\pi^\{\-1\}\(gH\)\.

The groupsGGandHHserve different geometrical roles\. The local right action ofHHpreserves the fibre, while the global left action ofGGacts transitively, meaning it moves points from one fibre to another\. The sections of vector bundles associated toG​→𝜋​G/HG\\overset\{\\pi\}\{\\to\}G/Hsimultaneously accommodate both these actions in a consistent way\.

###### Example 2\.10\.

LetVVbe a vector space andρ:H→GL​\(V\)\\rho:H\\to\\mathrm\{GL\}\(V\)a representation ofHH\. LetG×ρVG\\times\_\{\\rho\}Vbe the associated vector bundle with elements\[g,v\]=\{\(g​h,ρ​\(h−1\)​v\)\|h∈H\}\[g,v\]=\\\{\(gh,\\rho\(h^\{\-1\}\)v\)\\,\|\\,h\\in H\\\},\(g,v\)∈G×V\(g,v\)\\in G\\times V, and bundle projectionπρ​\(\[g,v\]\)=π​\(g\)=g​H\\pi\_\{\\rho\}\(\[g,v\]\)=\\pi\(g\)=gH\. The base space ofG×ρVG\\times\_\{\\rho\}VisG/HG/Hand its typical fibre isVV\. Given a sectionσ\\sigmaofGG, a sectionssofG×ρVG\\times\_\{\\rho\}Vcan be expressed ass​\(p\)=\[σ​\(p\),f​\(p\)\]s\(p\)=\[\\sigma\(p\),f\(p\)\], wheref:M→Vf:M\\to V\.

The fact that the value of the sections:M→Vs:M\\to Vis measured relative to a sectionσ\\sigmaof the principal bundle can be made manifest using an alternative perspective available for the case of principal bundles over homogeneous spaces\.

###### Definition 2\.11\(Mackey function\)\.

LetGGbe a Lie group with a closed Lie subgroupHH\. LetVVbe a vector spaceVVand letρ:H→GL​\(V\)\\rho:H\\to\\mathrm\{GL\}\(V\)be a representation ofHH\. AMackey functionis a mapk:G→Vk:G\\to Vsatisfyingk​\(g​h\)=ρ​\(h−1\)​k​\(g\)k\(gh\)=\\rho\(h^\{\-1\}\)k\(g\)for allh∈Hh\\in Handg∈Gg\\in G\.

The spaceℐ\\mathcal\{I\}of Mackey functions is a vector space isomorphic to the space of sections ofG×ρVG\\times\_\{\\rho\}V\[[22](https://arxiv.org/html/2605.11133#bib.bib45)\]\. Concretely, a section ofG×ρVG\\times\_\{\\rho\}Vcan be written as\[σ​\(p\),f​\(p\)\]\[\\sigma\(p\),f\(p\)\], wheref=k∘σf=k\\circ\\sigmaandk:G→Vk:G\\to Vis a Mackey function\. The action ofHHonVVthrough the representationρ\\rhocan be extended to an action ofGGthrough theinduced representationIndHG​ρ:G→End​\(ℐ\)\\mathrm\{Ind\}^\{G\}\_\{H\}\\rho:G\\to\\mathrm\{End\}\(\\mathcal\{I\}\)defined byIndHG​ρ​\(g\)​k​\(g′\)=k​\(g−1​g′\)\\mathrm\{Ind\}^\{G\}\_\{H\}\\rho\(g\)k\(g^\{\\prime\}\)=k\(g^\{\-1\}g^\{\\prime\}\), forg,g′∈Gg,g^\{\\prime\}\\in G\. This provides a well\-defined notion of a global action ofGGon the vector\-valued sections ofG×ρVG\\times\_\{\\rho\}V\.

## 3Steerable neural ODEs on homogeneous spaces

In this section, we introducesteerableneural ODEs on homogeneous spacesM=G/HM=G/Has a geometric extension of manifold neural ODEs \(NODEs\) incorporating general feature fields using parallel transport\. Our construction draws on the concepts of differential geometry introduced in[section˜2](https://arxiv.org/html/2605.11133#S2), in particular horizontal lifts and their properties\. The resulting model represents a novel approach to flow\-based machine learning models that accommodates features with non\-trivial transformation properties under the local action of the stabiliser subgroupHH\.

### 3\.1Motivation and general setup

We recall the definition of manifold NODEs\[[12](https://arxiv.org/html/2605.11133#bib.bib28),[28](https://arxiv.org/html/2605.11133#bib.bib53),[29](https://arxiv.org/html/2605.11133#bib.bib55)\], i\.e\., NODEs that transform points in a manifoldMMthrough the diffeomorphism defined by a flow onMM\.

###### Definition 3\.1\(Manifold NODE\)\.

Letϕ:M→T​M\\phi:M\\to TMbe a learnable vector field onMM\. Then for every pointp∈Mp\\in M, there exists a unique curveΦp:ℝ→M\\Phi\_\{p\}:\\mathbb\{R\}\\to Mthat solves the Cauchy problem

dd​t​Φp​\(t\)=ϕΦp​\(t\),Φp​\(0\)=p\.\\frac\{d\}\{dt\}\{\\Phi\}\_\{p\}\(t\)=\\phi\_\{\\Phi\_\{p\}\(t\)\}\\,,\\quad\\Phi\_\{p\}\(0\)=p\\,\.\(3\.1\)Themanifold NODEonMMis the diffeomorphismψ:M→M\\psi:M\\to Mdefined byψ​\(p\)=Φp​\(1\)\\psi\(p\)=\\Phi\_\{p\}\(1\)\. The pointp∈Mp\\in Mis theinputandψ​\(p\)∈M\\psi\(p\)\\in Mis theoutputof the model\.

A NODE can, therefore, be thought of as a curve on a manifoldMMwhose tangent vector at each point is determined by a machine learning model\. Geometrically, it is convenient to identify the manifold NODE with the flowΦ:ℝ×M→M\\Phi:\\mathbb\{R\}\\times M\\to Mobtained by collecting all integral curvesΦp​\(t\)\\Phi\_\{p\}\(t\)ofϕ\\phi, i\.e\.,Φ​\(t,p\)=Φp​\(t\)\\Phi\(t,p\)=\\Phi\_\{p\}\(t\)\.

Our objective is to generalise manifold NODEs so that each point on the manifoldMMcarries a feature valued in a vector spaceVV, and to describe how these features change as the points move with the flow onMMgenerated byϕ\\phi\. In the presence of a groupGGof global symmetries acting onMM, we also require that the features transform in a consistent way\. Previous work on symmetries of manifold NODEs \(e\.g\.,\[[21](https://arxiv.org/html/2605.11133#bib.bib44),[19](https://arxiv.org/html/2605.11133#bib.bib38),[1](https://arxiv.org/html/2605.11133#bib.bib3)\]\) considered the global action ofGGthat translates points inMM\. When features are attached to points ofMM, these features may also transform under an internal symmetry group\. For example, a tangent vector atp∈Mp\\in Mcan be rotated while remaining in the tangent spaceTp​MT\_\{p\}M\.

We assume thatGGis a Lie group acting smoothly onMM\. To ensure that the action does not distinguish between points inMM, we also assume that it is transitive, makingMMinto a homogeneousGG\-space,M=G/HM=G/H, withHHa closed subgroup ofGG\(see[section˜2](https://arxiv.org/html/2605.11133#S2)\)\. This setting naturally provides a notion of local transformations of the feature vectors, through a representationρ:H→GL​\(V\)\\rho:H\\to\\mathrm\{GL\}\(V\)of the stabiliser subgroupHH, which is consistent with the global action ofGG\.

In order to define an action on feature fields induced by the flow inM=G/HM=G/H, we proceed by leveraging the geometric perspective on feature fields onMMas sections of associated bundles\.

### 3\.2Feature fields and induced representations

We first give a rigorous geometric definition of a feature field, i\.e\., a smooth mapf:M→Vf:M\\to Vassigning to each pointppin the homogeneous spaceM=G/HM=G/Han element in some vector spaceVV\. Since such fields must be compatible with the global action ofGGonMMand with the local action of the stabiliser subgroupHH, we seek a vector bundle overMMthat encodes these symmetries\. The bundle with these properties is precisely the associated vector bundleG×ρVG\\times\_\{\\rho\}Vdefined in[example˜2\.10](https://arxiv.org/html/2605.11133#S2.Thmtheorem10), which inherits the global action ofGGfrom the principal bundleG​→𝜋​G/HG\\overset\{\\pi\}\{\\to\}G/Hand a local action ofHHthrough the representationρ:H→GL​\(V\)\\rho:H\\to\\mathrm\{GL\}\(V\)\. Under this construction, feature fields are naturally identified with smooth sections of the associated bundle\.

###### Definition 3\.2\(Feature fields\)\.

LetM=G/HM=G/Hbe a homogeneous space andVVbe a vector space carrying a representationρ:H→GL​\(V\)\\rho:H\\to\\mathrm\{GL\}\(V\)ofHH\. Afeature fieldonMMis a section of the associated vector bundleG×ρVG\\times\_\{\\rho\}V\.

In applications, it is often convenient to work with a local description of a feature field\. We therefore introduce an equivalent description in terms of local sections of the principal bundle and Mackey functions\.

###### Definition 3\.3\(Local feature fields\)\.

LetM=G/HM=G/Hbe a homogeneous space and letσ:M→G\\sigma:M\\to Gbe a local section of the principal bundleG​→𝜋​MG\\overset\{\\pi\}\{\\to\}M\. Alocal feature fieldrelative toσ\\sigmais a mapf:M→Vf:M\\to Vdefined byf=k∘σf=k\\circ\\sigma, wherek:G→Vk:G\\to Vis a Mackey function defined in[definition˜2\.11](https://arxiv.org/html/2605.11133#S2.Thmtheorem11)\.

\{restatable\}

lemmaMackeyLemma[Definitions˜3\.2](https://arxiv.org/html/2605.11133#S3.Thmtheorem2)and[3\.3](https://arxiv.org/html/2605.11133#S3.Thmtheorem3)are equivalent, establishing a one‑to‑one correspondence between Mackey functions and sections of the associated vector bundle\.

It follows from[Definition˜3\.3](https://arxiv.org/html/2605.11133#S3.Thmtheorem3)that, given a local sectionσ:M→G\\sigma:M\\to Gof the principal bundle, any sections:M→G×ρVs:M\\to G\\times\_\{\\rho\}V\(i\.e\., feature field\) of the associated bundle admits the local expression

s​\(p\)=\[σ​\(p\),f​\(p\)\]=\[σ​\(p\),k​\(σ​\(p\)\)\],s\(p\)=\\left\[\\sigma\(p\),f\(p\)\\right\]=\\left\[\\sigma\(p\),k\(\\sigma\(p\)\)\\right\]\\,,\(3\.2\)wherep∈Mp\\in M,k:G→Vk:G\\to Vis a uniquely determined Mackey function andf=k∘σf=k\\circ\\sigmais the local feature field in[definition˜3\.3](https://arxiv.org/html/2605.11133#S3.Thmtheorem3)\. The sectionssis independent of the choice ofσ\\sigma– a property inherited fromkk\.

While the point\[σ​\(p\),k​\(σ​\(p\)\)\]\[\\sigma\(p\),k\(\\sigma\(p\)\)\]in the total space of the bundleG×ρV→MG\\times\_\{\\rho\}V\\to Mis independent ofσ\\sigma, the vector in the fibreVVcorresponding to this point is not\. Indeed, the vectorf​\(p\)f\(p\)is measured relative to the sectionσ\\sigma, and fixingσ\\sigmaamounts to choosing a reference inVV\. This can be done through the canonical local trivialization introduced in[definition˜2\.5](https://arxiv.org/html/2605.11133#S2.Thmtheorem5), which is the mapφ:M×V→G×ρV\\varphi:M\\times V\\to G\\times\_\{\\rho\}Vdefined by

φ​\(p,v\)=\[σ​\(p\),v\],\\varphi\(p,v\)=\[\\sigma\(p\),v\],\(3\.3\)wherep∈Mp\\in Mandv∈Vv\\in V\. Since the mapφ\\varphiis a \(local\) diffeomorphism, the sectionsscan also be uniquely identified with the mapφ−1∘s:M→M×V\\varphi^\{\-1\}\\circ s:M\\to M\\times Vgiven by

\(φ−1∘s\)​\(p\)=\(p,f​\(p\)\)\.\(\\varphi^\{\-1\}\\circ s\)\(p\)=\(p,f\(p\)\)\.\(3\.4\)
Before proceeding, we need to describe the action of the global symmetry groupGGon sections of the associated bundleG×ρVG\\times\_\{\\rho\}V\. The left action ofGGonG×ρVG\\times\_\{\\rho\}Vis required to preserve linearity of the vector part, which forcesLg​\[g′,v\]=\[g​g′,v\]L\_\{g\}\[g^\{\\prime\},v\]=\[gg^\{\\prime\},v\]for allg,g′∈Gg,g^\{\\prime\}\\in G\. Since the projectionπ:G→G/H\\pi:G\\to G/Hcommutes with the left action, andHHacts transitively on the fibres, there is a unique element111The mapc:G×M→Hc:G\\times M\\to His called thecocycleassociated withσ\\sigma\. See\[[9](https://arxiv.org/html/2605.11133#bib.bib21)\]for a more detailed discussion\.c​\(g,p\)∈Hc\(g,p\)\\in Hfor everyg∈Gg\\in Gandp∈Mp\\in Msatisfying

Lg​σ​\(p\)=σ​\(g​p\)​c​\(g,p\)\.L\_\{g\}\\sigma\(p\)=\\sigma\(gp\)c\(g,p\)\\,\.\(3\.5\)The following lemma shows that the action ofGGon feature fields is given by the induced representation on Mackey functions\.

###### Lemma 3\.4\.

The action ofGGonG×ρVG\\times\_\{\\rho\}Vinduces an action on sectionsssof the forms​\(p\)=\[σ​\(p\),k​\(σ​\(p\)\)\]s\(p\)=\[\\sigma\(p\),k\(\\sigma\(p\)\)\], which is given by the induced representation acting on the Mackey functionkk\.

###### Proof\.

From the action onG×ρVG\\times\_\{\\rho\}V, the definition \([3\.5](https://arxiv.org/html/2605.11133#S3.E5)\), and the equivalence relations of the associated bundle, we get

Lg​s​\(p\)=\[Lg​σ​\(p\),k​\(σ​\(p\)\)\]=\[σ​\(g​p\),ρ​\(c​\(g,p\)\)​k​\(σ​\(p\)\)\]\.L\_\{g\}s\(p\)=\[L\_\{g\}\\sigma\(p\),k\(\\sigma\(p\)\)\]=\[\\sigma\(gp\),\\rho\(c\(g,p\)\)k\(\\sigma\(p\)\)\]\.\(3\.6\)Using the fact thatk​\(σ​\(p\)\)=k​\(Lg−1​Lg​σ​\(p\)\)=k​\(Lg−1​σ​\(g​p\)​c​\(g,p\)\)k\(\\sigma\(p\)\)=k\(L\_\{g^\{\-1\}\}L\_\{g\}\\sigma\(p\)\)=k\(L\_\{g^\{\-1\}\}\\sigma\(gp\)c\(g,p\)\)and the property of Mackey functions, we see that

Lg​s​\(p\)=\[σ​\(g​p\),k​\(Lg−1​σ​\(g​p\)\)\]\.L\_\{g\}s\(p\)=\[\\sigma\(gp\),k\(L\_\{g^\{\-1\}\}\\sigma\(gp\)\)\]\.\(3\.7\)Here,k​\(Lg−1​σ​\(g​p\)\)k\(L\_\{g^\{\-1\}\}\\sigma\(gp\)\)is exactly the induced representationIndHG​ρ\\mathrm\{Ind\}^\{G\}\_\{H\}\\rhoofggonk​\(g\)k\(g\)defined below[definition˜2\.11](https://arxiv.org/html/2605.11133#S2.Thmtheorem11)\. Thus,

Lg​s​\(p\)=\[σ​\(g​p\),IndHG​ρ​\(g\)​k​\(σ​\(g​p\)\)\]\.L\_\{g\}s\(p\)=\[\\sigma\(gp\),\\mathrm\{Ind\}^\{G\}\_\{H\}\\rho\(g\)k\(\\sigma\(gp\)\)\]\.\(3\.8\)∎

Finally, we want to define the left action ofGGonM×VM\\times Vin a way that is consistent with the left action onG×ρVG\\times\_\{\\rho\}V\. Through the canonical local trivialisationφ\\varphi, this can be obtained as

Lg​\(p,v\):=φ−1​\(\[Lg​σ​\(p\),v\]\)=\(g​p,ρ​\(c​\(g,p\)\)​v\)L\_\{g\}\(p,v\):=\\varphi^\{\-1\}\\big\(\[L\_\{g\}\\sigma\(p\),v\]\\big\)=\(gp,\\rho\(c\(g,p\)\)v\)\(3\.9\)for each\(p,v\)∈M×V\(p,v\)\\in M\\times V\.

We have now provided a rigorous geometric definition of a feature field, both from the perspective of a section of an associated bundle and from the equivalent local perspective in terms of Mackey functions\. In addition, we have derived how each feature field representative transforms under the left group action ofGG\. The results are summarised in[Table˜1](https://arxiv.org/html/2605.11133#S3.T1)\.

Feature field representativeTransformation under the left action ofg∈Gg\\in GSections:M→G×ρVs:M\\to G\\times\_\{\\rho\}V\[σ​\(p\),f​\(p\)\]↦\[Lg​σ​\(p\),f​\(p\)\]\[\\sigma\(p\),f\(p\)\]\\mapsto\[L\_\{g\}\\sigma\(p\),f\(p\)\]Local feature fieldf:M→Vf:M\\to Vf​\(p\)↦ρ​\(c​\(g,p\)\)​f​\(p\)f\(p\)\\mapsto\\rho\(c\(g,p\)\)f\(p\)Mackey functionk:G→Vk:G\\to Vk​\(σ​\(p\)\)↦IndHG​ρ​\(g\)​k​\(σ​\(g​p\)\)k\(\\sigma\(p\)\)\\mapsto\\mathrm\{Ind\}^\{G\}\_\{H\}\\rho\(g\)k\(\\sigma\(gp\)\)Local sectionφ−1∘s:M→M×V\\varphi^\{\-1\}\\circ s:M\\to M\\times V\(p,f​\(p\)\)↦\(g​p,ρ​\(c​\(g,p\)\)​f​\(p\)\)\(p,f\(p\)\)\\mapsto\(gp,\\rho\(c\(g,p\)\)f\(p\)\)Table 1:Summary of the four equivalent descriptions of a feature field introduced in[section˜3\.2](https://arxiv.org/html/2605.11133#S3.SS2), together with their transformation behaviours under the left action ofGG\.
### 3\.3Steerable NODEs through parallel transport

We now have the necessary constructions to accomplish the goal set forth in[section˜3\.1](https://arxiv.org/html/2605.11133#S3.SS1): to define a NODE model in which feature fields onM=G/HM=G/Hare transported along a flow in a way that respects the geometry and symmetries of the underlying space\. We refer to this novel class of models assteerableNODEs, since the feature vectors are not only translated using the global action ofGG, but simultaneouslysteeredusing the local action ofHH\. As we will see, this local action is determined by the transport flow and bundle geometry alone, but is independent of the value of the feature field\. This motivates the nomenclature in analogy with the original notion of steerability in CNNs\[[10](https://arxiv.org/html/2605.11133#bib.bib18)\]\.

