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

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

2007-07-12

arxiv.org/abs/0707.1860v1

Mathematics

Differential Geometry

10 pages

Scientific paper

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 ^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).

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