Integral geometry of complex space forms

TL;DR

This paper applies Alesker's valuation theory to complex space forms, providing algebraic insights and explicit formulas for integral geometry, including kinematic and tube formulas, in complex projective, hyperbolic, and Euclidean spaces.

Contribution

It offers a comprehensive algebraic framework for the integral geometry of complex space forms and computes explicit kinematic formulas for invariant valuations and curvature measures.

Findings

Derived new formulas for volumes of tubes around totally real submanifolds.

Showed stabilization of Lipschitz-Killing valuations on invariant angular curvature measures.

Provided algebraic descriptions of integral geometry in complex space forms.

Abstract

We show how Alesker's theory of valuations on manifolds gives rise to an algebraic picture of the integral geometry of any Riemannian isotropic space. We then apply this method to give a thorough account of the integral geometry of the complex space forms, i.e. complex projective space, complex hyperbolic space and complex euclidean space. In particular, we compute the family of kinematic formulas for invariant valuations and invariant curvature measures in these spaces. In addition to new and more efficient framings of the tube formulas of Gray and the kinematic formulas of Shifrin, this approach yields a new formula expressing the volumes of the tubes about a totally real submanifold in terms of its intrinsic Riemannian structure. We also show by direct calculation that the Lipschitz-Killing valuations stabilize the subspace of invariant angular curvature measures, suggesting the possibility that a similar phenomenon holds for all Riemannian manifolds. We conclude with a number of open questions and conjectures.