The Gauss-Bonnet-Grotemeyer Theorem in spaces of constant curvature

TL;DR

This paper generalizes Grotemeyer's variant of the Gauss-Bonnet theorem, which involves a normal moment integrand, to higher-dimensional hypersurfaces in space forms of constant curvature.

Contribution

It extends Grotemeyer's 1963 result from surfaces in Euclidean space to even-dimensional hypersurfaces in space forms of arbitrary constant curvature.

Findings

Generalized the integral identity to higher dimensions

Derived explicit formulas for space forms of constant curvature

Connected the result to classical topological invariants

Abstract

In 1963, K.P.Grotemeyer proved an interesting variant of the Gauss-Bonnet Theorem. Let M be an oriented closed surface in the Euclidean space R^3 with Euler characteristic \chi(M), Gauss curvature G and unit normal vector field n. Grotemeyer's identity replaces the Gauss-Bonnet integrand G by the normal moment <a,n>^2G, where $a$ is a fixed unit vector. Grotemeyer showed that the total integral of this integrand is (2/3)pi times chi(M). We generalize Grotemeyer's result to oriented closed even-dimesional hypersurfaces of dimension n in an (n+1) ndimensional space form N^{n+1}(k).