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.
