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Contact integral geometry
and the Heisenberg algebra
Dmitry Faifman |
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Geometry & Topology 23 (2019) 3041–3110
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Abstract |
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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. |
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Keywords
contact manifold, Crofton formula, Heisenberg algebra,
Lipschitz Killing curvatures, Weyl principle, intrinsic
volumes
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Mathematical Subject Classification 2010
Primary: 52A39, 53A55, 53C65, 53D10
Secondary: 53D05, 53D15
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References |
Publication
Received: 22 January 2018
Revised: 27 December 2018
Accepted: 29 January 2019
Published: 1 December 2019
Proposed: Leonid Polterovich
Seconded: Yasha Eliashberg, Dmitri Burago
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Authors |
| Dmitry Faifman | |
| Centre de Recherches
Mathématiques Université de Montréal Montréal, QC Canada |
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