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Curvature measures and the sub-Riemannian Gauss-Bonnet theorem

Davide Barilari, Eugenio Bellini, Andrea Pinamonti

TL;DR

This work develops a measure-theoretic Gauss–Bonnet framework for surfaces in 3D contact sub-Riemannian manifolds by leveraging a Riemannian approximation. The key insight is that the limit of the Gaussian curvature measures, after rescaling, decomposes into an absolutely continuous part away from characteristic points and a singular part concentrated at isolated characteristic points, captured by a measure $\mu_{-1}=d\alpha$ and Dirac masses weighted by characteristic indices. The analysis introduces the notion of finite order degeneracy for characteristic points, defines invariants $\Lambda^{(k)}(q)$ via horizontal kernel extensions, and provides local normal forms and an operative formula to compute these invariants. As consequences, the authors prove local integrability of $1/|X|$, establish an Euler characteristic formula in terms of the invariants, and illustrate the theory with the horizontal plane in the Heisenberg group, where the limiting measure includes a delta mass at the origin. The results unify and extend previous Riemannian-approximation approaches to sub-Riemannian Gaussian curvature and open pathways for analytic and geometric applications in SR geometry.

Abstract

We adopt a measure-theoretic perspective on the Riemannian approximation scheme proving a sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds. We show that the zero-order term in the limit is a singular measure supported on isolated characteristic points. In particular, this provides a unified interpretation of previous results. Moreover we give natural geometric conditions under which our result holds, namely when the surface admits characteristic points of finite order of degeneracy. This notion, which we introduce, extends the concept of mildly degenerate characteristic points for the Heisenberg group. As a byproduct, we prove that the mean curvature around an isolated characteristic point of finite order of degeneracy is locally integrable. In particular, this positively answers a question for analytic surfaces in every analytic 3D contact manifold.

Curvature measures and the sub-Riemannian Gauss-Bonnet theorem

TL;DR

This work develops a measure-theoretic Gauss–Bonnet framework for surfaces in 3D contact sub-Riemannian manifolds by leveraging a Riemannian approximation. The key insight is that the limit of the Gaussian curvature measures, after rescaling, decomposes into an absolutely continuous part away from characteristic points and a singular part concentrated at isolated characteristic points, captured by a measure and Dirac masses weighted by characteristic indices. The analysis introduces the notion of finite order degeneracy for characteristic points, defines invariants via horizontal kernel extensions, and provides local normal forms and an operative formula to compute these invariants. As consequences, the authors prove local integrability of , establish an Euler characteristic formula in terms of the invariants, and illustrate the theory with the horizontal plane in the Heisenberg group, where the limiting measure includes a delta mass at the origin. The results unify and extend previous Riemannian-approximation approaches to sub-Riemannian Gaussian curvature and open pathways for analytic and geometric applications in SR geometry.

Abstract

We adopt a measure-theoretic perspective on the Riemannian approximation scheme proving a sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds. We show that the zero-order term in the limit is a singular measure supported on isolated characteristic points. In particular, this provides a unified interpretation of previous results. Moreover we give natural geometric conditions under which our result holds, namely when the surface admits characteristic points of finite order of degeneracy. This notion, which we introduce, extends the concept of mildly degenerate characteristic points for the Heisenberg group. As a byproduct, we prove that the mean curvature around an isolated characteristic point of finite order of degeneracy is locally integrable. In particular, this positively answers a question for analytic surfaces in every analytic 3D contact manifold.

Paper Structure

This paper contains 29 sections, 28 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. Characteristic vector field and its singularities
  3. The Riemannian approximation
  4. Characteristic points of finite order
  5. Consequences: integrability
  6. Consequences: Euler characteristic
  7. An example: the horizontal plane in the Heisenberg group
  8. Structure of the paper
  9. Surfaces in 3D contact sub-Riemannian manifolds
  10. Contact sub-Riemannian manifolds
  11. Surfaces in three-dimensional contact sub-Riemannian manifolds
  12. Order of degeneracy of characteristic points. Proof of Proposition \ref{['prop:critical_curve_lambda_intro']}
  13. Well-posedness of the order. Proof of Proposition \ref{['prop:critical_curve_lambda_intro']}
  14. Normal forms and coordinate characterization. Proof of Theorem \ref{['thm:finite_order_intro']}
  15. Proof of Theorem \ref{['thm:finite_order_intro']}
  16. ...and 14 more sections

Key Result

Theorem 1.2

Let $(M,\mathcal{D},g)$ be a contact sub-Riemannian manifold and $S$ be an embedded, compact and orientable surface, with characteristic vector field $X$. Denote $K^\varepsilon$ and $\sigma^\varepsilon$ the Gaussian curvature and area form induced on $S$ by the metric $g^\varepsilon$, respectively. Furthermore the characteristic set is finite $\Sigma(S)=\{q_1,\dots,q_\ell\}\subset S$ and the foll

Theorems & Definitions (70)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Lemma 1.6
  • Proposition 1.7
  • Definition 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 60 more