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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
DOI: https://doi.org/10.1090/S0002-9939-1971-0273541-X
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 \[ (1)\qquad \begin {array}{*{20}{c}} {\int _M^{} {{{({\text {c}}\cdot {\text {e)}}}^m}GdV = {c_{n + m}}\chi (M)/{c_{m,}}} } & {{\text {for}}\;m = 0,2,4, \cdots ,} \\ { = 0,} & {{\text {for}}\;m = 1,3,5, \cdots ,} \\ \end {array} \] 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