Gauss-bonnet Theorem and Crofton Type Formulas in Complex Space Forms
نویسندگان
چکیده
We give an expression, in terms of the so-called Hermitian intrinsic volumes, for the measure of the set of complex r-planes intersecting a regular domain in any complex space form. Moreover, we obtain two different expressions for the Gauss-BonnetChern formula in complex space forms. One of them expresses 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. Finally, we express the average over the complex r-planes of the total Gauss curvature of the intersection with a domain.
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