Geometry & Topology

Contact integral geometry and the Heisenberg algebra

Dmitry Faifman

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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

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

Received: 22 January 2018
Revised: 27 December 2018
Accepted: 29 January 2019
First available in Project Euclid: 7 December 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52A39: Mixed volumes and related topics 53A55: Differential invariants (local theory), geometric objects 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx] 53D10: Contact manifolds, general
Secondary: 53D05: Symplectic manifolds, general 53D15: Almost contact and almost symplectic manifolds

contact manifold Crofton formula Heisenberg algebra Lipschitz Killing curvatures Weyl principle intrinsic volumes


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

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