Contact integral geometry and the Heisenberg algebra

Faifman, Dmitry¹

¹ Centre de Recherches Mathématiques, Université de Montréal, Montréal, QC, Canada

Geometry & topology, Tome 23 (2019) no. 6, pp. 3041-3110.

Résumé

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.

Détail

DOI : 10.2140/gt.2019.23.3041

Classification : 52A39, 53A55, 53C65, 53D10, 53D05, 53D15
Keywords: contact manifold, Crofton formula, Heisenberg algebra, Lipschitz Killing curvatures, Weyl principle, intrinsic volumes

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