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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On an integral formula of Gauss-Bonnet-Grotemeyer
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by Bang-yen Chen PDF
Proc. Amer. Math. Soc. 28 (1971), 208-212 Request permission

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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Additional Information
  • © Copyright 1971 American Mathematical Society
  • 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