Contact integral geometry and the Heisenberg algebra

TL;DR

This paper extends integral geometric concepts to contact manifolds and Heisenberg algebra structures, introducing universal valuations that measure curvature at tangency points and deriving Crofton formulas in contact and symplectic settings.

Contribution

It uncovers a canonical family of valuations for contact manifolds and Heisenberg algebra structures, generalizing Weyl's tube formula and Chern's work to new geometric contexts.

Findings

Established universal valuations for contact manifolds.

Derived Crofton formulas in contact and symplectic geometries.

Constructed symplectic-invariant distributions on Grassmannians.

Abstract

Generalizing Weyl's tube formula and building on Chern's work, Alesker reinterpreted the Lipschitz-Killing curvature integrals as a family of valuations (finitely-additive measures with good analytic properties), attached canonically to any Riemannian manifold, which is universal with respect to isometric embeddings. In this note, we uncover a similar structure for contact manifolds. Namely, we show that a contact manifold admits a canonical family of generalized valuations, which are universal under contact embeddings. Those valuations assign numerical invariants to even-dimensional submanifolds, which in a certain sense measure the curvature at points of tangency to the contact structure. Moreover, these valuations generalize to the class of manifolds equipped with the structure of a Heisenberg algebra on their cotangent bundle. Pursuing the analogy with Euclidean integral geometry, we construct symplectic-invariant distributions on Grassmannians to produce Crofton formulas on the contact sphere. Using closely related distributions, we obtain Crofton formulas also in the linear symplectic space.