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Canonical connections on sub-Riemannian manifolds with constant symbol

Erlend Grong

Abstract

As a tool to address the equivalence problem in sub-Riemannian geometry, we introduce a canonical choice of grading and compatible affine connection, available on any sub-Riemannian manifold with constant symbol. We completely compute these structures for contact manifolds of constant symbol, including the cases where the connections of Tanaka-Webster-Tanno are not defined. We also give an original intrinsic grading on sub-Riemannian (2,3,5)-manifolds, and use this to present the first flatness theorem in this setting.

Canonical connections on sub-Riemannian manifolds with constant symbol

Abstract

As a tool to address the equivalence problem in sub-Riemannian geometry, we introduce a canonical choice of grading and compatible affine connection, available on any sub-Riemannian manifold with constant symbol. We completely compute these structures for contact manifolds of constant symbol, including the cases where the connections of Tanaka-Webster-Tanno are not defined. We also give an original intrinsic grading on sub-Riemannian (2,3,5)-manifolds, and use this to present the first flatness theorem in this setting.

Paper Structure

This paper contains 32 sections, 13 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement:
  3. Sub-Riemmannian manifolds
  4. Carnot groups and algebras
  5. Definitions
  6. Isometries
  7. The free nilpotent algebra and induced inner products on Carnot algebras
  8. Nonholonomic tangent bundle and constant symbol
  9. Sub-Riemannian manifolds with constant symbols
  10. Compatible connections on the nonholonomic tangent bundle
  11. Graded sub-Riemannian structures
  12. Equiregular subbundles and grading
  13. Gradings and selectors
  14. Morimoto's Cartan connection as a grading and an affine connection
  15. Cartan connections
  16. ...and 17 more sections

Key Result

Theorem 1.1

The metric $\bar{g}$ and the connection $\nabla$ does not depend on the choice of local basis $X_1,X_2$ and can be defined globally. $(M,E,g)$ is locally isometric to the Cartan nilpotent group if and only if the curvature $R$ of $\nabla$ vanishes and the only non-zero parts of the torsion $T$ of $\

Theorems & Definitions (39)

  • Theorem 1.1: Flatness theorem for $(2,3,5)$-manifolds
  • Definition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Definition 4.1
  • Remark 4.2
  • ...and 29 more