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A sub-Riemannian Gauss-Bonnet theorem for surfaces in contact manifolds

Erlend Grong, Jorge Hidalgo, Sylvie Vega-Molino

Abstract

We obtain a sub-Riemannian version of the classical Gauss-Bonnet theorem. We consider subsurfaces of a three dimensional contact sub-Riemannian manifolds, and using a family of taming Riemannian metric, we obtain a pure sub-Riemannian result in the limit. In particular, we are able to recover topological information of the surface from the geometry around the characteristic set, i.e., the points where the tangent space to the surface and contact structure coincide. We both give a version for surfaces without boundary and surfaces with boundary.

A sub-Riemannian Gauss-Bonnet theorem for surfaces in contact manifolds

Abstract

We obtain a sub-Riemannian version of the classical Gauss-Bonnet theorem. We consider subsurfaces of a three dimensional contact sub-Riemannian manifolds, and using a family of taming Riemannian metric, we obtain a pure sub-Riemannian result in the limit. In particular, we are able to recover topological information of the surface from the geometry around the characteristic set, i.e., the points where the tangent space to the surface and contact structure coincide. We both give a version for surfaces without boundary and surfaces with boundary.
Paper Structure (17 sections, 12 theorems, 107 equations, 1 figure)

This paper contains 17 sections, 12 theorems, 107 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Relation to previous works
  3. Contact manifolds and the horizontal angle parameter
  4. Contact distributions and the Reeb vector field
  5. Surfaces and the horizontal angle parameter
  6. Local description
  7. A sufficient condition for \ref{['A']}
  8. Geometric identities and the variational metric
  9. Variational metrics and the Levi-Civita connection
  10. Expansion of the curvature
  11. Embedded surfaces and variational metric
  12. Proof of the Gauss-Bonnet formula
  13. The primitive of the curvature integral
  14. Proof of Theorem \ref{['th:main']}
  15. Computations of terms of order $-1/2$
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\Sigma\subseteq M$ be a compact $C^2$-surface without boundary such that Assumption A holds. Then

Figures (1)

  • Figure 1: The figure shows four cases where the boundary $\partial \Sigma$ in green intersects the characteristic foliation tangent to $E$ in blue. The three first are compatible with \ref{['Star']}. In the first it is only tangent at an isolated point, and its second derivative does not follow the curve in blue. The second and the third represent respectively a $C^1$- and a $C^2$-singularity. In the fourth picture, the boundary smoothly becomes tangent to $E$ which is not compatible with \ref{['Star']}.

Theorems & Definitions (28)

  • Theorem 1.1: Sub-Riemannian Gauss-Bonnet theorem
  • Corollary 1.2
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4: Characteristic vector fields
  • Example 2.5: The Euclidean unit sphere in the Heisenberg group
  • Proposition 2.6
  • ...and 18 more