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Contact integral geometry and the Heisenberg algebra

Dmitry Faifman

Geometry & Topology 23 (2019) 3041–3110

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. 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.

contact manifold, Crofton formula, Heisenberg algebra, Lipschitz Killing curvatures, Weyl principle, intrinsic volumes
Mathematical Subject Classification 2010
Primary: 52A39, 53A55, 53C65, 53D10
Secondary: 53D05, 53D15
Received: 22 January 2018
Revised: 27 December 2018
Accepted: 29 January 2019
Published: 1 December 2019
Proposed: Leonid Polterovich
Seconded: Yasha Eliashberg, Dmitri Burago
Dmitry Faifman
Centre de Recherches Mathématiques
Université de Montréal
Montréal, QC