The starting point is the manifold NODE in[definition˜3\.1](https://arxiv.org/html/2605.11133#S3.Thmtheorem1)determined by the vector fieldϕ\\phior, equivalently, the flowΦ\\Phigenerated byϕ\\phi\. A principalHH\-connectionω\\omegaonG→G/HG\\to G/Hprovides a consistent way to transport points in the principal bundle along this flow using parallel transport\. Feature fields are sections of the associated bundleG×ρVG\\times\_\{\\rho\}V, which inherits the notion of parallel transport fromG→G/HG\\to G/H\. The parallel transport in the associated bundle will define our steerable NODEs\.

Concretely, to describe parallel transport along the integral curvesΦp\\Phi\_\{p\}ofϕ\\phiin \([3\.1](https://arxiv.org/html/2605.11133#S3.E1)\), we use the ability to express a section ofG×ρVG\\times\_\{\\rho\}VoverΦp\\Phi\_\{p\}using the horizontal liftΦ~p\\tilde\{\\Phi\}\_\{p\}to the principalHH\-bundleGG, defined by

π∗​\(dd​t​Φ~p​\(t\)\)=dd​t​Φp​\(t\),ω​\(dd​t​Φ~p​\(t\)\)=0\.\\pi\_\{\*\}\\left\(\\frac\{d\}\{dt\}\\tilde\{\\Phi\}\_\{p\}\(t\)\\right\)=\\frac\{d\}\{dt\}\\Phi\_\{p\}\(t\)\\,,\\quad\\omega\\left\(\\frac\{d\}\{dt\}\\tilde\{\\Phi\}\_\{p\}\(t\)\\right\)=0\\,\.\(3\.10\)In terms of a sectionσ\\sigmaof the principal bundleG→G/HG\\to G/H, we haveΦ~p​\(t\)=σ​\(Φp​\(t\)\)​hp​\(t\)\\tilde\{\\Phi\}\_\{p\}\(t\)=\\sigma\(\\Phi\_\{p\}\(t\)\)h\_\{p\}\(t\), wherehp​\(t\)h\_\{p\}\(t\)is determined by \([2\.8](https://arxiv.org/html/2605.11133#S2.E8)\), and a section overΦp\\Phi\_\{p\}is

s​\(Φp​\(t\)\)=\[σ​\(Φp​\(t\)\),f​\(Φp​\(t\)\)\]=\[Φ~p​\(t\),ρ​\(hp−1​\(t\)\)​f​\(Φp​\(t\)\)\]\.s\(\\Phi\_\{p\}\(t\)\)=\\left\[\\sigma\(\\Phi\_\{p\}\(t\)\),f\(\\Phi\_\{p\}\(t\)\)\\right\]=\\left\[\\tilde\{\\Phi\}\_\{p\}\(t\),\\rho\(h\_\{p\}^\{\-1\}\(t\)\)f\(\\Phi\_\{p\}\(t\)\)\\right\]\\,\.\(3\.11\)According to[section˜2\.2](https://arxiv.org/html/2605.11133#S2.SS2), the section undergoes parallel transport alongΦp\\Phi\_\{p\}if

∇ϕs​\(Φp​\(t\)\)=\[Φ~p​\(t\),dd​t​\(ρ​\(hp−1​\(t\)\)​f​\(Φp​\(t\)\)\)\]=\[Φ~p​\(t\),0\],\\nabla\_\{\\phi\}s\(\\Phi\_\{p\}\(t\)\)=\\left\[\\tilde\{\\Phi\}\_\{p\}\(t\),\\frac\{d\}\{dt\}\\big\(\\rho\(h\_\{p\}^\{\-1\}\(t\)\)f\(\\Phi\_\{p\}\(t\)\)\\big\)\\right\]=\\left\[\\tilde\{\\Phi\}\_\{p\}\(t\),0\\right\]\\,,\(3\.12\)or, equivalently, ifρ​\(hp−1​\(t\)\)​f​\(Φp​\(t\)\)\\rho\(h\_\{p\}^\{\-1\}\(t\)\)f\(\\Phi\_\{p\}\(t\)\)is constant\.

Just as parallel transport in the principal bundleGGcan be described as the flowΦ~\\tilde\{\\Phi\}obtained through the horizontal lift as described in[section˜2\.2](https://arxiv.org/html/2605.11133#S2.SS2), so too can the parallel transport in the associated bundle be described by a horizontal flow\.

###### Definition 3\.5\(Parallel transport flow\)\.

LetM=G/HM=G/Hbe a homogeneous space,ϕ\\phia vector field onMM,ω\\omegaa principalHH\-connection onG→G/HG\\to G/H, andG×ρVG\\times\_\{\\rho\}Vthe associated bundle defined by the representationρ:H→GL​\(V\)\\rho:H\\to\\mathrm\{GL\}\(V\)\. LetΦ\\Phibe the flow inMMgenerated byϕ\\phi, andΦ~\\tilde\{\\Phi\}be the unique horizontal lift ofΦ\\Phiwith respect toω\\omega\. We define the mapΓΦ:ℝ×\(G×ρV\)→G×ρV\\Gamma\_\{\\Phi\}:\\mathbb\{R\}\\times\(G\\times\_\{\\rho\}V\)\\to G\\times\_\{\\rho\}Vby

ΓΦ​\(t,\[g,v\]\)=\[Φ~​\(t,g\),v\]\\Gamma\_\{\\Phi\}\(t,\[g,v\]\)=\[\\tilde\{\\Phi\}\(t,g\),v\]\(3\.13\)for everyg∈Gg\\in Gandv∈Vv\\in V\.

###### Lemma 3\.6\.

The mapΓΦ\\Gamma\_\{\\Phi\}in[definition˜3\.5](https://arxiv.org/html/2605.11133#S3.Thmtheorem5)is a horizontal flow inGGwith respect to the connection induced fromω\\omega, and describes the parallel transport inG×ρVG\\times\_\{\\rho\}Valong the flowΦ\\PhiinMM\.

###### Proof\.

We begin by showing thatΓΦ\\Gamma\_\{\\Phi\}is well\-defined\. We know that\(t,\[g,v\]\)=\(t,\[g​h,ρ​\(h−1\)​v\]\)\(t,\[g,v\]\)=\(t,\[gh,\\rho\(h^\{\-1\}\)v\]\)for allh∈Hh\\in H\. From the defining properties ofω\\omega, we also know thatΦ~​\(t,g​h\)=Φ~​\(t,g\)​h\\tilde\{\\Phi\}\(t,gh\)=\\tilde\{\\Phi\}\(t,g\)hfor allh∈Hh\\in H\. Thus, we have

ΓΦ​\(t,\[g​h,ρ​\(h−1\)​v\]\)=\[Φ~​\(t,g​h\),ρ​\(h−1\)​v\]=\[Φ~​\(t,g\),v\]=ΓΦ​\(t,\[g,v\]\),\\Gamma\_\{\\Phi\}\(t,\[gh,\\rho\(h^\{\-1\}\)v\]\)=\[\\tilde\{\\Phi\}\(t,gh\),\\rho\(h^\{\-1\}\)v\]=\[\\tilde\{\\Phi\}\(t,g\),v\]=\\Gamma\_\{\\Phi\}\(t,\[g,v\]\)\\,,\(3\.14\)andΓΦ\\Gamma\_\{\\Phi\}is well\-defined\.

SinceΦ~\\tilde\{\\Phi\}is a flow, it immediately follows thatΓΦ​\(0,\[g,v\]\)=\[g,v\]\\Gamma\_\{\\Phi\}\(0,\[g,v\]\)=\[g,v\]and thatΓΦ​\(s,ΓΦ​\(t,\[g,v\]\)\)=ΓΦ​\(s\+t,\[g,v\]\)\\Gamma\_\{\\Phi\}\(s,\\Gamma\_\{\\Phi\}\(t,\[g,v\]\)\)=\\Gamma\_\{\\Phi\}\(s\+t,\[g,v\]\)for allg∈Gg\\in G,v∈Vv\\in Vands,t∈ℝs,t\\in\\mathbb\{R\}, makingΓΦ\\Gamma\_\{\\Phi\}a flow on the associated bundle\.

Giveng∈Gg\\in G, letp=π​\(g\)p=\\pi\(g\)and setΦp​\(t\)=Φ​\(t,p\)\\Phi\_\{p\}\(t\)=\\Phi\(t,p\)\. BecauseΦ~\\tilde\{\\Phi\}is the horizontal lift ofΦ\\Phi, the horizontal lift ofΦp​\(t\)\\Phi\_\{p\}\(t\)throughggisΦ~p​\(t\)=Φ~​\(t,g\)\\tilde\{\\Phi\}\_\{p\}\(t\)=\\tilde\{\\Phi\}\(t,g\), and

ΓΦ​\(t,\[g,v\]\)=\[Φ~p​\(t\),v\]\.\\Gamma\_\{\\Phi\}\(t,\[g,v\]\)=\[\\tilde\{\\Phi\}\_\{p\}\(t\),v\]\.\(3\.15\)Comparing with[eq\.˜3\.11](https://arxiv.org/html/2605.11133#S3.E11), this is exactly the form of a parallel transported section where the elementvvcorresponds to the constant quantityρ​\(hp−1​\(t\)\)​f​\(Φp​\(t\)\)\\rho\(h\_\{p\}^\{\-1\}\(t\)\)f\(\\Phi\_\{p\}\(t\)\)\. It follows thatΓΦ\\Gamma\_\{\\Phi\}is a flow that transports the element\[g,v\]\[g,v\]alongΦp\\Phi\_\{p\}by parallel transport\.

Moreover, the mapt↦\[Φ~p​\(t\),v\]t\\mapsto\[\\tilde\{\\Phi\}\_\{p\}\(t\),v\]defines a curve inG×ρVG\\times\_\{\\rho\}VwhoseGG\-component is horizontal with respect toω\\omegaand whoseVV\-component remains constant\. Taking this parallel transport to define an induced connection on the associated bundle, the curve is manifestly horizontal\. In this sense,ΓΦ\\Gamma\_\{\\Phi\}is a horizontal flow\. ∎

The flowΓΦ\\Gamma\_\{\\Phi\}provides a geometrically principled way of transporting feature fields \(i\.e\., sections ofG×ρVG\\times\_\{\\rho\}V\) along the flow corresponding to the manifold NODE \([3\.1](https://arxiv.org/html/2605.11133#S3.E1)\)\. IfΦp\\Phi\_\{p\}is a solution curve of a NODE defined by the flowΦ\\Phigenerated byϕ\\phi, andhp​\(t\)∈Hh\_\{p\}\(t\)\\in His the element inHHwith initial conditionhp​\(0\)=eh\_\{p\}\(0\)=ethat defines the horizontal liftΦ~p​\(t\)=σ​\(Φp​\(t\)\)​hp​\(t\)\\tilde\{\\Phi\}\_\{p\}\(t\)=\\sigma\(\\Phi\_\{p\}\(t\)\)h\_\{p\}\(t\), then

ΓΦ​\(1,\[σ​\(p\),v\]\)=\[Φ~p​\(1\),v\]=\[σ​\(Φp​\(1\)\),ρ​\(hp​\(1\)\)​v\]\\Gamma\_\{\\Phi\}\(1,\[\\sigma\(p\),v\]\)=\[\\tilde\{\\Phi\}\_\{p\}\(1\),v\]=\[\\sigma\(\\Phi\_\{p\}\(1\)\),\\rho\(h\_\{p\}\(1\)\)v\]\(3\.16\)is the result of parallel transport of the sections​\(p\)=\[σ​\(p\),v\]s\(p\)=\[\\sigma\(p\),v\]along the trajectoryΦp\\Phi\_\{p\}frompptoΦp​\(1\)\\Phi\_\{p\}\(1\)\. Furthermore, sinceΓΦ\\Gamma\_\{\\Phi\}is a flow and the local trivialisationφ\\varphiis a \(local\) diffeomorphism, the map

\(p,v\)​⟼𝜑​\[σ​\(p\),v\]↦ΓΦ​\(1,\[σ​\(p\),v\]\)​⟼φ−1​\(Φp​\(1\),ρ​\(hp​\(1\)\)​v\)\(p,v\)\\overset\{\\varphi\}\{\\longmapsto\}\[\\sigma\(p\),v\]\\mapsto\\Gamma\_\{\\Phi\}\(1,\[\\sigma\(p\),v\]\)\\overset\{\\varphi^\{\-1\}\}\{\\longmapsto\}\(\\Phi\_\{p\}\(1\),\\rho\(h\_\{p\}\(1\)\)v\)\(3\.17\)is a diffeomorphism that maps the initial data\(p,v\)∈M×V\(p,v\)\\in M\\times Vto the output\(Φp​\(1\),ρ​\(hp​\(1\)\)​v\)\(\\Phi\_\{p\}\(1\),\\rho\(h\_\{p\}\(1\)\)v\)by transporting the base pointppalong the flowΦ\\Phiand simultaneously steering the feature vectorvvthrough the action ofρ​\(hp​\(1\)\)\\rho\(h\_\{p\}\(1\)\)\. This diffeomorphism defines the steerable NODE\.

###### Definition 3\.7\(Steerable NODEs on homogeneous spaces\)\.

LetM=G/HM=G/Hbe a homogeneous space,ρ:H→GL​\(V\)\\rho:H\\to\\mathrm\{GL\}\(V\)be a representation ofHH, and letσ\\sigmabe a local section of the principalHH\-bundleGG\. Furthermore, letϕ:M→T​M\\phi:M\\to TMbe a learnable vector field onMMandω∈Ω1​\(G,𝔥\)\\omega\\in\\Omega^\{1\}\(G,\\mathfrak\{h\}\)a learnable principalHH\-connection onGG\. Then for every pointp∈Mp\\in M, there exist unique curvesΦp:ℝ→M\\Phi\_\{p\}:\\mathbb\{R\}\\to Mandhp:ℝ→Hh\_\{p\}:\\mathbb\{R\}\\to Hthat solve the parallel transport ODEs

d​Φp​\(t\)d​t=ϕ\(Φp​\(t\)\),\\displaystyle\\frac\{d\\Phi\_\{p\}\(t\)\}\{dt\}=\\phi\_\{\(\\Phi\_\{p\}\(t\)\)\},\(3\.18\)d​hpd​t=−\(Rhp​\(t\)\)∗​ω​\(σ∗​ϕ\(Φp​\(t\)\)\),\\displaystyle\\frac\{dh\_\{p\}\}\{dt\}=\-\(R\_\{h\_\{p\}\(t\)\}\)\_\{\*\}\\omega\\left\(\\sigma\_\{\*\}\\phi\_\{\(\\Phi\_\{p\}\(t\)\)\}\\right\),\(3\.19\)with initial conditionsΦp​\(0\)=p\\Phi\_\{p\}\(0\)=pandhp​\(0\)=eh\_\{p\}\(0\)=e\. Thesteerable NODEonM×VM\\times Vis the diffeomorphismΨ:M×V→M×V\\Psi:M\\times V\\to M\\times Vdefined byΨ​\(p,v\)=\(Φp​\(1\),ρ​\(hp​\(1\)\)​v\)\\Psi\(p,v\)=\(\\Phi\_\{p\}\(1\),\\rho\(h\_\{p\}\(1\)\)v\)\. The point\(p,v\)∈M×V\(p,v\)\\in M\\times Vis theinputandΨ​\(p,v\)∈M×V\\Psi\(p,v\)\\in M\\times Vis theoutputof the model\.

The pointwise formulation of steerable NODEs in[definition˜3\.7](https://arxiv.org/html/2605.11133#S3.Thmtheorem7)shows that we can indeed interpret the model as a coupled system of \(neural\) ODEs, where the gradients are determined by the vector fieldϕ\\phiand the connectionω\\omega, and where we recover the manifold NODE whenHHis trivial\. We emphasise that bothϕ\\phiandω\\omegaare learnable and can be parametrised, e\.g\., by neural networks during implementation\. Given an input\(p,v\)\(p,v\), measured with respect to the sectionσ\\sigmaof the principal bundleGG, we first integrate \([3\.18](https://arxiv.org/html/2605.11133#S3.E18)\) to obtain the curveΦp​\(t\)\\Phi\_\{p\}\(t\)onM=G/HM=G/Halong which to transport the feature vector\. Note that this is a manifold NODE onMMindependent of the connection\. Subsequently, we integrate \([3\.19](https://arxiv.org/html/2605.11133#S3.E19)\), which depends on both the vector fieldϕ\\phiand the connectionω\\omega, to obtain the steering functionhp​\(t\)h\_\{p\}\(t\)\. The model outputΨ\(p,v\)=\(Φp\(1\),ρ\(hp\(1\)v\)\\Psi\(p,v\)=\(\\Phi\_\{p\}\(1\),\\rho\(h\_\{p\}\(1\)v\)can then be compared to a target, and the parameters definingϕ\\phiandω\\omegaupdated to minimise the discrepancy\. In particular, it is clear that the available frameworks for training NODEs \(e\.g\., the adjoint sensitivity method\[[7](https://arxiv.org/html/2605.11133#bib.bib13)\]or the flow matching framework\[[27](https://arxiv.org/html/2605.11133#bib.bib52)\]\) are then directly applicable to steerable NODEs\.

The feature fields are sections ofG×ρVG\\times\_\{\\rho\}V, and are therefore inherently local since, in general, this bundle admits no global sections\. This is also reflected in the steerable NODE and its dependence on the local sectionσ\\sigmaofGG\. A global extension to all ofMMrequires using an atlas of charts coveringMMand the transition functions defining the bundlesGGandG×ρVG\\times\_\{\\rho\}V\. This is standard practice in differential geometry \(see, e\.g\.,\[[30](https://arxiv.org/html/2605.11133#bib.bib56),[24](https://arxiv.org/html/2605.11133#bib.bib48)\]\) and we will not go into detail here\.

The steerable NODE can be interpreted geometrically as a model that learns a flow in the associated bundleG×ρVG\\times\_\{\\rho\}Vthat transforms the feature vectorf​\(p\)∈Vf\(p\)\\in Vatp∈Mp\\in Mby transporting it along a flow inMMin a way that is manifestly compatible with the global symmetryGGofMM\. The flow inMMis determined by the choice of vector field, and the transport can be accomplished in different ways corresponding to the choice of the connectionω\\omega\. Consequently, training the steerable NODE amounts to learning both where to transportf​\(p\)f\(p\)and in which of the possible ways to perform the transport\.

