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

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

2009-04-02

arxiv.org/abs/0904.0336v2

Mathematics

Differential Geometry

24 pages, to appear in Israel Journal of Mathematics

Scientific paper

10.1007/s11856-011-0083-8

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.

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