The Gauss-Bonnet theorem and Crofton type formulas in complex space forms
Differential Geometry
2011-07-21 v2 Metric Geometry
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.
Cite
@article{arxiv.0904.0336,
title = {The Gauss-Bonnet theorem and Crofton type formulas in complex space forms},
author = {Judit Abardia and Eduardo Gallego and Gil Solanes},
journal= {arXiv preprint arXiv:0904.0336},
year = {2011}
}
Comments
24 pages, to appear in Israel Journal of Mathematics