The Gauss-Bonnet-Grotemeyer Theorem in space forms

Eric L. Grinberg; Haizhong Li

doi:10.3934/ipi.2010.4.655

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2010, Volume 4, Issue 4: 655-664. Doi: 10.3934/ipi.2010.4.655

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The Gauss-Bonnet-Grotemeyer Theorem in space forms

Eric L. Grinberg^1, and
Haizhong Li^2,

1.
Department of Mathematics & Statistics, University of New Hampshire, Durham, NH 03824
2.
Department of Mathematical Sciences, Tsinghua University, 100084, Beijing

Received: December 31, 2008

Revised: June 30, 2009

Published: October 2010

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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 $\mathbb R^3$ with Euler characteristic $\chi(M)$, Gauss curvature $G$ and unit normal vector field $\vec n$. Grotemeyer's identity replaces the Gauss-Bonnet integrand $G$ by the normal moment $ ( \vec a \cdot \vec n )^2G$, where $a$ is a fixed unit vector: $ \int_M(\vec a\cdot \vec n)^2 Gdv=\frac{2 \pi}{3}\chi(M) $. We generalize Grotemeyer's result to oriented closed even-dimensional hypersurfaces of dimension $n$ in an $(n+1)$-dimensional space form $N^{n+1}(k)$.

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Mathematics Subject Classification: Primary: 53C42; Secondary 53A10.

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