[Figure˜2](https://arxiv.org/html/2605.11133#S3.F2)provides a visual illustration of a steerable NODE, showing how the coupled system of \(neural\) ODEs interact to transport pointspponM=G/HM=G/Halong the solution curveΦp\\Phi\_\{p\}, while simultaneously steering the associated featuresf​\(p\)f\(p\)through the induced action of the solution curvehph\_\{p\}onHH\.

![Refer to caption](https://arxiv.org/html/2605.11133v1/x3.png)Figure 2:A steerable NODE consists of a coupled system of ODEs evolving on different manifolds\. The integral curveΦp\\Phi\_\{p\}\(bottom\) is defined on the homogeneous spaceM=G/HM=G/Hby a learnable vector fieldϕ\\phi\(blue\)\. The curvehph\_\{p\}\(top\) is defined on the closed Lie subgroupHHand is determined by a learnable connectionω\\omega\(green\)\. Feature vectorsv∈Vv\\in V, viewed as elements in the fibre of the associated bundleG×ρVG\\times\_\{\\rho\}V, are transported along the curveΦp\\Phi\_\{p\}and simultaneouslysteeredby the representationρ​\(hp​\(t\)\)\\rho\(h\_\{p\}\(t\)\)for eacht∈\[0,1\]t\\in\[0,1\]\. The resulting map\(p,v\)↦\(Φp​\(1\),ρ​\(hp​\(1\)\)​v\)\(p,v\)\\mapsto\(\\Phi\_\{p\}\(1\),\\rho\(h\_\{p\}\(1\)\)v\)defines the steerable NODE\. Notably, the curveΦp\\Phi\_\{p\}depends only onϕ\\phi, whereas the curvehph\_\{p\}, which determines the transformation of the feature component, depends on bothϕ\\phiand the connectionω\\omega\. Geometrically, the steering byρ​\(hp​\(t\)\)\\rho\(h\_\{p\}\(t\)\)describes a parallel transport along the curveΦp\\Phi\_\{p\}in the associated bundleG×ρVG\\times\_\{\\rho\}V\.We end this section by providing a concrete example of a steerable NODE on a topologically trivial homogeneous space with fixed vector fieldϕ\\phiand connectionω\\omega\.

###### Example 3\.9\.

LetG=ℝ2×U​\(1\)G=\\mathbb\{R\}^\{2\}\\times U\(1\)be the trivialH=U​\(1\)H=U\(1\)bundle overM=G/H=ℝ2M=G/H=\\mathbb\{R\}^\{2\}with the globalGG\-action given by translations\. A point inGGis\(x,y,θ\)\(x,y,\\theta\)with\(x,y\)∈ℝ2\(x,y\)\\in\\mathbb\{R\}^\{2\}andθ∈ℝ/2​π​ℤ\\theta\\in\\mathbb\{R\}/2\\pi\\mathbb\{Z\}, and the action ofg=\(α,β,φ\)∈Gg=\(\\alpha,\\beta,\\varphi\)\\in Gis

Lg​\(x,y,θ\)=\(x\+α,y\+β,θ\+φ​\(mod​2​π\)\)\.L\_\{g\}\(x,y,\\theta\)=\(x\+\\alpha,y\+\\beta,\\theta\+\\varphi\\,\(\\mathrm\{mod\}\\,2\\pi\)\)\\,\.\(3\.20\)The induced action onM=ℝ2M=\\mathbb\{R\}^\{2\}is

Lg​\(x,y\)=\(x\+α,y\+β\),L\_\{g\}\(x,y\)=\(x\+\\alpha,y\+\\beta\)\\,,\(3\.21\)with stabiliserH=\{h=\(0,0,φ\)\|φ∈ℝ/2​π​ℤ\}H=\\\{h=\(0,0,\\varphi\)\|\\varphi\\in\\mathbb\{R\}/2\\pi\\mathbb\{Z\}\\\}and right actionRh​\(x,y,θ\)=\(x,y,θ\+φ​\(mod​2​π\)\)R\_\{h\}\(x,y,\\theta\)=\(x,y,\\theta\+\\varphi\\,\(\\mathrm\{mod\}\\,2\\pi\)\)\. Note that the adjoint actionAdh\\mathrm\{Ad\}\_\{h\}is trivial sinceHHis abelian, and that\(Rh\)∗=id\(R\_\{h\}\)\_\{\*\}=\\mathrm\{id\}\.

We consider the vector fieldϕ=∂x\\phi=\\partial\_\{x\}, with integral curveΦp​\(t\)=p\+\(t,0\)\\Phi\_\{p\}\(t\)=p\+\(t,0\)through the pointp∈ℝ2p\\in\\mathbb\{R\}^\{2\}\. To define a steerable NODE, we also need a connection onGG\. We takeω=π​d​x\+d​y\+d​θ\\omega=\\pi dx\+dy\+d\\theta, which is clearly a 1\-form onGGtaking values in𝔥=ℝ\\mathfrak\{h\}=\\mathbb\{R\}and indeed a principal connection, since:

1. \(i\)It acts trivially on the vertical subspace, ω​\(∂θ\)=π​d​x​\(∂θ\)\+d​y​\(∂θ\)\+d​θ​\(∂θ\)=1\.\\omega\(\\partial\_\{\\theta\}\)=\\pi dx\(\\partial\_\{\\theta\}\)\+dy\(\\partial\_\{\\theta\}\)\+d\\theta\(\\partial\_\{\\theta\}\)=1\\,\.\(3\.22\)
2. \(ii\)It isHH\-equivariant, \(Rh\)∗​ω=π​\(Rh\)∗​d​x\+\(Rh\)∗​d​y\+\(Rh\)∗​d​θ=ω=Adh−1​ω\.\(R\_\{h\}\)^\{\*\}\\omega=\\pi\(R\_\{h\}\)^\{\*\}dx\+\(R\_\{h\}\)^\{\*\}dy\+\(R\_\{h\}\)^\{\*\}d\\theta=\\omega=\\mathrm\{Ad\}\_\{h^\{\-1\}\}\\omega\\,\.\(3\.23\)

We choose a section ofGGto beσ​\(x,y\)=\(x,y,χ\)\\sigma\(x,y\)=\(x,y,\\chi\), for some constantχ∈ℝ/2​π​ℤ\\chi\\in\\mathbb\{R\}/2\\pi\\mathbb\{Z\}, which yieldsσ∗​ϕ=∂x\\sigma\_\{\*\}\\phi=\\partial\_\{x\}\. Inserting this into \([3\.19](https://arxiv.org/html/2605.11133#S3.E19)\), we obtain

d​hpd​t=−ω​\(σ∗​ϕ\)=−ω​\(∂x\)=−π,\\frac\{dh\_\{p\}\}\{dt\}=\-\\omega\(\\sigma\_\{\*\}\\phi\)=\-\\omega\(\\partial\_\{x\}\)=\-\\pi\\,,\(3\.24\)where we have again used the triviality of the adjointHH\-action\. Solving this equation yieldshp​\(t\)=−π​th\_\{p\}\(t\)=\-\\pi t, where we have enforced the initial conditionhp​\(0\)=0h\_\{p\}\(0\)=0\. The horizontal lift is given by

Φ~p​\(t\)=σ​\(Φp​\(t\)\)​h​\(t\)=\(x\+t,y,χ−π​t​\(mod​2​π\)\),\\tilde\{\\Phi\}\_\{p\}\(t\)=\\sigma\(\\Phi\_\{p\}\(t\)\)h\(t\)=\(x\+t,y,\\chi\-\\pi t\\,\(\\mathrm\{mod\}\\,2\\pi\)\)\\,,\(3\.25\)and the corresponding lift of the vector field isϕ~=∂x−π​∂θ\\tilde\{\\phi\}=\\partial\_\{x\}\-\\pi\\partial\_\{\\theta\}\. Indeed, this is horizontal,ω​\(ϕ~\)=π−π=0\\omega\(\\tilde\{\\phi\}\)=\\pi\-\\pi=0, and projects toϕ\\phi,π∗​ϕ~=∂x\\pi\_\{\*\}\\tilde\{\\phi\}=\\partial\_\{x\}\.

For the associated vector bundle, we choose a non\-trivial representation ofH=U​\(1\)H=U\(1\)onV=ℝ2V=\\mathbb\{R\}^\{2\}defined by rotations

ρ​\(φ\)=R​\(φ\)=\(cos⁡\(φ\)−sin⁡\(φ\)sin⁡\(φ\)cos⁡\(φ\)\)\.\\rho\(\\varphi\)=R\(\\varphi\)=\\left\(\\begin\{array\}\[\]\{cc\}\\cos\(\\varphi\)&\-\\sin\(\\varphi\)\\\\ \\sin\(\\varphi\)&\\cos\(\\varphi\)\\end\{array\}\\right\)\\,\.\(3\.26\)Fixing a pair\(p,v\)\(p,v\)inM×V=ℝ2×ℝ2M\\times V=\\mathbb\{R\}^\{2\}\\times\\mathbb\{R\}^\{2\}, the steerable NODE defined by the vector fieldϕ\\phiand the connectionω\\omegais then given by[definition˜3\.7](https://arxiv.org/html/2605.11133#S3.Thmtheorem7)as

Ψ​\(p,v\)=\(Φp​\(1\),ρ​\(hp​\(1\)\)​v\)=\(x\+1,y,R​\(π\)​v\),\\Psi\(p,v\)=\(\\Phi\_\{p\}\(1\),\\rho\(h\_\{p\}\(1\)\)v\)=\(x\+1,y,R\(\\pi\)v\)\\,,\(3\.27\)where we writep=\(x,y\)p=\(x,y\)and have used thathp​\(1\)=−π=π∈ℝ/2​π​ℤh\_\{p\}\(1\)=\-\\pi=\\pi\\in\\mathbb\{R\}/2\\pi\\mathbb\{Z\}\.

## 4Equivariance of steerable NODEs

In the previous section, we constructed steerable NODEs on the homogeneous spaceM=G/HM=G/H, where the feature fields transform in a representationρ\\rhoof the stabilising subgroupHH\. This guarantees equivariance with respect to the localHH\-action\. Beyond this, the global symmetry groupGGalso has a natural action on the feature maps through the induced representationIndHG​\(ρ\)\\mathrm\{Ind\}\_\{H\}^\{G\}\(\\rho\)\. This raises the question whether steerable NODEs can be made equivariant with respect to the global action ofGG, and what additional constraints such equivariance would impose on the model\.

To address these questions, we first need a clear definition of equivariance of steerable NODEs\. For ordinary manifold NODEs onM=G/HM=G/Hin[definition˜3\.1](https://arxiv.org/html/2605.11133#S3.Thmtheorem1), which correspond to the case of trivialHHin the steerable framework, equivariance is defined as follows\.

###### Definition 4\.1\.

A manifold neural ODEψ:M→M\\psi:M\\to MisGG\-equivariantifψ​\(g​p\)=Lg​ψ​\(p\)\\psi\(gp\)=L\_\{g\}\\psi\(p\)for allp∈Mp\\in Mandg∈Gg\\in G\.

It is well known that equivariance of the diffeomorphismψ\\psi, equivariance of the corresponding flow,Lg​Φ​\(t,p\)=Φ​\(t,g​p\)L\_\{g\}\\Phi\(t,p\)=\\Phi\(t,gp\), and invariance of the generating vector field,\(Lg\)∗​ϕp=ϕg​p\(L\_\{g\}\)\_\{\*\}\\phi\_\{p\}=\\phi\_\{gp\}, are equivalent\[[21](https://arxiv.org/html/2605.11133#bib.bib44),[19](https://arxiv.org/html/2605.11133#bib.bib38),[1](https://arxiv.org/html/2605.11133#bib.bib3)\]\.

A notion of equivariance of the steerable NODEs in[definition˜3\.7](https://arxiv.org/html/2605.11133#S3.Thmtheorem7)can be obtained in an analogous way, using the left action ofGGonM×VM\\times Vintroduced in[section˜3\.2](https://arxiv.org/html/2605.11133#S3.SS2)\.

###### Definition 4\.2\.

A steerable NODEΨ:M×V→M×V\\Psi:M\\times V\\to M\\times Von the homogeneous spaceM=G/HM=G/HisGG\-equivariantifLg​Ψ​\(p,v\)=Ψ​\(Lg​\(p,v\)\)L\_\{g\}\\Psi\(p,v\)=\\Psi\(L\_\{g\}\(p,v\)\)for allp∈Mp\\in M,v∈Vv\\in Vandg∈Gg\\in G\.

### 4\.1Equivariance via horizontal flows

In this section, we provide sufficient conditions for equivariance of steerable NODEs under the global symmetry groupGG\. More specifically, in[theorem˜4\.6](https://arxiv.org/html/2605.11133#S4.Thmtheorem6)we show how a steerable NODE inherits the equivariance of the NODE on the base manifoldM=G/HM=G/Hif the connection defining parallel transport is compatible with the left action ofGG\.

Equivariance of the steerable NODE is conveniently described in terms of the flows associated with the manifold NODE onM=G/HM=G/H\. As discussed in[Section˜2\.2](https://arxiv.org/html/2605.11133#S2.SS2), given a connectionω\\omega, the flowΦ:ℝ×M→M\\Phi:\\mathbb\{R\}\\times M\\to Mcorresponding to the NODEψ:M→M\\psi:M\\to Mlifts uniquely to a horizontal flowΦ~:ℝ×G→G\\tilde\{\\Phi\}:\\mathbb\{R\}\\times G\\to GonGG, which is equivariant with respect to the right action ofHH\. In the following[lemma](https://arxiv.org/html/2605.11133#S4.Thmtheorem4), we show that if the flowΦ\\PhiisGG\-equivariant, so is the horizontal lift, provided that the connection respects the group structure ofGG\.

###### Lemma 4\.4\.

LetΦ\\Phibe a flow on the homogeneous spaceM=G/HM=G/Hgenerated by a vector fieldϕ:M→T​M\\phi:M\\to TM, and letω\\omegabe a fixed principalHH\-connection onGG\. If the vector fieldϕ\\phiand the connectionω\\omegaare bothGG\-invariant, then the horizontal liftΦ~\\tilde\{\\Phi\}ofΦ\\PhiisGG\-equivariant and the corresponding horizontal liftϕ~\\tilde\{\\phi\}ofϕ\\phiisGG\-invariant\.

###### Proof\.

SinceΦ~\\tilde\{\\Phi\}is the flow generated byϕ~\\tilde\{\\phi\}, it suffices to prove that\(Lg\)∗​ϕ~g′=ϕ~g​g′\(L\_\{g\}\)\_\{\*\}\\tilde\{\\phi\}\_\{g^\{\\prime\}\}=\\tilde\{\\phi\}\_\{gg^\{\\prime\}\}for allg,g′∈Gg,g^\{\\prime\}\\in Gif\(Lg\)∗​ϕp=ϕg​p\(L\_\{g\}\)\_\{\*\}\\phi\_\{p\}=\\phi\_\{gp\}and\(Lg\)∗​ωg′=ωg−1​g′\(L\_\{g\}\)^\{\*\}\\omega\_\{g^\{\\prime\}\}=\\omega\_\{g^\{\-1\}g^\{\\prime\}\}for allg,g′∈Gg,g^\{\\prime\}\\in Gandp∈Mp\\in M\(see, e\.g\.,\[[1](https://arxiv.org/html/2605.11133#bib.bib3), Theorem 3\.4\]\)\. We do this by showing that\(Lg\)∗​ϕ~\(L\_\{g\}\)\_\{\*\}\\tilde\{\\phi\}is a horizontalHH\-equivariant lift ofϕ\\phi\.

First, we use the fact that the projectionπ:G→G/H\\pi:G\\to G/Hcommutes with the left action ofGG, which implies thatπ∗∘\(Lg\)∗=\(Lg\)∗∘π∗\\pi\_\{\*\}\\circ\(L\_\{g\}\)\_\{\*\}=\(L\_\{g\}\)\_\{\*\}\\circ\\pi\_\{\*\}\. For allg,g′∈Gg,g^\{\\prime\}\\in Gwe then have that

π∗​\(\(Lg\)∗​ϕ~g′\)=\(Lg\)∗​\(π∗​ϕ~g′\)=\(Lg\)∗​ϕπ​\(g′\)=ϕLg​π​\(g′\)=ϕπ​\(g​g′\),\\pi\_\{\*\}\\left\(\(L\_\{g\}\)\_\{\*\}\\tilde\{\\phi\}\_\{g^\{\\prime\}\}\\right\)=\(L\_\{g\}\)\_\{\*\}\\left\(\\pi\_\{\*\}\\tilde\{\\phi\}\_\{g^\{\\prime\}\}\\right\)=\(L\_\{g\}\)\_\{\*\}\\phi\_\{\\pi\(g^\{\\prime\}\)\}=\\phi\_\{L\_\{g\}\\pi\(g^\{\\prime\}\)\}=\\phi\_\{\\pi\(gg^\{\\prime\}\)\}\\,,\(4\.1\)where we have used the fact thatϕ~\\tilde\{\\phi\}is a lift ofϕ\\phi, and thatϕ\\phiisGG\-invariant\. Consequently,\(Lg\)∗​ϕ~\(L\_\{g\}\)\_\{\*\}\\tilde\{\\phi\}is a lift ofϕ\\phi\. Furthermore, the lift\(Lg\)∗​ϕ~\(L\_\{g\}\)\_\{\*\}\\tilde\{\\phi\}is horizontal by the invariance ofω\\omegasince, for allg,g′∈Gg,g^\{\\prime\}\\in G,

ωg​g′​\(\(Lg\)∗​ϕ~g′\)=\(Lg\)∗​ωg​g′​\(ϕ~g′\)=ωg′​\(ϕ~g′\)=0\.\\omega\_\{gg^\{\\prime\}\}\\left\(\(L\_\{g\}\)\_\{\*\}\\tilde\{\\phi\}\_\{g^\{\\prime\}\}\\right\)=\(L\_\{g\}\)^\{\*\}\\omega\_\{gg^\{\\prime\}\}\(\\tilde\{\\phi\}\_\{g^\{\\prime\}\}\)=\\omega\_\{g^\{\\prime\}\}\(\\tilde\{\\phi\}\_\{g^\{\\prime\}\}\)=0\\,\.\(4\.2\)Finally, equivariance of\(Lg\)∗​ϕ~\(L\_\{g\}\)\_\{\*\}\\tilde\{\\phi\}under the right action ofHHis immediate\. For allg,g′∈Gg,g^\{\\prime\}\\in Gandh∈Hh\\in Hwe have

\(Rh\)∗​\(\(Lg\)∗​ϕ~g′\)=\(Lg\)∗​\(\(Rh\)∗​ϕ~g′\)=\(Lg\)∗​ϕ~g′​h\.\(R\_\{h\}\)\_\{\*\}\\left\(\(L\_\{g\}\)\_\{\*\}\\tilde\{\\phi\}\_\{g^\{\\prime\}\}\\right\)=\(L\_\{g\}\)\_\{\*\}\\left\(\(R\_\{h\}\)\_\{\*\}\\tilde\{\\phi\}\_\{g^\{\\prime\}\}\\right\)=\(L\_\{g\}\)\_\{\*\}\\tilde\{\\phi\}\_\{g^\{\\prime\}h\}\\,\.\(4\.3\)
Consequently,Lg​ϕ~L\_\{g\}\\tilde\{\\phi\}is a horizontal,HH\-equivariant lift ofϕ\\phi\. Sinceϕ~\\tilde\{\\phi\}is the unique vector field onGGwith these properties, we must haveLg​ϕ~=ϕ~L\_\{g\}\\tilde\{\\phi\}=\\tilde\{\\phi\}for allg∈Gg\\in G\. ∎

The parallel transport in the associated bundleG×ρVG\\times\_\{\\rho\}Vis induced by the parallel transport defined by the horizontal liftΦ~\\tilde\{\\Phi\}\. Equivariance under the global action ofGGis therefore directly inherited by the parallel transport flowΓΦ\\Gamma\_\{\\Phi\}defined in[definition˜3\.5](https://arxiv.org/html/2605.11133#S3.Thmtheorem5)\.

