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Sub-Riemannian structures and non-transitive Cartan geometries via Lie groupoids

Ivan Beschastnyi, Francesco Cattafi, João Nuno Mestre

Abstract

In this paper we discuss how to associate a suitable non-transitive version of a Cartan connection to sub-Riemannian manifolds of corank 1 (including contact and quasi-contact sub-Riemannian manifolds) with non-necessarily constant sub-Riemannian symbols. In particular, we recast the variation of the sub-Riemannian symbols into a suitable "type" map, which is constant if and only if the symbols are constant. We then consider the (non-transitive) groupoid of sub-Riemannian symmetries and investigate its smoothness, properness, regularity, and other properties in relation with the type map. Last, we describe how to build a "non-transitive" analogue of a Cartan connection on top of such (Lie) groupoid, obtained as the sum of a tautological form with a multiplicative Ehresmann connection. We conclude by illustrating our results on concrete examples in dimension 5.

Sub-Riemannian structures and non-transitive Cartan geometries via Lie groupoids

Abstract

In this paper we discuss how to associate a suitable non-transitive version of a Cartan connection to sub-Riemannian manifolds of corank 1 (including contact and quasi-contact sub-Riemannian manifolds) with non-necessarily constant sub-Riemannian symbols. In particular, we recast the variation of the sub-Riemannian symbols into a suitable "type" map, which is constant if and only if the symbols are constant. We then consider the (non-transitive) groupoid of sub-Riemannian symmetries and investigate its smoothness, properness, regularity, and other properties in relation with the type map. Last, we describe how to build a "non-transitive" analogue of a Cartan connection on top of such (Lie) groupoid, obtained as the sum of a tautological form with a multiplicative Ehresmann connection. We conclude by illustrating our results on concrete examples in dimension 5.

Paper Structure

This paper contains 20 sections, 16 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Cartan geometries and local invariants
  3. Cartan geometries and Cartan groupoids
  4. Cartan geometries and sub-Riemannian structures
  5. Main results and structure of the paper
  6. Further motivations and future applications
  7. Notations and conventions
  8. Acknowledgements
  9. Sub-Riemannian structures
  10. Distributions
  11. Corank-one, contact and quasi-contact structures
  12. Corank-one sub-Riemannian structures and their type map
  13. Groupoids and sub-Riemannian structures
  14. Groupoids and distributions
  15. Groupoids and sub-Riemannian structures
  16. ...and 5 more sections

Key Result

Theorem 2.14

Let $(M,D,g)$ and $(M',D',g')$ be corank-one sub-Riemannian manifolds. Given any $x, y \in M$, the following properties are equivalent:

Theorems & Definitions (69)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6
  • Remark 2.7: dimensional constraints
  • Remark 2.8: coorientable structures
  • Remark 2.9: contact vs symplectic structures
  • Definition 2.10
  • ...and 59 more