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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On an integral formula of Gauss-Bonnet-Grotemeyer

Author: Bang-yen Chen
Journal: Proc. Amer. Math. Soc. 28 (1971), 208-212
MSC: Primary 53.72
MathSciNet review: 0273541
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Abstract: Let $ e(p)$ and $ G(p)$ be the unit outer normal and the Gauss-Kronecker curvature of an oriented closed even-dimensional hypersurface $ M$ of dimension $ n$ in $ {E^{n + 1}}$. Then for a fixed unit vector $ c$ in $ {E^{n + 1}}$, we have

$\displaystyle (1)\qquad \begin{array}{*{20}{c}} {\int_M^{} {{{({\text{c}}\cdot{... ...2,4, \cdots ,} \\ { = 0,} & {{\text{for}}\;m = 1,3,5, \cdots ,} \\ \end{array} $

where $ {\text{c}} \cdot {\text{e}}$ denotes the inner product of $ c$ and $ e$ the area of $ m$-dimensional unit sphere, and $ \chi (M)$ the Euler characteristic of $ M$.

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Keywords: Closed hypersurface, differential form, Gauss-Kronecker curvature, Euler characteristic, Gauss-Bonnet-Grotemeyer formula
Article copyright: © Copyright 1971 American Mathematical Society