###### Lemma 4\.5\.

LetΦ\\Phibe the flow on the homogeneous spaceM=G/HM=G/Hgenerated by a vector fieldϕ:M→T​M\\phi:M\\to TM, and letω\\omegabe a principalHH\-connection onGG\. The horizontal liftΓΦ:ℝ×\(G×ρV\)→G×ρV\\Gamma\_\{\\Phi\}:\\mathbb\{R\}\\times\(G\\times\_\{\\rho\}V\)\\to G\\times\_\{\\rho\}VofΦ\\Phiin[definition˜3\.5](https://arxiv.org/html/2605.11133#S3.Thmtheorem5), defining parallel transport in the associated bundleG×ρVG\\times\_\{\\rho\}V, isGG\-equivariant if the vector fieldϕ\\phiand the connectionω\\omegaare bothGG\-invariant\.

###### Proof\.

By[Lemma˜4\.4](https://arxiv.org/html/2605.11133#S4.Thmtheorem4), the horizontal liftΦ~\\tilde\{\\Phi\}of the flowΦ\\Phito the principal bundleGGisGG\-equivariant\. The left action ofGGonG×ρVG\\times\_\{\\rho\}Vis given byLg​\[g′,v\]=\[g​g′,v\]L\_\{g\}\[g^\{\\prime\},v\]=\[gg^\{\\prime\},v\], which implies

Lg​ΓΦ​\(t,\[g′,v\]\)\\displaystyle L\_\{g\}\\Gamma\_\{\\Phi\}\(t,\[g^\{\\prime\},v\]\)=\\displaystyle=Lg​\[Φ~​\(t,g′\),v\]=\[Lg​Φ~​\(t,g′\),v\]=\[Φ~​\(t,g​g′\),v\]\\displaystyle L\_\{g\}\\left\[\\tilde\{\\Phi\}\(t,g^\{\\prime\}\),v\\right\]=\\left\[L\_\{g\}\\tilde\{\\Phi\}\(t,g^\{\\prime\}\),v\\right\]=\\left\[\\tilde\{\\Phi\}\(t,gg^\{\\prime\}\),v\\right\]=\\displaystyle=ΓΦ​\(t,\[g​g′,v\]\)=ΓΦ​\(t,Lg​\[g′,v\]\)\\displaystyle\\Gamma\_\{\\Phi\}\(t,\[gg^\{\\prime\},v\]\)=\\Gamma\_\{\\Phi\}\(t,L\_\{g\}\[g^\{\\prime\},v\]\)for everyg,g′∈Gg,g^\{\\prime\}\\in Gandv∈Vv\\in V\. Consequently, the flowΓΦ\\Gamma\_\{\\Phi\}isGG\-equivariant\. ∎

We are now ready to prove the main theorem establishingGG\-equivariance of the steerable NODE under the conditions that the vector field generating the flow inG/HG/Hand theHH\-principal connection defining parallel transport are bothGG\-invariant\.

###### Theorem 4\.6\.

LetM=G/HM=G/Hbe a homogeneous space, letϕ:M→T​M\\phi:M\\to TMbe a vector field onMM, and letω\\omegabe a principalHH\-connection onGG\. The steerable NODE defined byϕ\\phiandω\\omegaisGG\-equivariant if the vector fieldϕ\\phiand the connectionω\\omegaare bothGG\-invariant\.

###### Proof\.

The steerable NODEΨ:M×V→M×V\\Psi:M\\times V\\to M\\times Vcan be expressed in terms of the flowΓΦ\\Gamma\_\{\\Phi\}and the canonical local trivialisationϕ\\phiwith respect to the local sectionσ\\sigmaas

Ψ​\(p,v\)=φ−1​\(ΓΦ​\(1,φ​\(p,v\)\)\)\\Psi\(p,v\)=\\varphi^\{\-1\}\\left\(\\Gamma\_\{\\Phi\}\(1,\\varphi\(p,v\)\)\\right\)\\,\(4\.4\)for anyp∈Mp\\in Mandv∈Vv\\in V\. Note that this indeed defines a diffeomorphism, since bothφ\\varphiandΓΦ\\Gamma\_\{\\Phi\}are diffeomorphisms\. We also note that the left action ofGGonM×VM\\times Vis given by \([3\.9](https://arxiv.org/html/2605.11133#S3.E9)\) as,

Lg​\(p,v\)=φ−1​\(Lg​\(φ​\(p,v\)\)\),L\_\{g\}\(p,v\)=\\varphi^\{\-1\}\\left\(L\_\{g\}\(\\varphi\(p,v\)\)\\right\)\\,,\(4\.5\)for anyp∈Mp\\in Mandv∈Vv\\in V\. Consequently,[Lemma˜4\.5](https://arxiv.org/html/2605.11133#S4.Thmtheorem5)implies

Lg∘Ψ\\displaystyle L\_\{g\}\\circ\\Psi=\\displaystyle=\(φ−1∘Lg∘φ\)∘\(φ−1∘ΓΦ1∘φ\)=φ−1∘Lg∘ΓΦ1∘φ\\displaystyle\\left\(\\varphi^\{\-1\}\\circ L\_\{g\}\\circ\\varphi\\right\)\\circ\\left\(\\varphi^\{\-1\}\\circ\\Gamma^\{1\}\_\{\\Phi\}\\circ\\varphi\\right\)=\\varphi^\{\-1\}\\circ L\_\{g\}\\circ\\Gamma^\{1\}\_\{\\Phi\}\\circ\\varphi=\\displaystyle=φ−1∘ΓΦ1∘Lg∘φ=\(φ−1∘ΓΦ1∘φ\)∘\(φ−1∘Lg∘φ\)\\displaystyle\\varphi^\{\-1\}\\circ\\Gamma^\{1\}\_\{\\Phi\}\\circ L\_\{g\}\\circ\\varphi=\\left\(\\varphi^\{\-1\}\\circ\\Gamma^\{1\}\_\{\\Phi\}\\circ\\varphi\\right\)\\circ\\left\(\\varphi^\{\-1\}\\circ L\_\{g\}\\circ\\varphi\\right\)=\\displaystyle=Ψ∘Lg,\\displaystyle\\Psi\\circ L\_\{g\}\\,,where we use the notationΓΦ1​\(\[p,v\]\)=ΓΦ​\(1,\[p,v\]\)\\Gamma^\{1\}\_\{\\Phi\}\(\[p,v\]\)=\\Gamma\_\{\\Phi\}\(1,\[p,v\]\)\. This completes the proof\. ∎

We note that the converse of[Theorem˜4\.6](https://arxiv.org/html/2605.11133#S4.Thmtheorem6)is not true\. Equivariance of the steerable NODE flow implies invariance of the vector fieldϕ\\phigenerating the corresponding flow onM=G/HM=G/H\. However, it does not guarantee invariance ofω\\omega, only that a one\-dimensional subspace of the horizontal componentHg​G⊂Tg​GH\_\{g\}G\\subset T\_\{g\}Gis preserved by the left actionLgL\_\{g\}\. This is illustrated by the following example\.

###### Example 4\.8\.

We return to the setting in[Example˜3\.9](https://arxiv.org/html/2605.11133#S3.Thmtheorem9), whereG=ℝ2×U​\(1\)G=\\mathbb\{R\}^\{2\}\\times U\(1\)is the trivialH=U​\(1\)H=U\(1\)bundle overM=G/H=ℝ2M=G/H=\\mathbb\{R\}^\{2\}with the globalGG\-action given by translations\. Letω=d​θ\+f​\(y\)​d​y\\omega=d\\theta\+f\(y\)dy, for some functionf:ℝ→ℝf:\\mathbb\{R\}\\to\\mathbb\{R\}, which is clearly a 1\-form onGGtaking values in𝔥=ℝ\\mathfrak\{h\}=\\mathbb\{R\}\. This is a principal connection onGG, because:

1. \(i\)It acts trivially on the vertical subspace, since ω​\(∂θ\)=d​θ​\(∂θ\)\+f​\(y\)​d​y​\(∂θ\)=1\+f​\(y\)⋅0=1\.\\omega\(\\partial\_\{\\theta\}\)=d\\theta\(\\partial\_\{\\theta\}\)\+f\(y\)dy\(\\partial\_\{\\theta\}\)=1\+f\(y\)\\cdot 0=1\\,\.\(4\.6\)
2. \(ii\)It isHH\-equivariant, since withh=\(0,0,φ\)∈Hh=\(0,0,\\varphi\)\\in Hwe haveRh​\(x,y,θ\)=\(x,y,θ\+φ​\(mod​2​π\)\)R\_\{h\}\(x,y,\\theta\)=\(x,y,\\theta\+\\varphi\\,\(\\mathrm\{mod\}\\,2\\pi\)\)which implies \(Rh\)∗​d​θ=d​θ,\(Rh\)∗​\(f​\(y\)​d​y\)=f​\(y\)​d​y\.\(R\_\{h\}\)^\{\*\}d\\theta=d\\theta\\,,\\quad\(R\_\{h\}\)^\{\*\}\(f\(y\)dy\)=f\(y\)dy\\,\.\(4\.7\)Consequently,\(Rh\)∗​ω=ω=Adh−1​ω\(R\_\{h\}\)^\{\*\}\\omega=\\omega=\\mathrm\{Ad\}\_\{h^\{\-1\}\}\\omega, where we have used thatAdh\\mathrm\{Ad\}\_\{h\}is trivial forHHabelian\.

However, the connectionω\\omegais notGG\-invariant in general, since withg=\(0,β,0\)∈Gg=\(0,\\beta,0\)\\in Gwe haveLg​\(x,y,θ\)=\(x,y\+β,θ\)L\_\{g\}\(x,y,\\theta\)=\(x,y\+\\beta,\\theta\)and

\(Lg\)∗​ω=d​θ\+f​\(y\+β\)​d​y\.\(L\_\{g\}\)^\{\*\}\\omega=d\\theta\+f\(y\+\\beta\)dy\\,\.\(4\.8\)Thus,ω\\omegais invariant only ifffis constant\. Takingffnon\-constant gives a principal connection which is notGG\-invariant\.

We now construct a horizontal,GG\-equivariant flow onGGthat projects to aGG\-equivariant flow onM=G/HM=G/H\. LetΦ~\\tilde\{\\Phi\}be the flow generated byϕ~=∂x\\tilde\{\\phi\}=\\partial\_\{x\}\. From \([3\.20](https://arxiv.org/html/2605.11133#S3.E20)\), we have\(Lg\)∗​ϕ~=ϕ~\(L\_\{g\}\)\_\{\*\}\\tilde\{\\phi\}=\\tilde\{\\phi\}soϕ~\\tilde\{\\phi\}isGG\-invariant andΦ~\\tilde\{\\Phi\}isGG\-equivariant\. Furthermore, we have

ω​\(ϕ~\)=ω​\(∂x\)=d​θ​\(∂x\)\+f​\(y\)​d​y​\(∂x\)=0,\\omega\(\\tilde\{\\phi\}\)=\\omega\(\\partial\_\{x\}\)=d\\theta\(\\partial\_\{x\}\)\+f\(y\)dy\(\\partial\_\{x\}\)=0\\,,\(4\.9\)which means thatϕ~\\tilde\{\\phi\}is horizontal with respect toω\\omega\. Finally,ϕ~\\tilde\{\\phi\}projects toϕ=π∗​ϕ~=π∗​\(∂x\)=∂x\\phi=\\pi\_\{\*\}\\tilde\{\\phi\}=\\pi\_\{\*\}\(\\partial\_\{x\}\)=\\partial\_\{x\}\. From \([3\.21](https://arxiv.org/html/2605.11133#S3.E21)\), we have\(Lg\)∗​ϕ=ϕ\(L\_\{g\}\)\_\{\*\}\\phi=\\phi, soϕ\\phiis aGG\-invariant vector field onMMgenerating aGG\-equivariant flowΦ\\Phi\. Consequently,Φ~\\tilde\{\\Phi\}has the desired properties\.

Extending to the parallel transport flowΓΦ\\Gamma\_\{\\Phi\}on the associated bundleG×ρVG\\times\_\{\\rho\}V, withρ:U​\(1\)→G​L​\(V\)\\rho:U\(1\)\\to GL\(V\), we have

ΓΦ​\(t,\[g,v\]\)=\[Φ~​\(t,g\),v\]\.\\Gamma\_\{\\Phi\}\(t,\[g,v\]\)=\[\\tilde\{\\Phi\}\(t,g\),v\]\\,\.\(4\.10\)Equivariance ofΦ~\\tilde\{\\Phi\}implies equivariance ofΓΦ\\Gamma\_\{\\Phi\}; see the proof of[Lemma˜4\.5](https://arxiv.org/html/2605.11133#S4.Thmtheorem5)\. Moreover,ΓΦ\\Gamma\_\{\\Phi\}is horizontal by construction\. Consequently, the steerable NODEΨ\\Psiobtained fromΓΦ\\Gamma\_\{\\Phi\}isGG\-equivariant\.

This illustrates why the converse of[Theorem˜4\.6](https://arxiv.org/html/2605.11133#S4.Thmtheorem6)is not true\. Equivariance of the flowΦ~\\tilde\{\\Phi\}in thexx\-direction forcesω\\omegato be invariant under the corresponding translations, but it can still fail to be invariant underyy\-translations and therefore under the full groupGG\.

### 4\.2Equivariance in the local formulation

We now consider equivariance of steerable NODEs in their local description, i\.e\., mapsΨ:M×V→M×V\\Psi:M\\times V\\to M\\times Vdefined by

Ψ​\(p,v\)=\(Φp​\(1\),ρ​\(hp​\(1\)\)​v\),\\Psi\(p,v\)=\(\\Phi\_\{p\}\(1\),\\rho\(h\_\{p\}\(1\)\)v\)\\,,\(4\.11\)for all\(p,v\)∈M×V\(p,v\)\\in M\\times V, whereΦp\\Phi\_\{p\}andhph\_\{p\}are given by[definition˜3\.7](https://arxiv.org/html/2605.11133#S3.Thmtheorem7)for some vector fieldϕ\\phiand connectionω\\omega\. SinceΨ\\Psiacts on the spaceM×VM\\times V, we must consider the left action onM×VM\\times V\. Recall that according to \([3\.9](https://arxiv.org/html/2605.11133#S3.E9)\) this action is given by

Lg​\(p,v\)=\(g​p,ρ​\(c​\(g,p\)\)​v\),L\_\{g\}\(p,v\)=\(gp,\\rho\(c\(g,p\)\)v\),\(4\.12\)wherec:G×M→Hc:G\\times M\\to His the map defined in \([3\.5](https://arxiv.org/html/2605.11133#S3.E5)\)\. In the following[lemma](https://arxiv.org/html/2605.11133#S4.Thmtheorem9), we show that the equivariance ofΨ\\Psiunder this action can be characterised by the equivariance of the vector fieldϕ\\phitogether with a condition involving the mapcc\.

###### Lemma 4\.9\.

LetΨ:M×V→M×V\\Psi:M\\times V\\to M\\times Vbe a steerable NODE defined by the vector fieldϕ:M→T​M\\phi:M\\to TMand the principalHH\-connectionω\\omegaonGG\. ThenΨ\\PsiisGG\-equivariant if and only if the vector fieldϕ\\phiisGG\-invariant and

c​\(g,Φp​\(1\)\)​hp​\(1\)=hg​p​\(1\)​c​\(g,p\)​k​\(g,p\)c\(g,\\Phi\_\{p\}\(1\)\)\\,h\_\{p\}\(1\)=h\_\{gp\}\(1\)\\,c\(g,p\)\\,k\(g,p\)\(4\.13\)holds for allg∈Gg\\in G,p∈Mp\\in Mand somek​\(g,p\)∈ker⁡ρk\(g,p\)\\in\\ker\\rho\.

###### Proof\.

Using the definition of the left action ofGGonM×VM\\times Vin \([3\.9](https://arxiv.org/html/2605.11133#S3.E9)\), we compute

Lg​Ψ​\(p,v\)=\(Lg​Φp​\(1\),ρ​\(c​\(g,Φp​\(1\)\)​hp​\(1\)\)​v\)L\_\{g\}\\Psi\(p,v\)=\(L\_\{g\}\\Phi\_\{p\}\(1\),\\rho\(c\(g,\\Phi\_\{p\}\(1\)\)h\_\{p\}\(1\)\)v\)\(4\.14\)and

Ψ​\(Lg​\(p,v\)\)=\(Φg​p​\(1\),ρ​\(hg​p​\(1\)​c​\(g,p\)\)​v\)\\Psi\(L\_\{g\}\(p,v\)\)=\\big\(\\Phi\_\{gp\}\(1\),\\rho\(h\_\{gp\}\(1\)c\(g,p\)\)v\\big\)\(4\.15\)forg∈Gg\\in G,p∈Mp\\in M, andv∈Vv\\in V\. These expressions agree for allg∈Gg\\in G,p∈Mp\\in M, andv∈Vv\\in Vif and only ifϕ\\phiisGG\-invariant andρ​\(c​\(g,Φp​\(1\)\)​hp​\(1\)\)​v=ρ​\(hg​p​\(1\)​c​\(g,p\)\)​v\\rho\(c\(g,\\Phi\_\{p\}\(1\)\)h\_\{p\}\(1\)\)v=\\rho\(h\_\{gp\}\(1\)c\(g,p\)\)vholds for allg∈Gg\\in G,p∈Mp\\in Mandv∈Vv\\in V\. The second condition is equivalent to

c​\(g,Φp​\(1\)\)​hp​\(1\)​\(hg​p​\(1\)​c​\(g,p\)\)−1∈ker⁡ρ,c\(g,\\Phi\_\{p\}\(1\)\)h\_\{p\}\(1\)\\left\(h\_\{gp\}\(1\)c\(g,p\)\\right\)^\{\-1\}\\in\\ker\{\\rho\},\(4\.16\)yielding \([4\.13](https://arxiv.org/html/2605.11133#S4.E13)\)\. ∎

The local description of equivariance provided in[lemma˜4\.9](https://arxiv.org/html/2605.11133#S4.Thmtheorem9)can be visualised by the commutative diagram in[fig\.˜3](https://arxiv.org/html/2605.11133#S4.F3)\.

