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Controllability and diffeomorphism groups on manifolds with boundary

Erlend Grong, Alexander Schmeding

TL;DR

The paper investigates the diffeomorphism groups of compact manifolds with smooth boundary, revealing a fibre-bundle structure from a short exact sequence obtained by restricting diffeomorphisms to the boundary and proving the existence of smooth local sections. It extends Agrachev-style controllability results to manifolds with boundary by incorporating a boundary-control mechanism and showing that Diff(M) and Diff^{∂,id}(M) are generated by the exponential map image. The authors develop a robust Lie-group framework via embedding into the double, construct a boundary thickening with a continuous extension operator, and prove a localisation lemma that permits global generation from local data. They also provide a counter-example illustrating the necessity of boundary-control conditions and discuss sub-Riemannian implications for manifolds with boundary, underscoring the practical relevance for numerical and ML contexts.

Abstract

In this article we consider diffeomorphism groups of manifolds with smooth boundary. We show that the diffeomorphism groups of the manifold and its boundary fit into a short exact sequence which admits local sections. In other words, they form an infinite-dimensional fibre bundle. Manifolds with boundary are of interest in numerical analysis and with a view towards applications in machine learning we establish controllability results for families of vector fields. This generalises older results due to Agrachev and Caponigro in the boundary-less case. Our results show in particular that the diffeomorphism group of a manifold with smooth boundary is generated by the image of the exponential map.

Controllability and diffeomorphism groups on manifolds with boundary

TL;DR

The paper investigates the diffeomorphism groups of compact manifolds with smooth boundary, revealing a fibre-bundle structure from a short exact sequence obtained by restricting diffeomorphisms to the boundary and proving the existence of smooth local sections. It extends Agrachev-style controllability results to manifolds with boundary by incorporating a boundary-control mechanism and showing that Diff(M) and Diff^{∂,id}(M) are generated by the exponential map image. The authors develop a robust Lie-group framework via embedding into the double, construct a boundary thickening with a continuous extension operator, and prove a localisation lemma that permits global generation from local data. They also provide a counter-example illustrating the necessity of boundary-control conditions and discuss sub-Riemannian implications for manifolds with boundary, underscoring the practical relevance for numerical and ML contexts.

Abstract

In this article we consider diffeomorphism groups of manifolds with smooth boundary. We show that the diffeomorphism groups of the manifold and its boundary fit into a short exact sequence which admits local sections. In other words, they form an infinite-dimensional fibre bundle. Manifolds with boundary are of interest in numerical analysis and with a view towards applications in machine learning we establish controllability results for families of vector fields. This generalises older results due to Agrachev and Caponigro in the boundary-less case. Our results show in particular that the diffeomorphism group of a manifold with smooth boundary is generated by the image of the exponential map.
Paper Structure (19 sections, 16 theorems, 73 equations)

This paper contains 19 sections, 16 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Sequences of diffeomorphism groups on manifolds with boundary
  3. Preliminaries on manifolds with smooth boundary
  4. Spaces and manifolds of smooth mappings
  5. Diffeomorphism groups of manifolds with boundary
  6. Diffeomorphism groups as a principle bundle
  7. Construction of a boundary thickening
  8. Construction of a local section
  9. The section as a (local) group morphism
  10. Proof of the controllability results
  11. Why the results are not geometric
  12. Localisation lemma and proof of controllability
  13. A lemma on flows of vector fields
  14. Localisation lemma for an interior point
  15. Localisation lemma for a boundary point
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $M$ be a smooth compact manifold with smooth boundary $\partial M$. Then the short exact sequence eq:sequence admits local sections in the sense that there exists a smooth section of the restriction map $\mathop{\mathrm{Diff}}\nolimits(M) \rightarrow \mathop{\mathrm{Diff}}\nolimits(\partial M)$

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3: $C^\infty (M,N)$ is a canonical manifold
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 24 more