## Geometry & Topology

### Contact integral geometry and the Heisenberg algebra

Dmitry Faifman

#### 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. 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.

#### Article information

Source
Geom. Topol., Volume 23, Number 6 (2019), 3041-3110.

Dates
Revised: 27 December 2018
Accepted: 29 January 2019
First available in Project Euclid: 7 December 2019

https://projecteuclid.org/euclid.gt/1575687771

Digital Object Identifier
doi:10.2140/gt.2019.23.3041

Mathematical Reviews number (MathSciNet)
MR4039185

Zentralblatt MATH identifier
07142694

#### Citation

Faifman, Dmitry. Contact integral geometry and the Heisenberg algebra. Geom. Topol. 23 (2019), no. 6, 3041--3110. doi:10.2140/gt.2019.23.3041. https://projecteuclid.org/euclid.gt/1575687771

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