![Refer to caption](https://arxiv.org/html/2605.11133v1/x4.png)Figure 3:An equivariant steerable NODE has the following commutative diagram\. It is clear from the diagram thatLg​Φp​\(1\)=Φg​p​\(1\)L\_\{g\}\\Phi\_\{p\}\(1\)=\\Phi\_\{gp\}\(1\)must hold\. Moreover, it follows from how featuresf​\(p\)f\(p\)are transformed underΨ\\Psi\([definition˜3\.7](https://arxiv.org/html/2605.11133#S3.Thmtheorem7)\) and howLgL\_\{g\}acts on local feature fields \([Table˜1](https://arxiv.org/html/2605.11133#S3.T1)\) thatf​\(Φp​\(1\)\)=ρ​\(hp​\(1\)\)​f​\(p\)f\(\\Phi\_\{p\}\(1\)\)=\\rho\(h\_\{p\}\(1\)\)f\(p\),f​\(g​p\)=ρ​\(c​\(g,p\)\)​f​\(p\)f\(gp\)=\\rho\(c\(g,p\)\)f\(p\), andf​\(Φg​p​\(1\)\)=ρ​\(c​\(g,Φp​\(1\)\)​hp​\(1\)\)​f​\(p\)=ρ​\(hg​p​\(1\)​c​\(g,p\)\)​f​\(p\)f\(\\Phi\_\{gp\}\(1\)\)=\\rho\(c\(g,\\Phi\_\{p\}\(1\)\)h\_\{p\}\(1\)\)f\(p\)=\\rho\(h\_\{gp\}\(1\)c\(g,p\)\)f\(p\), where the latter condition is \([4\.13](https://arxiv.org/html/2605.11133#S4.E13)\)\. Compare with\[[1](https://arxiv.org/html/2605.11133#bib.bib3), Figure 8\]\.The local equivariance condition in[Lemma˜4\.9](https://arxiv.org/html/2605.11133#S4.Thmtheorem9)is automatically satisfied under the geometric assumptions of[theorem˜4\.6](https://arxiv.org/html/2605.11133#S4.Thmtheorem6)\. Supposeϕ\\phiandω\\omegaareGG\-invariant\. From the construction ofΨ\\Psi, the curveshph\_\{p\}andhg​ph\_\{gp\}satisfyhp​\(0\)=hg​p​\(0\)=eh\_\{p\}\(0\)=h\_\{gp\}\(0\)=eand define two unique horizontal liftsΦ~p​\(t\)=σ​\(Φp​\(t\)\)​hp​\(t\)\\tilde\{\\Phi\}\_\{p\}\(t\)=\\sigma\(\\Phi\_\{p\}\(t\)\)h\_\{p\}\(t\)andΦ~g​p​\(t\)=σ​\(Φg​p​\(t\)\)​hg​p​\(t\)\\tilde\{\\Phi\}\_\{gp\}\(t\)=\\sigma\(\\Phi\_\{gp\}\(t\)\)h\_\{gp\}\(t\)overΦp\\Phi\_\{p\}andΦg​p\\Phi\_\{gp\}, respectively\.

Sinceω\\omegaisGG\-invariant, the curveLg​Φ~pL\_\{g\}\\tilde\{\\Phi\}\_\{p\}must also be horizontal for allg∈Gg\\in Gandp∈Mp\\in M\. Thus, for eachppandgg, there exists a unique elementh​\(g,p\)∈Hh\(g,p\)\\in Hsuch thatLg​Φ~p​\(t\)=Φ~g​p​\(t\)​h​\(g,p\)L\_\{g\}\\tilde\{\\Phi\}\_\{p\}\(t\)=\\tilde\{\\Phi\}\_\{gp\}\(t\)h\(g,p\)for alltt\.

Left translation gives

Lg​Φ~p​\(t\)=σ​\(Lg​Φp​\(t\)\)​c​\(g,Φp​\(t\)\)​hp​\(t\)\.L\_\{g\}\\tilde\{\\Phi\}\_\{p\}\(t\)=\\sigma\(L\_\{g\}\\Phi\_\{p\}\(t\)\)c\(g,\\Phi\_\{p\}\(t\)\)h\_\{p\}\(t\)\\,\.\(4\.17\)In particular, we have

Lg​Φ~p​\(0\)=Φ~g​p​\(0\)​c​\(g,p\),L\_\{g\}\\tilde\{\\Phi\}\_\{p\}\(0\)=\\tilde\{\\Phi\}\_\{gp\}\(0\)c\(g,p\)\\,,\(4\.18\)which impliesh​\(g,p\)=c​\(g,p\)h\(g,p\)=c\(g,p\)\. SubstitutingΦ~p​\(t\)=σ​\(Φp​\(t\)\)​hp​\(t\)\\tilde\{\\Phi\}\_\{p\}\(t\)=\\sigma\(\\Phi\_\{p\}\(t\)\)h\_\{p\}\(t\)andΦ~g​p​\(t\)=σ​\(ug​p​\(t\)\)​hg​p​\(t\)\\tilde\{\\Phi\}\_\{gp\}\(t\)=\\sigma\(u\_\{gp\}\(t\)\)h\_\{gp\}\(t\)yields

c​\(g,Φp​\(t\)\)​hp​\(t\)=hg​p​\(t\)​c​\(g,p\)\.c\(g,\\Phi\_\{p\}\(t\)\)\\,h\_\{p\}\(t\)=h\_\{gp\}\(t\)\\,c\(g,p\)\.\(4\.19\)Fort=1t=1, this corresponds to \([4\.13](https://arxiv.org/html/2605.11133#S4.E13)\) withk=ek=e\.

We conclude this section with an example that illustrates the local description of equivariance\.

###### Example 4\.10\.

We return again to[Example˜3\.9](https://arxiv.org/html/2605.11133#S3.Thmtheorem9), and the steerable NODE defined byϕ=∂x\\phi=\\partial\_\{x\}andω=π​d​x\+d​y\+d​θ\\omega=\\pi dx\+dy\+d\\theta\. Invariance ofϕ\\phiandω\\omegais immediate sinceGGacts by translations\. To determine whetherΨ​\(x,y,v\)=\(x\+1,y,R​\(π\)​v\)\\Psi\(x,y,v\)=\(x\+1,y,R\(\\pi\)v\)is equivariant, in the sense of[definition˜4\.2](https://arxiv.org/html/2605.11133#S4.Thmtheorem2), we need to determine the functionc​\(g,p\)c\(g,p\)for the sectionσ​\(x,y\)=\(x,y,χ\)\\sigma\(x,y\)=\(x,y,\\chi\)\. Withg=\(α,β,φ\)∈Gg=\(\\alpha,\\beta,\\varphi\)\\in G, we findc​\(g,p\)=φc\(g,p\)=\\varphi, independently ofp∈Mp\\in M\.

Forp=\(x,y\)∈ℝ2p=\(x,y\)\\in\\mathbb\{R\}^\{2\}andv∈ℝ2v\\in\\mathbb\{R\}^\{2\}, we then have

Lg​\(p,v\)=\(g​p,ρ​\(c​\(g,p\)\)​v\)=\(x\+α,y\+β,R​\(φ\)​v\),L\_\{g\}\(p,v\)=\(gp,\\rho\(c\(g,p\)\)v\)=\(x\+\\alpha,y\+\\beta,R\(\\varphi\)v\)\\,,\(4\.20\)and consequently

Ψ​\(Lg​\(p,v\)\)=\(x\+α\+1,y\+β,R​\(π\)​R​\(φ\)​v\)\.\\Psi\(L\_\{g\}\(p,v\)\)=\(x\+\\alpha\+1,y\+\\beta,R\(\\pi\)R\(\\varphi\)v\)\\,\.\(4\.21\)Interchanging the order ofΨ\\PsiandLgL\_\{g\}, we obtain

Lg​\(Ψ​\(p,v\)\)=Lg​\(x\+1,y,R​\(π\)\)=\(x\+1\+α,y\+β,R​\(φ\)​R​\(π\)​v\)\.L\_\{g\}\(\\Psi\(p,v\)\)=L\_\{g\}\(x\+1,y,R\(\\pi\)\)=\(x\+1\+\\alpha,y\+\\beta,R\(\\varphi\)R\(\\pi\)v\)\\,\.\(4\.22\)Clearly, sinceHHis abelian,R​\(φ\)​R​\(π\)=R​\(π\)​R​\(φ\)R\(\\varphi\)R\(\\pi\)=R\(\\pi\)R\(\\varphi\)and we haveLg∘Ψ=Ψ∘LgL\_\{g\}\\circ\\Psi=\\Psi\\circ L\_\{g\}as expected from[theorem˜4\.6](https://arxiv.org/html/2605.11133#S4.Thmtheorem6)\.

We can also verify equivariance ofΨ\\Psiby checking the conditions in[Lemma˜4\.9](https://arxiv.org/html/2605.11133#S4.Thmtheorem9)\. The vector fieldϕ\\phiis invariant by construction andc​\(g,p\)=φc\(g,p\)=\\varphi\. The maphph\_\{p\}was computed in[example˜3\.9](https://arxiv.org/html/2605.11133#S3.Thmtheorem9)to behp​\(t\)=−π​th\_\{p\}\(t\)=\-\\pi t, independent of the pointpp\. Thus

c​\(g,Φp​\(t\)\)​hp​\(t\)=hg​p​\(t\)​c​\(g,p\)c\(g,\\Phi\_\{p\}\(t\)\)h\_\{p\}\(t\)=h\_\{gp\}\(t\)c\(g,p\)\(4\.23\)holds, which is the condition in \([4\.13](https://arxiv.org/html/2605.11133#S4.E13)\) withk=ek=e\.

### 4\.3Invariant connections and Wang’s theorem

We have seen that steerable NODEs are equivariant if both the vector fieldϕ\\phiand the connectionω\\omegaare invariant\. When we treat these as learnable quantities, we therefore need to understand how to parametrise the spaces of invariant vector fields and principal connections\. In\[[1](https://arxiv.org/html/2605.11133#bib.bib3)\], it was shown that invariant vector fields onMMcan be parametrised using the differential invariants of the action ofGGonMM\. For invariant principal connections onGG, a corresponding parametrisation can be accomplished using a theorem by Wang\. Proposition A of\[[33](https://arxiv.org/html/2605.11133#bib.bib71)\]provides, as a special case, the following classification ofGG\-invariant connections onGGas a principalHH\-bundle overM=G/HM=G/H\.

\{restatable\}

\[Wang’s theorem\[[33](https://arxiv.org/html/2605.11133#bib.bib71)\]\]theoremWangTheorem TheGG\-invariant connectionsω\\omegaonGGare in one\-to\-one correspondence with linear mapsΛ:𝔤→𝔥\\Lambda:\\mathfrak\{g\}\\to\\mathfrak\{h\}that satisfy

- \(i\)Λ∘Adh=Adh∘Λ,∀h∈H\\Lambda\\circ\\mathrm\{Ad\}\_\{h\}=\\mathrm\{Ad\}\_\{h\}\\circ\\Lambda,\\quad\\forall h\\in H,
- \(ii\)Λ\|𝔥=id\|𝔥\\left\.\\Lambda\\right\|\_\{\\mathfrak\{h\}\}=\\left\.\\mathrm\{id\}\\right\|\_\{\\mathfrak\{h\}\},

whereAd\\mathrm\{Ad\}is the adjoint action ofHHon its Lie algebra𝔥\\mathfrak\{h\}\.

More concretely, the theorem provides explicit constructions of the linear mapΛ\\Lambdaand invariant connectionω\\omega\. Given aGG\-invariant connectionω\\omega, the associated linear map is defined as

Λ:=ωe\.\\Lambda:=\\omega\_\{e\}\\,\.\(4\.24\)Conversely, given the linear mapΛ\\Lambda, the corresponding invariant connection is defined by

ωg:=\(Lg−1\)∗​Λ,\\omega\_\{g\}:=\(L\_\{g^\{\-1\}\}\)^\{\*\}\\Lambda\\,,\(4\.25\)for allg∈Gg\\in G\. In[appendix˜B](https://arxiv.org/html/2605.11133#A2), we provide a self\-contained proof of[section˜4\.3](https://arxiv.org/html/2605.11133#S4.SS3)\.

We will now illustrate how this construction allows us to parametrise the most general invariant connections in two familiar examples\.

###### Example 4\.11\.

We first consider the simplest possible exampleG=ℝ2G=\\mathbb\{R\}^\{2\},H=ℝH=\\mathbb\{R\},M=G/H=ℝM=G/H=\\mathbb\{R\}, where both left and right actions are translations\. The Lie algebras are𝔤=ℝ2\\mathfrak\{g\}=\\mathbb\{R\}^\{2\},𝔥=ℝ\\mathfrak\{h\}=\\mathbb\{R\}, and the adjoint action ofHHis trivial since it is abelian,Adh=id𝔥\\mathrm\{Ad\}\_\{h\}=\\mathrm\{id\}\_\{\\mathfrak\{h\}\}\. We write an element of𝔤\\mathfrak\{g\}asX=\[α​β\]TX=\[\\alpha\\,\\,\\beta\]^\{T\}and an element of𝔥\\mathfrak\{h\}asY=βY=\\beta, and their images under the exponential map as

g=exp⁡\(X\)=\(α,β\),h=exp⁡\(Y\)=β\.g=\\exp\(X\)=\(\\alpha,\\beta\)\\,,\\quad h=\\exp\(Y\)=\\beta\\,\.\(4\.26\)
We now construct allGG\-invariant principal connections onGGusing Wang’s theorem\. A linear mapΛ:𝔤→𝔥\\Lambda:\\mathfrak\{g\}\\to\\mathfrak\{h\}is parametrised as matrixΛ=\[a​b\]\\Lambda=\[a\\,\\,b\]and acts onXXas

Λ​\(X\)=\[a​b\]​\[α​β\]T=a​α\+b​β\.\\Lambda\(X\)=\[a\\,\\,b\]\[\\alpha\\,\\,\\beta\]^\{T\}=a\\alpha\+b\\beta\\,\.\(4\.27\)We want to describe the most general mapΛ\\Lambdasatisfying the conditions onΛ\\Lambdain[section˜4\.3](https://arxiv.org/html/2605.11133#S4.SS3)\. Condition \(i\) is trivial sinceAd=id𝔥\\mathrm\{Ad\}=\\mathrm\{id\}\_\{\\mathfrak\{h\}\}\. Acting onX=\[0​β\]T∈𝔥⊂𝔤X=\[0\\,\\,\\beta\]^\{T\}\\in\\mathfrak\{h\}\\subset\\mathfrak\{g\}yields

Λ​\(X\)=\[a​b\]​\[0​β\]T=b​β\.\\Lambda\(X\)=\[a\\,\\,b\]\[0\\,\\,\\beta\]^\{T\}=b\\beta\\,\.\(4\.28\)Condition \(ii\) amounts toΛ​\(X\)=β\\Lambda\(X\)=\\beta, which is equivalent tob=1b=1\. Consequently, the linear maps satisfying both \(i\) and \(ii\) are

Λ=\[a​1\],a∈ℝ,\\Lambda=\[a\\,\\,1\]\\,,\\quad a\\in\\mathbb\{R\}\\,,\(4\.29\)which is a 1\-parameter family of maps\. The corresponding 1\-parameter family ofGG\-invariant connections is obtained by first defining

ωe:=Λ\.\\omega\_\{e\}:=\\Lambda\.\(4\.30\)WithX=\[α​β\]TX=\[\\alpha\\,\\,\\beta\]^\{T\}, we then have

ωe​\(X\)=Λ​\(X\)=\[a​1\]​\[α​β\]T=a​α\+β∈𝔥\.\\omega\_\{e\}\(X\)=\\Lambda\(X\)=\[a\\,\\,1\]\[\\alpha\\,\\,\\beta\]^\{T\}=a\\alpha\+\\beta\\in\\mathfrak\{h\}\\,\.\(4\.31\)Subsequently, we defineω∈Ω1​\(G,𝔥\)\\omega\\in\\Omega^\{1\}\(G,\\mathfrak\{h\}\)as

ωg:=\(Lg−1\)∗​ωe,∀g∈G\.\\omega\_\{g\}:=\(L\_\{g\}^\{\-1\}\)^\{\*\}\\omega\_\{e\},\\ \\ \\ \\forall g\\in G\\,\.\(4\.32\)BecauseGGacts as translations we have\(Lg\)∗=id\(L\_\{g\}\)^\{\*\}=\\mathrm\{id\}, implying thatω\\omegais a constant form onGG\.

We conclude this example by providing some intuition for the 1\-parameter family of invariant connections defined by \([4\.32](https://arxiv.org/html/2605.11133#S4.E32)\) and \([4\.29](https://arxiv.org/html/2605.11133#S4.E29)\)\. To this end, we first consider the horizontal subspace defined by the connectionω\\omegacorresponding to the linear mapΛ=\[a​1\]\\Lambda=\[a\\,\\,1\]\. At the identity, this subspace is defined as the kernelHe​G=ker⁡ωeH\_\{e\}G=\\ker\\omega\_\{e\}, which forv=\[α​β\]v=\[\\alpha\\,\\,\\beta\]amounts to

ωe​\(v\)=a​α\+β=0\.\\omega\_\{e\}\(v\)=a\\alpha\+\\beta=0\\,\.\(4\.33\)Consequently,He​G⊂Te​G≅ℝ2H\_\{e\}G\\subset T\_\{e\}G\\cong\\mathbb\{R\}^\{2\}is the straight lineβ=−a​α\\beta=\-a\\alpha, which corresponds to the expected 1\-parameter family of possible decompositionsTe​G=He​G⊕Ve​GT\_\{e\}G=H\_\{e\}G\\oplus V\_\{e\}Gobtained by specifying a slopeaaofHe​GH\_\{e\}G\. Since the connectionω\\omegais constant, the decompositionTg​G=Hg​G⊕Vg​GT\_\{g\}G=H\_\{g\}G\\oplus V\_\{g\}Gis identical for allg∈Gg\\in G, corresponding to a constant slope throughoutGG\. As expected, this defines an invariant decomposition of the tangent bundleT​GTG\.

