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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$.

References [Enhancements On Off] (What's this?)

  • [1] Bang-yen Chen, On the total curvature of immersed manifolds. I. An inequality of Fenchel-Borsuk-Willmore, Amer. J. Math. 93 (1971), 148–162. MR 0278240
  • [2] Shiing-shen Chern, Integral formulas for hypersurfaces in Euclidean space and their applications to uniqueness theorems, J. Math. Mech. 8 (1959), 947–955. MR 0114170
  • [3] K. P. Grotemeyer, Über das Normalenbündel differenzierbarer Mannigfaltigkeiten, Ann. Acad. Sci. Fenn. Ser. A I No. 336/15 (1963), 12 (German). MR 0163265

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0273541-X
Keywords: Closed hypersurface, differential form, Gauss-Kronecker curvature, Euler characteristic, Gauss-Bonnet-Grotemeyer formula
Article copyright: © Copyright 1971 American Mathematical Society