The Gauss-Bonnet theorem and Crofton type formulas in complex space forms

TL;DR

This paper derives formulas relating curvature, topology, and geometric measures in complex space forms, extending classical integral geometry results with new versions of the Gauss-Bonnet-Chern theorem.

Contribution

It introduces new Crofton type formulas and variation formulas in complex space forms, connecting hermitian intrinsic volumes with intersection measures and curvature integrals.

Findings

Derived measure of complex planes intersecting domains in complex space forms.

Established two versions of the Gauss-Bonnet-Chern formula involving intrinsic volumes.

Provided variation formulas in the integral geometry of complex space forms.

Abstract

We compute the measure with multiplicity of the set of complex planes intersecting a compact domain in a complex space form. The result is given in terms of the so-called hermitian intrinsic volumes. Moreover, we obtain two different versions for the Gauss-Bonnet-Chern formula in complex space forms. One of them gives the Gauss curvature integral in terms of the Euler characteristic, and some hermitian intrinsic volumes. The other one, which is shorter, involves the measure of complex hyperplanes meeting the domain. As a tool, we obtain variation formulas in integral geometry of complex space forms.