Finally, we consider the vector fieldϕ=∂x\\phi=\\partial\_\{x\}onMM, which isGG\-invariant by virtue of\(Lg\)∗=id\(L\_\{g\}\)\_\{\*\}=\\mathrm\{id\}\. The integral curve through the origin inMMisΦ0​\(t\)=t\\Phi\_\{0\}\(t\)=tand its horizontal lift through\(0,0\)∈G\(0,0\)\\in GisΦ~0​\(t\)=\(t,−a​t\)\\tilde\{\\Phi\}\_\{0\}\(t\)=\(t,\-at\), since

Φ~˙0​\(t\)=dd​t​Φ~0​\(t\)=\[1−a\]\\dot\{\\tilde\{\\Phi\}\}\_\{0\}\(t\)=\\frac\{d\}\{dt\}\\tilde\{\\Phi\}\_\{0\}\(t\)=\[1\\,\\,\-a\]\(4\.34\)which satisfiesΛ​\(Φ~˙0​\(t\)\)=\[a​1\]​\[1−a\]T=0\\Lambda\(\\dot\{\\tilde\{\\Phi\}\}\_\{0\}\(t\)\)=\[a\\,\\,1\]\[1\\,\\,\-a\]^\{T\}=0\. The parallel transport of\(0,0\)\(0,0\)alongu​\(t\)u\(t\)is consequently\(1,−a\)\(1,\-a\)\. The generalisation to an arbitrary initial point\(x,y\)∈G\(x,y\)\\in Gis straightforward\.

###### Example 4\.12\.

We consider the caseG=SO​\(3\)G=\\mathrm\{SO\}\(3\),H=SO​\(2\)H=\\mathrm\{SO\}\(2\)andM=G/H=SO​\(3\)/SO​\(2\)≅S2M=G/H=\\mathrm\{SO\}\(3\)/\\mathrm\{SO\}\(2\)\\cong S^\{2\}\. The canonical projectionπ:G→M\\pi:G\\to Msends a matrixR∈SO​\(3\)R\\in\\mathrm\{SO\}\(3\)to its first column\. To makeπ\\piinvariant under the right action ofHH, we embedHHinGGas the subgroup of rotations about the first coordinate axis, i\.e\.,

H↪G,\(a11a12a21a22\)↦\(1000a11a120a21a22\)\.H\\hookrightarrow G,\\qquad\\begin\{pmatrix\}a\_\{11\}&a\_\{12\}\\\\ a\_\{21\}&a\_\{22\}\\end\{pmatrix\}\\mapsto\\begin\{pmatrix\}1&0&0\\\\ 0&a\_\{11\}&a\_\{12\}\\\\ 0&a\_\{21\}&a\_\{22\}\\end\{pmatrix\}\\,\.\(4\.35\)
The Lie algebra𝔤=𝔰​𝔬​\(3\)\\mathfrak\{g\}=\\mathfrak\{so\}\(3\)is spanned by

X1=\(00000−1010\),X2=\(001000−100\),X3=\(0−10100000\),X\_\{1\}=\\begin\{pmatrix\}0&0&0\\\\ 0&0&\-1\\\\ 0&1&0\\end\{pmatrix\},\\ \\ X\_\{2\}=\\begin\{pmatrix\}0&0&1\\\\ 0&0&0\\\\ \-1&0&0\\end\{pmatrix\},\\ \\ X\_\{3\}=\\begin\{pmatrix\}0&\-1&0\\\\ 1&0&0\\\\ 0&0&0\\end\{pmatrix\},\(4\.36\)while the Lie algebra𝔥\\mathfrak\{h\}is spanned byX1X\_\{1\}\. Thus, a general element of𝔤\\mathfrak\{g\}can be written as

X=a1​X1\+a2​X2\+a3​X3,a1,a2,a3∈ℝ,X=a\_\{1\}X\_\{1\}\+a\_\{2\}X\_\{2\}\+a\_\{3\}X\_\{3\},\\qquad a\_\{1\},a\_\{2\},a\_\{3\}\\in\\mathbb\{R\},\(4\.37\)and an element in𝔥\\mathfrak\{h\}has the formY=b​X1Y=bX\_\{1\}, whereb∈ℝb\\in\\mathbb\{R\}\. The exponentiation ofYYgives a general element inHHof the form

h=exp⁡\(Y\)=exp⁡\(b​X1\)=\(1000cos⁡\(b\)−sin⁡\(b\)0sin⁡\(b\)cos⁡\(b\)\)\.h=\\exp\(Y\)=\\exp\(bX\_\{1\}\)=\\begin\{pmatrix\}1&0&0\\\\ 0&\\cos\(b\)&\-\\sin\(b\)\\\\ 0&\\sin\(b\)&\\cos\(b\)\\end\{pmatrix\}\.\(4\.38\)
A general linear mappingΛ:𝔤→𝔥\\Lambda:\\mathfrak\{g\}\\to\\mathfrak\{h\}is given by

Λ​\(X\)=Λ​\(a1​X1\+a2​X2\+a3​X3\)=\(a1​c1\+a2​c2\+a3​c3\)​X1,\\Lambda\(X\)=\\Lambda\(a\_\{1\}X\_\{1\}\+a\_\{2\}X\_\{2\}\+a\_\{3\}X\_\{3\}\)=\(a\_\{1\}c\_\{1\}\+a\_\{2\}c\_\{2\}\+a\_\{3\}c\_\{3\}\)X\_\{1\},\(4\.39\)wherec1,c2,c3∈ℝc\_\{1\},c\_\{2\},c\_\{3\}\\in\\mathbb\{R\}are constants definingΛ\\Lambda\. To impose the conditions in[section˜4\.3](https://arxiv.org/html/2605.11133#S4.SS3), we first compute the adjoint actionAdh:𝔤→𝔤\\mathrm\{Ad\}\_\{h\}:\\mathfrak\{g\}\\to\\mathfrak\{g\}defined by

Adh​\(X\)=h​X​h−1,\\mathrm\{Ad\}\_\{h\}\(X\)=hXh^\{\-1\}\\,,\(4\.40\)where we use matrix notation for the induced action ofHHon𝔤\\mathfrak\{g\}\.

Using orthogonality, we immediately obtainAdh​\(X1\)=X1\\mathrm\{Ad\}\_\{h\}\(X\_\{1\}\)=X\_\{1\},Adh​\(X2\)=cos⁡\(b\)​X2\+sin⁡\(b\)​X3\\mathrm\{Ad\}\_\{h\}\(X\_\{2\}\)=\\cos\(b\)X\_\{2\}\+\\sin\(b\)X\_\{3\}, andAdh​\(X3\)=−sin⁡\(b\)​X2\+cos⁡\(b\)​X3\\mathrm\{Ad\}\_\{h\}\(X\_\{3\}\)=\-\\sin\(b\)X\_\{2\}\+\\cos\(b\)X\_\{3\}\. Thus,

Adh​\(X\)=a1​X1\+\(a2​cos⁡\(b\)−a3​sin⁡\(b\)\)​X2\+\(a2​sin⁡\(b\)\+a3​cos⁡\(b\)\)​X3\.\\mathrm\{Ad\}\_\{h\}\(X\)=a\_\{1\}X\_\{1\}\+\(a\_\{2\}\\cos\(b\)\-a\_\{3\}\\sin\(b\)\)X\_\{2\}\+\(a\_\{2\}\\sin\(b\)\+a\_\{3\}\\cos\(b\)\)X\_\{3\}\.\(4\.41\)ApplyingΛ\\Lambdagives

Λ​\(Adh​\(X\)\)=\(a1​c1\+\(a2​cos⁡\(b\)−a3​sin⁡\(b\)\)​c2\+\(a2​sin⁡\(b\)\+a3​cos⁡\(b\)\)​c3\)​X1\.\\Lambda\(\\mathrm\{Ad\}\_\{h\}\(X\)\)=\\left\(a\_\{1\}c\_\{1\}\+\(a\_\{2\}\\cos\(b\)\-a\_\{3\}\\sin\(b\)\)c\_\{2\}\+\(a\_\{2\}\\sin\(b\)\+a\_\{3\}\\cos\(b\)\)\\,c\_\{3\}\\right\)X\_\{1\}\.\(4\.42\)Moreover, sinceAdh\\mathrm\{Ad\}\_\{h\}acts trivially on𝔥\\mathfrak\{h\},

Adh​\(Λ​\(X\)\)=Λ​\(X\)=\(a1​c1\+a2​c2\+a3​c3\)​X1\.\\mathrm\{Ad\}\_\{h\}\(\\Lambda\(X\)\)=\\Lambda\(X\)=\(a\_\{1\}c\_\{1\}\+a\_\{2\}c\_\{2\}\+a\_\{3\}c\_\{3\}\)X\_\{1\}\\,\.\(4\.43\)For condition \(i\) in[section˜4\.3](https://arxiv.org/html/2605.11133#S4.SS3)to hold, \([4\.42](https://arxiv.org/html/2605.11133#S4.E42)\) and \([4\.43](https://arxiv.org/html/2605.11133#S4.E43)\) must be equal for allhh, forcingc2=c3=0c\_\{2\}=c\_\{3\}=0\. Condition \(ii\) of[section˜4\.3](https://arxiv.org/html/2605.11133#S4.SS3)requires thatΛ\\Lambdarestricted to𝔥\\mathfrak\{h\}is the identity mapping\. SinceΛ​\(X1\)=c1​X1\\Lambda\(X\_\{1\}\)=c\_\{1\}X\_\{1\}, this impliesc1=1c\_\{1\}=1\.

We conclude that there is a unique linear mapΛ:𝔤→𝔥\\Lambda:\\mathfrak\{g\}\\to\\mathfrak\{h\}satisfying both conditions in[section˜4\.3](https://arxiv.org/html/2605.11133#S4.SS3):

Λ​\(a1​X1\+a2​X2\+a3​X3\)=a1​X1\.\\Lambda\(a\_\{1\}X\_\{1\}\+a\_\{2\}X\_\{2\}\+a\_\{3\}X\_\{3\}\)=a\_\{1\}X\_\{1\}\\,\.\(4\.44\)By[section˜4\.3](https://arxiv.org/html/2605.11133#S4.SS3), this determines a uniqueGG\-invariant connectionω\\omegagiven by

ωg=\(Lg−1\)∗​Λ,\\omega\_\{g\}=\(L\_\{g^\{\-1\}\}\)^\{\*\}\\Lambda\\,,\(4\.45\)for eachg∈Gg\\in G\.

## 5Relation to existing NODE models

Having defined steerable NODEs on homogeneous spacesM=G/HM=G/Hand derived conditions for equivariance under the global symmetry groupGG, we now turn to the relation of these models to the existing literature on NODEs and their applications\. First, we consider the relation to previous constructions where an action on tangent vectors onMMis induced by the flow\. Second, we show how our steerable NODEs extend and generalise existing work on continuous normalizing flows on Lie groups\. Finally, we provide a detailed discussion of steerable NODEs in the specific caseM=S2M=S^\{2\}to illustrate our framework in a familiar geometric setting common in applications\.

### 5\.1Parallel transport and the push\-forward

In our previous work\[[1](https://arxiv.org/html/2605.11133#bib.bib3)\], we showed that a manifold NODEψ:M→M\\psi:M\\to Minduces an action on tangent vector fieldsX:M→T​MX:M\\to TMthrough the push\-forward\. This is analogous to the construction of CNFs, which instead induce an action through the push\-forward of densities\. To demonstrate how the induced action on vector fields is related to steerable NODEs, in the setting of homogeneous spaces, we now show an example of how the push\-forward can be expressed as parallel transport along a flow for a specific connectionω\\omega\.

###### Example 5\.1\.

We return to the setting in[example˜4\.11](https://arxiv.org/html/2605.11133#S4.Thmtheorem11), whereG=ℝ2G=\\mathbb\{R\}^\{2\}acts by translations onM=G/H=ℝM=G/H=\\mathbb\{R\}and letψ:M→M\\psi:M\\to Mbe the manifold NODE defined by the vector fieldϕ=∂x\\phi=\\partial\_\{x\}\. Since\(Lg\)∗=id\(L\_\{g\}\)\_\{\*\}=\\mathrm\{id\}, the vector fieldϕ\\phiis invariant, and the manifold NODEψ\\psiis equivariant\. The integral curves ofϕ\\phiareΦx​\(t\)=x\+t\\Phi\_\{x\}\(t\)=x\+t, for everyx∈Mx\\in M, and, consequently, we haveΨ​\(x\)=x\+1\\Psi\(x\)=x\+1\. Applying the action induced by the push\-forward ofψ\\psito a vectorw∈Tx​M≅ℝw\\in T\_\{x\}M\\cong\\mathbb\{R\}in the tangent space toMMyields

ψ∗​\(w\)=d​ψ​\(w\)=w\.\\psi\_\{\*\}\(w\)=d\\psi\(w\)=w\\,\.\(5\.1\)
To express the induced action using a steerable NODEΨ\\Psi, we first takeV≅ℝV\\cong\\mathbb\{R\}so that the feature vectors have the correct dimension to be able to represent tangent vectors\. In the steerable NODE context, the equivalent condition to \([5\.1](https://arxiv.org/html/2605.11133#S5.E1)\), that the transformation of the feature vector should be trivial, must be accompanied by a choice of sectionσ:M→G\\sigma:M\\to Gof the principal bundle relative to which the vector component should be constant\. This is due to the fact that, unlike the situation for the push\-forward acting on the tangent space, there is no canonical reference in the vector spaceVVin the steerable NODE setting222This can be understood as the vector spaceV≅ℝV\\cong\\mathbb\{R\}being disconnected from its geometrical origin representing tangent vectors\. Establishing the connection betweenVVandTx​MT\_\{x\}Mrequires a choice of section relating the two\.\.

We consider the section

σ​\(x\)=\(x,f​\(x\)\),x∈M,\\sigma\(x\)=\(x,f\(x\)\)\\,,\\quad x\\in M\\,,\(5\.2\)wheref:ℝ→ℝf:\\mathbb\{R\}\\to\\mathbb\{R\}is a smooth function\. Withϕ=∂x\\phi=\\partial\_\{x\}, we then have

σ∗​ϕΦx​\(t\)=\(1,f′​\(x\+t\)\),\\sigma\_\{\*\}\\phi\_\{\\Phi\_\{x\}\(t\)\}=\(1,f^\{\\prime\}\(x\+t\)\)\\,,\(5\.3\)and, using\(Rh\)∗=id\(R\_\{h\}\)\_\{\*\}=\\mathrm\{id\}for translationsh∈Hh\\in H, we obtain the parallel transport equation

d​hx​\(t\)d​t=−ω​\(σ∗​ϕΦx​\(t\)\)=−\(a​\(x\+t\)\+f′​\(x\+t\)\),hx​\(0\)=0\.\\frac\{dh\_\{x\}\(t\)\}\{dt\}=\-\\omega\(\\sigma\_\{\*\}\\phi\_\{\\Phi\_\{x\}\(t\)\}\)=\-\(a\(x\+t\)\+f^\{\\prime\}\(x\+t\)\)\\,,\\quad h\_\{x\}\(0\)=0\\,\.\(5\.4\)Here, we have used the fact that any connectionω\\omega, corresponding to a decomposition of the tangent space of the total spaceGG, can be expressed asωg=\[a​\(π​\(g\)\)​1\]\\omega\_\{g\}=\[a\(\\pi\(g\)\)\\,\\,1\],g∈Gg\\in G, for some smooth functiona:M→ℝa:M\\to\\mathbb\{R\}\. This decomposition is equivariant under right translations byh∈Hh\\in H, as required, sinceaadepends only on the pointπ​\(g\)\\pi\(g\)inMM\.

The condition that the steerable NODE preserves the feature vector \(relative to the sectionσ\\sigma\) then amounts tohx​\(1\)=0h\_\{x\}\(1\)=0\. From \([5\.4](https://arxiv.org/html/2605.11133#S5.E4)\), we see that this condition is in particular satisfied if

a​\(x\)=−f′​\(x\),∀x∈M\.a\(x\)=\-f^\{\\prime\}\(x\)\\,,\\quad\\forall x\\in M\\,\.\(5\.5\)Furthermore, if the section isGG\-invariant,σ​\(g​x\)=σ​\(x\)\\sigma\(gx\)=\\sigma\(x\)for anyg∈Gg\\in G, we havea​\(x\)=−f′​\(x\)=0a\(x\)=\-f^\{\\prime\}\(x\)=0for anyx∈Mx\\in M, which corresponds to an invariant connection, as expected\.

### 5\.2Continuous normalizing flows onG/HG/H

In this section, we demonstrate that our framework captures continuous normalizing flows on Lie groups\. In particular, we connect to the normalizing flows for lattice gauge theories studied in\[[15](https://arxiv.org/html/2605.11133#bib.bib30)\]\. We demonstrate that our framework provides a generalisation of this to non\-trivial stabiliser subgroupsH⊂GH\\subset G\.

#### 5\.2\.1Continuous normalizing flows on a Lie groupGG

Continuous normalizing flows \(CNFs\) provide a framework for learning probability densities via diffeomorphic transformations generated by NODEs\. Recently, CNFs have been applied to lattice gauge theories, where the data take values in Lie groups and are subject to gauge symmetries\[[15](https://arxiv.org/html/2605.11133#bib.bib30)\]\. In this section, we show that these constructions can be interpreted as a special case of steerable neural ODEs on homogeneous spaces, corresponding to the situation where the stabiliser subgroupHHis trivial\.

LetGGbe a compact Lie group equipped with Haar measureμG\\mu\_\{G\}, and letp0:G→ℝ≥0p\_\{0\}:G\\to\\mathbb\{R\}\_\{\\geq 0\}be an initial probability density\. In lattice gauge theories, one usually takesG=S​U​\(N\)G=SU\(N\), but there is no need to make that restriction here\. A CNF onGGis defined by a vector fieldϕ:G→T​G\\phi:G\\to TGand the associated Cauchy problem

g˙​\(t\)=ϕ​\(g​\(t\)\),g​\(0\)=g0∈G,\\dot\{g\}\(t\)=\\phi\(g\(t\)\)\\,,\\qquad g\(0\)=g\_\{0\}\\in G\\,,\(5\.6\)whereg​\(t\)∈Gg\(t\)\\in Gis the solution curve of the ODE on the groupGG, generated byϕ\\phi\. In the notation of section[3](https://arxiv.org/html/2605.11133#S3)we haveg​\(t\)=Φg0​\(t\)g\(t\)=\\Phi\_\{g\_\{0\}\}\(t\)\.

The solution defines a flowΦ:ℝ×G→G\\Phi:\\mathbb\{R\}\\times G\\to G, and the time\-evolved densityptp\_\{t\}satisfies the continuity equation

dd​t​log⁡pt​\(g​\(t\)\)=−divG⁡ϕ​\(g​\(t\)\),\\frac\{d\}\{dt\}\\log p\_\{t\}\(g\(t\)\)=\-\\operatorname\{div\}\_\{G\}\\phi\(g\(t\)\),\(5\.7\)wheredivG\\operatorname\{div\}\_\{G\}denotes the divergence with respect to the Haar measureμG\\mu\_\{G\}, defined by the equationℒϕ​μG=\(divG⁡ϕ\)​μG\\mathcal\{L\}\_\{\\phi\}\\mu\_\{G\}=\(\\operatorname\{div\}\_\{G\}\\phi\)\\mu\_\{G\}\.

In\[[15](https://arxiv.org/html/2605.11133#bib.bib30)\], the vector fieldϕ\\phiis constructed to be equivariant under gauge transformations, ensuring that the induced flow preserves gauge symmetry of the target distribution\. Importantly, the flow acts only on the group\-valued variables themselves, without transporting additional feature fields\. Moreover, to see the precise connection to\[[15](https://arxiv.org/html/2605.11133#bib.bib30)\], one should choose

ϕ​\(g​\(t\)\)=Zθ​\(g\)​g​\(t\),\\phi\(g\(t\)\)=Z\_\{\\theta\}\(g\)g\(t\)\\,,\(5\.8\)whereZθ:G→𝔤Z\_\{\\theta\}:G\\to\\mathfrak\{g\}is a neural network with parametersθ\\theta\. If we setG=S​U​\(N\)G=SU\(N\)and fix some basis\{Ta\}\\\{T\_\{a\}\\\}of the Lie algebra𝔰​𝔲​\(N\)\\mathfrak\{su\}\(N\)we can write the divergence in[5\.7](https://arxiv.org/html/2605.11133#S5.E7)explicitly as∑a∂aZθa​\(g\)\\sum\_\{a\}\\partial\_\{a\}Z^\{a\}\_\{\\theta\}\(g\)\. Moreover, in a lattice gauge theory setting, the group elementggwould be replaced by a collection of elements\{ge\}e∈E\\\{g\_\{e\}\\\}\_\{e\\in E\}, wheree∈Ee\\in Elabels the edges of the lattice\.

#### 5\.2\.2Connection with NODEs onG/HG/H

This setting fits naturally into our framework by viewing the Lie groupGGas a homogeneous spaceM=G/HM=G/HwithH=\{e\}H=\\\{e\\\}\. In this case, the principal bundleG→G/HG\\to G/His trivial, and the base manifold is canonically identified withGGitself\. A manifold neural ODE onMMin the sense of Definition[3\.1](https://arxiv.org/html/2605.11133#S3.Thmtheorem1)is therefore exactly an ODE of the form \([5\.6](https://arxiv.org/html/2605.11133#S5.E6)\)\.

SinceHHis trivial, there is no non\-trivial local symmetry acting on the fibres, and the associated bundle with fibreVVreduces to the product bundle

G×V→G\.G\\times V\\to G\\,\.\(5\.9\)Feature fields are simply mapsf:G→Vf:G\\to V, and there is no need to introduce a representationρ\\rhoor a connection to ensure consistency under local gauge transformations\.

From this perspective, the CNFs of\[[15](https://arxiv.org/html/2605.11133#bib.bib30)\]correspond precisely to manifold NODEs onGG, without steering of additional features\. The equivariance constraints imposed onϕ\\phiin that work ensure that the flowΦ\\Phirespects the relevant global symmetries\.

In the language of principal bundles, the vector fields used in\[[15](https://arxiv.org/html/2605.11133#bib.bib30)\]can be interpreted as left\- or right\-invariant vector fields onGG, or more generally, as equivariant mapsϕ:G→T​G\\phi:G\\to TGsatisfying

ϕ​\(g​g′\)=\(Lg\)∗​ϕ​\(g′\)∀g,g′∈G\.\\phi\(gg^\{\\prime\}\)=\(L\_\{g\}\)\_\{\*\}\\phi\(g^\{\\prime\}\)\\qquad\\forall g,g^\{\\prime\}\\in G\\,\.\(5\.10\)This is the condition required for the diffeomorphism induced byΦ\\Phito beGG\-equivariant in the sense of[Definition˜4\.1](https://arxiv.org/html/2605.11133#S4.Thmtheorem1)\. Since the bundle is trivial in this case, there is no distinction between vertical and horizontal directions, and no additional geometric structure is required to define the flow\.

While the lattice gauge theory literature refers to this as “gauge equivariance” \(stemming from the redundancy in the physical description\), in our geometric framework on the homogeneous spaceM=GM=G, this manifests as a global symmetry under the group action ofGGon itself\. This is precisely the condition required for the induced diffeomorphismΦ\\Phito beGG\-equivariant in the sense of[Definition˜4\.1](https://arxiv.org/html/2605.11133#S4.Thmtheorem1)\.

Although no explicit feature transport is introduced in standard CNFs, probability densities can be understood as geometric objects induced by the diffeomorphismΦ\\Phi\. Equation \([5\.7](https://arxiv.org/html/2605.11133#S5.E7)\) arises from the Jacobian determinant of the flow and corresponds to the push\-forward of the density underΦ\\Phi\. In this sense, CNFs already make implicit use of induced actions, but restricted to scalar\-valued densities rather than general feature fields\.

Our framework becomes richer whenHHis non\-trivial\. In this case, the base manifold is a quotient spaceM=G/HM=G/H, and features transform in a representationρ:H→GL​\(V\)\\rho:H\\to\\mathrm\{GL\}\(V\)\. Feature fields are sections of the associated bundleG×ρVG\\times\_\{\\rho\}V, and their evolution along a NODE trajectory is governed by parallel transport with respect to a connectionω\\omega\. From a physics perspective, this corresponds to systems with internal degrees of freedom that transform under a local symmetry group\. For instance, it could correspond to some gauge covariant matter fields that transform with respect toHH\. Another concrete interpretation might be a molecular dipole moment attached to a rotating rigid body, extending existing flow\-based protein generation models\[[34](https://arxiv.org/html/2605.11133#bib.bib75),[5](https://arxiv.org/html/2605.11133#bib.bib9),[32](https://arxiv.org/html/2605.11133#bib.bib31)\]to incorporate non\-scalar features with consistent transformation properties\.

In such settings, the base flow onG/HG/Hdescribes the evolution of the physical configuration, while parallel transport in the associated bundle encodes how internal features are rotated or transformed consistently along the trajectory\.

We conclude that CNFs on Lie groups correspond to the simplest case of steerable NODEs, whereHHis trivial, and no internal feature transport is required\. Our construction therefore provides a geometric generalisation of CNFs to homogeneous spaces with non\-trivial stabilisers\.

### 5\.3Example: Steerable NODEs onS2S^\{2\}

We now illustrate our framework in the special case of the homogeneous space

M=SO​\(3\)/SO​\(2\)≃S2\.M=\\mathrm\{SO\}\(3\)/\\mathrm\{SO\}\(2\)\\simeq S^\{2\}\\,\.\(5\.11\)This example highlights the role of invariant connections and clarifies the interpretation of steerable NODEs with non\-trivial stabiliser subgroup\. See also Example[4\.12](https://arxiv.org/html/2605.11133#S4.Thmtheorem12)for more details on this setting\.

The groupG=SO​\(3\)G=\\mathrm\{SO\}\(3\)acts transitively on the unit sphereS2S^\{2\}by rotations\. Fix a reference pointp0∈S2p\_\{0\}\\in S^\{2\}\. The stabiliser subgroup ofp0p\_\{0\}isH=SO​\(2\)H=\\mathrm\{SO\}\(2\)\. For example,p0p\_\{0\}can be chosen so thatS​O​\(2\)SO\(2\)acts as rotations about thexx\-axis\. This identifiesS2S^\{2\}with the homogeneous spaceG/HG/H\.

The principal bundle

π:SO​\(3\)→S2\\pi:\\mathrm\{SO\}\(3\)\\to S^\{2\}\(5\.12\)maps a rotation matrixRRto its action onp0p\_\{0\}, i\.e\.,π​\(R\)=R​p0\\pi\(R\)=Rp\_\{0\}\. Geometrically, a point in the total space corresponds to a choice of local frame whosexx\-axis is aligned with the point on the sphere\.

LetVVbe a vector space carrying a representation

ρ:SO​\(2\)→GL​\(V\)\.\\rho:\\mathrm\{SO\}\(2\)\\to\\mathrm\{GL\}\(V\)\\,\.\(5\.13\)Feature fields onS2S^\{2\}transforming underρ\\rhoare sections of the associated bundle

SO​\(3\)×ρV→S2\.\\mathrm\{SO\}\(3\)\\times\_\{\\rho\}V\\to S^\{2\}\\,\.\(5\.14\)
To transport features along trajectories onS2S^\{2\}, we require a connection on the principal bundleSO​\(3\)→S2\\mathrm\{SO\}\(3\)\\to S^\{2\}\. SinceS2S^\{2\}is a homogeneous space, it is natural to restrict attention toGG\-invariant connections\. Recall from[Section˜4\.3](https://arxiv.org/html/2605.11133#S4.SS3)that by Wang’s theorem,GG\-invariant connections are in one\-to\-one correspondence with linear maps

Λ:𝔰​𝔬​\(3\)→𝔰​𝔬​\(2\)\\Lambda:\\mathfrak\{so\}\(3\)\\to\\mathfrak\{so\}\(2\)\(5\.15\)satisfying suitable equivariance and normalisation conditions\. In this case, there exists a*unique*GG\-invariant principal connection, corresponding to the reductive decomposition

𝔰​𝔬​\(3\)=𝔰​𝔬​\(2\)⊕𝔪,\\mathfrak\{so\}\(3\)=\\mathfrak\{so\}\(2\)\\oplus\\mathfrak\{m\},\(5\.16\)where𝔪\\mathfrak\{m\}is spanned by generators rotatingp0p\_\{0\}into the tangent plane ofS2S^\{2\}\.

To proceed, letϕ:S2→T​S2\\phi:S^\{2\}\\to TS^\{2\}be a \(possibly equivariant\) vector field defining a NODE on the sphere:

p˙​\(t\)=ϕ​\(p​\(t\)\),p​\(0\)=p0\.\\dot\{p\}\(t\)=\\phi\(p\(t\)\),\\qquad p\(0\)=p\_\{0\}\\,\.\(5\.17\)Given a horizontal liftp~​\(t\)∈SO​\(3\)\\tilde\{p\}\(t\)\\in\\mathrm\{SO\}\(3\)satisfying

π​\(p~​\(t\)\)=p​\(t\),ω​\(p~˙​\(t\)\)=0,\\pi\(\\tilde\{p\}\(t\)\)=p\(t\)\\,,\\qquad\\omega\(\\dot\{\\tilde\{p\}\}\(t\)\)=0,\(5\.18\)the evolution of a feature vectorv∈Vv\\in Vis given by parallel transport,

dd​t​v​\(t\)=0,f​\(p​\(t\)\)=\[p~​\(t\),v​\(t\)\]\.\\frac\{d\}\{dt\}v\(t\)=0,\\qquad f\(p\(t\)\)=\[\\tilde\{p\}\(t\),v\(t\)\]\\,\.\(5\.19\)In local coordinates, this yields a transformation

f​\(p​\(t\)\)=ρ​\(h​\(t\)\)​f​\(p​\(0\)\),f\(p\(t\)\)=\\rho\(h\(t\)\)f\(p\(0\)\)\\,,\(5\.20\)whereh​\(t\)∈SO​\(2\)h\(t\)\\in\\mathrm\{SO\}\(2\)is determined by the connection and the base\-space trajectory\.

IfV=ℝV=\\mathbb\{R\}andρ\\rhois the trivial representation, the associated bundle reduces to a trivial line bundle and parallel transport is trivial\. In this case, the steerable NODE reduces to a standard manifold NODE onS2S^\{2\}, recovering the setting of continuous normalizing flows on homogeneous spaces without internal structure\. For non\-trivialρ\\rho, even though the connection is uniquely determined by invariance, the model captures genuinely new behaviour that cannot be expressed using base\-space flows alone\.

## 6Conclusions

In this paper, we consider a geometric extension of neural ODEs on homogeneous spacesM=G/HM=G/H, which incorporates features transforming in a representationρ\\rhoof the stabiliser subgroupHH\. In this setting, a manifold NODE transports points along integral curves of a learnable vector field\. Feature fields, on the other hand, are interpreted as sections of the associated bundleG×ρVG\\times\_\{\\rho\}V\. We show that the natural mechanism for transporting such features along NODE trajectories is parallel transport with respect to a principal connection on the bundleG→G/HG\\to G/H\. This leads to[Definition˜3\.7](https://arxiv.org/html/2605.11133#S3.Thmtheorem7)of a steerable NODE as a coupled geometric system consisting of a base flow onG/HG/Htogether with a parallel transport equation steering the feature field through a learnable principal connection\.

The resulting construction provides a geometrically consistent notion of feature transport that generalises both the push\-forward of tangent vectors and the density evolution appearing in continuous normalizing flows\. A central structural result is that equivariance of the steerable NODE with respect to the globalGG\-action is guaranteed if both the base vector field and the principal connection areGG\-invariant \([Theorem˜4\.6](https://arxiv.org/html/2605.11133#S4.Thmtheorem6)\)\. Using Wang’s theorem, we obtain an explicit classification ofGG\-invariant principal connections onG→G/HG\\to G/Hwhich, together with our previous results for equivariant manifold NODEs, yields a concrete parametrisation of all admissible equivariant steerable NODE models\.

Conceptually, our analysis shows that extending NODEs to general feature fields on homogeneous spaces necessarily introduces principal connections as learnable geometric structures\. This reveals a fundamental distinction between scalar transport \(e\.g\., densities\) and general vector\-valued feature transport: while scalar quantities may be transported canonically, non\-scalar features require additional geometric data to define their evolution in a manner consistent with both local and global symmetries\.

The framework developed here opens several directions for further research\. In this work, we characterise invariant connections in order to ensure global equivariance of the steerable NODE\. A natural next step is to investigate the learning dynamics of such connections in practice\. In particular, it would be interesting to explore explicit parametrisations of invariant connections via Wang’s theorem for specific Lie groups and to understand the inductive bias induced by different classes of connections\. Allowing for non\-invariant connections may also be relevant in settings where symmetry is only approximate\.

We emphasise that, although the global construction is well\-defined, its local realisation depends on a choice of section of the principal bundle\. In this sense, the formulation is gauge\-dependent at the level of coordinates\. A systematic investigation of gauge\-equivariance, transition functions between local trivialisations, and numerically stable implementations across charts would further clarify the interplay between geometric structure and practical computation\.

For equivariant manifold NODEs, universality results have been established under appropriate assumptions\. An important open question is whether steerable NODEs with learnable connections retain comparable approximation properties for spaces of equivariant bundle maps or flows on associated bundles\. While homogeneous spaces provide a clean and structured setting, many applications involve manifolds with local but not global homogeneous symmetry\. Extending the present framework to more general principal bundles that do not arise from global quotients would significantly broaden its scope\.

A particularly intriguing direction is to extend the present framework beyond smooth homogeneous spaces to discrete geometric settings such as combinatorial complexes\. In recent years, graph and sheaf neural networks have demonstrated that feature fields on graphs and cell complexes can be modelled as sections of vector bundles\[[6](https://arxiv.org/html/2605.11133#bib.bib79)\]and cellular sheaves\[[17](https://arxiv.org/html/2605.11133#bib.bib77),[3](https://arxiv.org/html/2605.11133#bib.bib78)\], where restriction maps encode local consistency constraints analogous to parallel transport\. From this perspective, cellular sheaves may be viewed as discrete counterparts of vector bundles, and their restriction maps as a combinatorial analogue of connection data\. It would therefore be natural to investigate whether a theory of steerable NODEs can be formulated in this discrete setting, combining continuous\-time dynamics on cochains with sheaf\-consistent feature evolution\. Such a development could provide a unified geometric framework bridging continuous gauge\-theoretic models and discrete topological deep learning, and may open new avenues for modelling dynamical processes on graphs and higher\-order networks\.

In the more applied direction, we also showed that continuous normalizing flows on Lie groups arise as a special case of our construction when the stabiliser subgroup is trivial\. This suggests the possibility of combining density evolution with feature transport, leading to gauge\-equivariant generative models on homogeneous spaces\. Such models may be of particular interest in physics\-informed machine learning and geometric generative modelling\.

Finally, it would be interesting to explore potential applications within geometric deep learning, such as data defined on spheres and rotation groups\. This occurs for instance in molecular modelling \(e\.g\.,\[[34](https://arxiv.org/html/2605.11133#bib.bib75),[5](https://arxiv.org/html/2605.11133#bib.bib9)\]\) or climate science \(e\.g\.,\[[26](https://arxiv.org/html/2605.11133#bib.bib50)\]\), as well as gauge\-equivariant models \(e\.g\.,\[[13](https://arxiv.org/html/2605.11133#bib.bib29),[15](https://arxiv.org/html/2605.11133#bib.bib30)\]\) in lattice gauge theory\. In these contexts, steerable NODEs offer a principled mechanism for coupling state evolution and feature transformation in continuous time while respecting symmetry constraints\.

We hope to report on progress on these interesting directions in the future\.

## Acknowledgements

The work of E\.A\., D\.P\., and F\.O\. was supported by the Wallenberg AI, Autonomous Systems and Software Program \(WASP\) funded by the Knut and Alice Wallenberg Foundation\. The work of F\.O\. was partially funded by the Swedish Research Council under grant agreement no\. 2025\-05053\.

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## Appendix AProof of[Definition˜3\.3](https://arxiv.org/html/2605.11133#S3.Thmtheorem3)

In[section˜3\.2](https://arxiv.org/html/2605.11133#S3.SS2)we introduced two definitions of a feature field\. Here, we prove the lemma stating that these two definitions are equivalent\.\\MackeyLemma\*

###### Proof\.

We start by showing that a local feature field defines a section of the associated bundle\. Letσ:G/H→G\\sigma:G/H\\to Gbe a \(local\) section of the principal bundleG​→𝜋​G/HG\\overset\{\\pi\}\{\\to\}G/H, and letf=k∘σf=k\\circ\\sigma, wherek:G→Vk:G\\to Vis a Mackey function \([definition˜2\.11](https://arxiv.org/html/2605.11133#S2.Thmtheorem11)\)\. Defines:M→G×ρVs:M\\to G\\times\_\{\\rho\}Vby

s​\(p\)=\[σ​\(p\),f​\(p\)\]=\[σ​\(p\),k​\(σ​\(p\)\)\],s\(p\)=\\left\[\\sigma\(p\),f\(p\)\\right\]=\\left\[\\sigma\(p\),k\(\\sigma\(p\)\)\\right\]\\,,\(A\.1\)forp∈Mp\\in M\. The Mackey condition ensures thatssis independent of the choice of principal section: ifσ′\\sigma^\{\\prime\}is another section withσ′​\(p\)=σ​\(p\)​h​\(p\)\\sigma^\{\\prime\}\(p\)=\\sigma\(p\)h\(p\)for someh​\(p\)∈Hh\(p\)\\in H, then

\[σ′​\(p\),k​\(σ′​\(p\)\)\]\\displaystyle\\left\[\\sigma^\{\\prime\}\(p\),k\(\\sigma^\{\\prime\}\(p\)\)\\right\]=\\displaystyle=\[σ​\(p\)​h​\(p\),k​\(σ​\(p\)​h​\(p\)\)\]\\displaystyle\\left\[\\sigma\(p\)h\(p\),k\(\\sigma\(p\)h\(p\)\)\\right\]=\\displaystyle=\[σ​\(p\)​h​\(p\),ρ​\(h​\(p\)−1\)​k​\(σ​\(p\)\)\]=\[σ​\(p\),k​\(σ​\(p\)\)\]\.\\displaystyle\\left\[\\sigma\(p\)h\(p\),\\rho\(h\(p\)^\{\-1\}\)k\(\\sigma\(p\)\)\\right\]=\\left\[\\sigma\(p\),k\(\\sigma\(p\)\)\\right\]\\,\.Thusssis well\-defined, smooth, and independent of the choice of local sectionσ\\sigma\.

To show the converse, i\.e\., that every sectionssof the associated bundle defines a Mackey function, let\{Ui\}\\\{U\_\{i\}\\\}be an open cover ofM=G/HM=G/Hand denote byσi:Ui→G\\sigma\_\{i\}:U\_\{i\}\\to Gthe sections defined locally onUiU\_\{i\}, related by the bundle cocyclesci​j:Ui∩Uj→Hc\_\{ij\}:U\_\{i\}\\cap U\_\{j\}\\to Hsatisfyingσi​\(p\)=σj​\(p\)​ci​j​\(p\)\\sigma\_\{i\}\(p\)=\\sigma\_\{j\}\(p\)c\_\{ij\}\(p\)for allp∈Ui∩Ujp\\in U\_\{i\}\\cap U\_\{j\}\. For allg∈π−1​\(Ui\)g\\in\\pi^\{\-1\}\(U\_\{i\}\), there is a unique elementhi​\(g\)∈Hh\_\{i\}\(g\)\\in Hsuch thatg=σi​\(π​\(g\)\)​hi​\(g\)g=\\sigma\_\{i\}\(\\pi\(g\)\)h\_\{i\}\(g\)\.

Lets:M→G×ρVs:M\\to G\\times\_\{\\rho\}Vbe a smooth section\. Then for eachp∈Uip\\in U\_\{i\}

s​\(p\)=\[σi​\(p\),vi​\(p\)\],s\(p\)=\[\\sigma\_\{i\}\(p\),v\_\{i\}\(p\)\]\\,,\(A\.2\)wherevi​\(p\)∈Vv\_\{i\}\(p\)\\in V\. Now, definek:G→Vk:G\\to Vby

k​\(g\):=ρ​\(hi​\(g\)−1\)​vi​\(π​\(g\)\)k\(g\):=\\rho\(h\_\{i\}\(g\)^\{\-1\}\)v\_\{i\}\(\\pi\(g\)\)\(A\.3\)for eachg∈π−1​\(Ui\)g\\in\\pi^\{\-1\}\(U\_\{i\}\), wherehi​\(g\)h\_\{i\}\(g\)is the unique element satisfyingg=σi​\(π​\(g\)\)​hi​\(g\)g=\\sigma\_\{i\}\(\\pi\(g\)\)h\_\{i\}\(g\)\. This is, indeed, a Mackey function, since the right action of an elementh∈Hh\\in Honggcorresponds to a new element in the same fibre:

k​\(g​h\)=ρ​\(h−1\)​k​\(g\)\.k\(gh\)=\\rho\(h^\{\-1\}\)k\(g\)\\,\.It is also well\-defined on overlaps: Ifp∈Ui∩Ujp\\in U\_\{i\}\\cap U\_\{j\},

s​\(p\)=\[σi​\(p\),vi​\(p\)\]=\[σj​\(p\),vj​\(p\)\],s\(p\)=\[\\sigma\_\{i\}\(p\),v\_\{i\}\(p\)\]=\[\\sigma\_\{j\}\(p\),v\_\{j\}\(p\)\]\\,,\(A\.4\)from which a simple calculation shows thatvj​\(p\)=ρ​\(ci​j​\(p\)−1\)​vi​\(p\)v\_\{j\}\(p\)=\\rho\(c\_\{ij\}\(p\)^\{\-1\}\)v\_\{i\}\(p\)\. Furthermore, ifg=σi​\(p\)​hi​\(g\)g=\\sigma\_\{i\}\(p\)h\_\{i\}\(g\), there is also a unique elementhj​\(g\)∈Hh\_\{j\}\(g\)\\in Hsatisfyingg=σj​\(p\)​hj​\(g\)g=\\sigma\_\{j\}\(p\)h\_\{j\}\(g\)\. A similar calculation yieldshi​\(g\)=ci​j​\(p\)​hj​\(g\)h\_\{i\}\(g\)=c\_\{ij\}\(p\)h\_\{j\}\(g\)\. It follows that

ρ​\(hj​\(g\)−1\)​vj​\(p\)=ρ​\(hi​\(g\)−1\)​ρ​\(ci​j​\(p\)\)​vj​\(p\)=ρ​\(hi​\(g\)−1\)​vi​\(p\),\\rho\(h\_\{j\}\(g\)^\{\-1\}\)v\_\{j\}\(p\)=\\rho\(h\_\{i\}\(g\)^\{\-1\}\)\\rho\(c\_\{ij\}\(p\)\)v\_\{j\}\(p\)=\\rho\(h\_\{i\}\(g\)^\{\-1\}\)v\_\{i\}\(p\)\\,,\(A\.5\)showing thatkkis well\-defined on overlaps\.

With the definition ofkk, \([A\.2](https://arxiv.org/html/2605.11133#A1.E2)\) can be written as

s​\(p\)=\[σi​\(p\),k​\(σi​\(p\)\)\]\.s\(p\)=\[\\sigma\_\{i\}\(p\),k\(\\sigma\_\{i\}\(p\)\)\]\\,\.\(A\.6\)Ifp∈Ui∩Ujp\\in U\_\{i\}\\cap U\_\{j\}, this can be written as

s​\(p\)=\[σj​\(p\)​\(ci​j​\(p\)\)−1,k​\(σi​\(p\)\)\]=\[σj​\(p\),ρ​\(\(ci​j​\(p\)\)−1\)​k​\(σi​\(p\)\)\]\.s\(p\)=\[\\sigma\_\{j\}\(p\)\(c\_\{ij\}\(p\)\)^\{\-1\},k\(\\sigma\_\{i\}\(p\)\)\]=\[\\sigma\_\{j\}\(p\),\\rho\(\(c\_\{ij\}\(p\)\)^\{\-1\}\)k\(\\sigma\_\{i\}\(p\)\)\]\\,\.\(A\.7\)Becausekkis a Mackey function andσi​\(p\)​ci​j​\(p\)=σj​\(p\)\\sigma\_\{i\}\(p\)c\_\{ij\}\(p\)=\\sigma\_\{j\}\(p\), it follows that

s​\(p\)=\[σj​\(p\),k​\(σj​\(p\)\)\],s\(p\)=\[\\sigma\_\{j\}\(p\),k\(\\sigma\_\{j\}\(p\)\)\]\\,,\(A\.8\)showing that the definition ofkkis consistent with the transformation behaviour of the section\.

The Mackey functionkkis smooth on eachπ−1​\(Ui\)\\pi^\{\-1\}\(U\_\{i\}\)becausehih\_\{i\},σi\\sigma\_\{i\}andviv\_\{i\}are smooth\. Since they also agree on overlaps,kkis a globally smooth function\.

Lastly, note that if two Mackey functionskkandk′k^\{\\prime\}are defined for the same section, thenk​\(σi​\(p\)\)=k′​\(σi​\(p\)\)k\(\\sigma\_\{i\}\(p\)\)=k^\{\\prime\}\(\\sigma\_\{i\}\(p\)\)for allp∈Uip\\in U\_\{i\}\. Since every elementg∈π−1​\(Ui\)g\\in\\pi^\{\-1\}\(U\_\{i\}\)satisfiesg=σi​\(π​\(g\)\)​hi​\(g\)g=\\sigma\_\{i\}\(\\pi\(g\)\)h\_\{i\}\(g\)for some uniquehi​\(g\)∈Hh\_\{i\}\(g\)\\in H, the Mackey condition yields

k​\(g\)=ρ​\(hi​\(g\)−1\)​k​\(σi​\(p\)\)=ρ​\(hi​\(g\)−1\)​k′​\(σi​\(p\)\)=k′​\(g\)\.k\(g\)=\\rho\(h\_\{i\}\(g\)^\{\-1\}\)k\(\\sigma\_\{i\}\(p\)\)=\\rho\(h\_\{i\}\(g\)^\{\-1\}\)k^\{\\prime\}\(\\sigma\_\{i\}\(p\)\)=k^\{\\prime\}\(g\)\.\(A\.9\)Thus, the Mackey function constructed from the sectionssis unique\. ∎

## Appendix BProof of Wang’s theorem forG→G/HG\\to G/H

In this appendix, we provide a self\-contained proof of[section˜4\.3](https://arxiv.org/html/2605.11133#S4.SS3)\. The theorem is a special case of the more general result due to Wang\[[33](https://arxiv.org/html/2605.11133#bib.bib71)\], in the situation when we considerGGas a principalHH\-bundle over the homogeneous spaceG/HG/H\.

###### Proof\.

First, given a linear mapΛ:𝔤→𝔥\\Lambda:\\mathfrak\{g\}\\to\\mathfrak\{h\}satisfying conditions \(i\) and \(ii\), we define

ωe:=Λ\\omega\_\{e\}:=\\Lambda\(B\.1\)andω∈Ω1​\(G,𝔥\)\\omega\\in\\Omega^\{1\}\(G,\\mathfrak\{h\}\)by

ωg:=\(Lg−1\)∗​ωe=\(Lg−1\)∗​Λ\.\\omega\_\{g\}:=\\left\(L\_\{g^\{\-1\}\}\\right\)^\{\*\}\\omega\_\{e\}=\\left\(L\_\{g^\{\-1\}\}\\right\)^\{\*\}\\Lambda\.\(B\.2\)For anya,g∈Ga,g\\in GandX∈Tg​GX\\in T\_\{g\}G, we have

\(\(La\)∗​ω\)g​\(X\)\\displaystyle\\left\(\(L\_\{a\}\)^\{\*\}\\omega\\right\)\_\{g\}\(X\)=\\displaystyle=ωa​g​\(\(La\)∗​X\)=ωe​\(\(L\(a​g\)−1\)∗​\(La\)∗​X\)\\displaystyle\\omega\_\{ag\}\\left\(\(L\_\{a\}\)\_\{\*\}X\\right\)=\\omega\_\{e\}\\left\(\\left\(L\_\{\(ag\)^\{\-1\}\}\\right\)\_\{\*\}\(L\_\{a\}\)\_\{\*\}X\\right\)=\\displaystyle=ωe​\(\(Lg−1\)∗​X\)=ωg​\(X\)\.\\displaystyle\\omega\_\{e\}\\left\(\\left\(L\_\{g^\{\-1\}\}\\right\)\_\{\*\}X\\right\)=\\omega\_\{g\}\(X\)\.which shows that\(La\)∗​ω=ω\(L\_\{a\}\)^\{\*\}\\omega=\\omegafor alla∈Ga\\in G, i\.e\.,ω\\omegaisGG\-invariant\. GivenA∈𝔥A\\in\\mathfrak\{h\}andu∈Gu\\in G, the fundamental vector fieldA\#​\(u\)A^\{\\\#\}\(u\)is

A\#​\(u\)=dd​t​u​exp⁡\(t​A\)\|t=0=\(Lu\)∗​A\.A^\{\\\#\}\(u\)=\\left\.\\frac\{d\}\{dt\}u\\,\\exp\(tA\)\\right\|\_\{t=0\}=\(L\_\{u\}\)\_\{\*\}A\.\(B\.3\)This implies

ωu​\(A\#​\(u\)\)=ωu​\(\(Lu\)∗​A\)=ωe​\(A\)=Λ​\(A\)=A,\\omega\_\{u\}\(A^\{\\\#\}\(u\)\)=\\omega\_\{u\}\(\(L\_\{u\}\)\_\{\*\}A\)=\\omega\_\{e\}\(A\)=\\Lambda\(A\)=A,\(B\.4\)with the last equality coming from condition \(ii\), which shows thatω\\omegasatisfies condition \(i\) in[definition˜2\.6](https://arxiv.org/html/2605.11133#S2.Thmtheorem6)\. Furthermore, withg∈Gg\\in G,h∈Hh\\in H, andX∈Tg​GX\\in T\_\{g\}G, we have

\(\(Rh\)∗​ω\)g​\(X\)\\displaystyle\\left\(\\left\(R\_\{h\}\\right\)^\{\*\}\\omega\\right\)\_\{g\}\(X\)=\\displaystyle=ωg​h​\(\(Rh\)∗​X\)=ωe​\(\(L\(g​h\)−1\)∗​\(Rh\)∗​X\)\\displaystyle\\omega\_\{gh\}\\left\(\(R\_\{h\}\)\_\{\*\}X\\right\)=\\omega\_\{e\}\\left\(\\left\(L\_\{\(gh\)^\{\-1\}\}\\right\)\_\{\*\}\(R\_\{h\}\)\_\{\*\}X\\right\)=\\displaystyle=Λ​\(\(Lh−1\)∗​\(Lg−1\)∗​\(Rh\)∗​X\)=Λ​\(\(Lh−1\)∗​\(Rh\)∗​\(Lg−1\)∗​X\)\\displaystyle\\Lambda\(\\left\(L\_\{h^\{\-1\}\}\\right\)\_\{\*\}\\left\(L\_\{g^\{\-1\}\}\\right\)\_\{\*\}\(R\_\{h\}\)\_\{\*\}X\)=\\Lambda\(\\left\(L\_\{h^\{\-1\}\}\\right\)\_\{\*\}\(R\_\{h\}\)\_\{\*\}\\left\(L\_\{g^\{\-1\}\}\\right\)\_\{\*\}X\)=\\displaystyle=Λ​\(Adh−1∘\(Lg−1\)∗​X\)=Adh−1∘Λ​\(\(Lg−1\)∗​X\)\\displaystyle\\Lambda\(\\mathrm\{Ad\}\_\{h^\{\-1\}\}\\circ\\left\(L\_\{g^\{\-1\}\}\\right\)\_\{\*\}X\)=\\mathrm\{Ad\}\_\{h^\{\-1\}\}\\circ\\Lambda\(\\left\(L\_\{g^\{\-1\}\}\\right\)\_\{\*\}X\)=\\displaystyle=Adh−1∘ωg​\(X\)\.\\displaystyle\\mathrm\{Ad\}\_\{h^\{\-1\}\}\\circ\\omega\_\{g\}\(X\)\.where we have used condition \(i\)\. Consequently,ω\\omegaalso satisfies condition \(ii\) in[definition˜2\.6](https://arxiv.org/html/2605.11133#S2.Thmtheorem6), making it aGG\-invariant Ehresmann connection onGG\.

Conversely, given aGG\-invariant Ehresmann connectionω∈Ω1​\(G,𝔥\)\\omega\\in\\Omega^\{1\}\(G,\\mathfrak\{h\}\)we define a mapΛ:𝔤→𝔥\\Lambda:\\mathfrak\{g\}\\to\\mathfrak\{h\}by

Λ:=ωe\.\\Lambda:=\\omega\_\{e\}\.\(B\.5\)Linearity ofΛ\\Lambdafollows from the linearity ofω\\omega\. We note thatωg​\(X\)=ωe​\(\(Lg−1\)∗​X\)\\omega\_\{g\}\(X\)=\\omega\_\{e\}\(\(L\_\{g^\{\-1\}\}\)\_\{\*\}X\)for allg∈Gg\\in GandX∈Tg​GX\\in T\_\{g\}Gsinceω\\omegaisGG\-invariant\. Combined with condition \(ii\) in[definition˜2\.6](https://arxiv.org/html/2605.11133#S2.Thmtheorem6), this implies

Adh∘Λ​\(X\)\\displaystyle\\mathrm\{Ad\}\_\{h\}\\circ\\Lambda\(X\)=\\displaystyle=Adh∘ωe​\(X\)=\(\(Rh−1\)∗​ω\)e​\(X\)=ωh−1​\(\(Rh−1\)∗​X\)\\displaystyle\\mathrm\{Ad\}\_\{h\}\\circ\\omega\_\{e\}\(X\)=\(\(R\_\{h^\{\-1\}\}\)^\{\*\}\\omega\)\_\{e\}\(X\)=\\omega\_\{h^\{\-1\}\}\(\(R\_\{h^\{\-1\}\}\)\_\{\*\}X\)=\\displaystyle=ωe​\(\(Lh\)∗​\(Rh−1\)∗​X\)=ωe​\(Adh​X\)=Λ∘Adh​\(X\),\\displaystyle\\omega\_\{e\}\(\(L\_\{h\}\)\_\{\*\}\(R\_\{h^\{\-1\}\}\)\_\{\*\}X\)=\\omega\_\{e\}\(\\mathrm\{Ad\}\_\{h\}X\)=\\Lambda\\circ\\mathrm\{Ad\}\_\{h\}\(X\)\\,,for anyh∈Hh\\in HandX∈𝔤X\\in\\mathfrak\{g\}, i\.e\.,Λ\\Lambdasatisfies \(i\)\. Furthermore, for anyA∈HA\\in H, we haveA=A\#​\(e\)A=A^\{\\\#\}\(e\)and

Λ​\(A\)=ωe​\(A\)=ωe​\(A\#​\(e\)\)=A\.\\Lambda\(A\)=\\omega\_\{e\}\(A\)=\\omega\_\{e\}\(A^\{\\\#\}\(e\)\)=A\.\(B\.6\)which implies thatΛ\\Lambdaalso satisfies condition \(ii\)\.

Finally, we verify that the correspondence described above is one\-to\-one\. Given the linear mapΛ:𝔤→𝔥\\Lambda:\\mathfrak\{g\}\\to\\mathfrak\{h\}satisfying conditions \(i\) and \(ii\), we constructedω\\omegabyωg:=\(Lg−1\)∗​Λ\\omega\_\{g\}:=\(L\_\{g^\{\-1\}\}\)^\{\*\}\\Lambda\. In particular,Λ=\(Le−1\)∗​Λ=ωe\\Lambda=\(L\_\{e^\{\-1\}\}\)^\{\*\}\\Lambda=\\omega\_\{e\}\. Conversely, ifω\\omegais aGG\-invariant connection and we defineΛ:=ωe\\Lambda:=\\omega\_\{e\}, then byGG\-invariance

ωg​\(X\)=ωe​\(\(Lg−1\)∗​X\)=Λ​\(\(Lg−1\)∗​X\)=\(Lg−1\)∗​Λ​\(X\),\\omega\_\{g\}\(X\)=\\omega\_\{e\}\(\(L\_\{g^\{\-1\}\}\)\_\{\*\}X\)=\\Lambda\(\(L\_\{g^\{\-1\}\}\)\_\{\*\}X\)=\(L\_\{g^\{\-1\}\}\)^\{\*\}\\Lambda\(X\)\\,,\(B\.7\)for allg∈Gg\\in GandX∈Tg​GX\\in T\_\{g\}G\. We conclude that the correspondence of the two constructions is one\-to\-one\. ∎