On an integral formula for closed hypersurfaces of the sphere

Author:
Chorng-shi Houh

Journal:
Proc. Amer. Math. Soc. **35** (1972), 234-237

MSC:
Primary 53C45

MathSciNet review:
0296867

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Abstract: In a compact oriented hypersurface of the sphere the integral formula is proved where is the *r*th mean curvature, *e* is the unit normal of in . Some applications are considered.

**[1]**Bang-yen Chen,*Some integral formulas for hypersurfaces in Euclidean spaces*, Nagoya Math. J.**43**(1971), 117–125. MR**0293549****[2]**S. S. Chern,*Minimal submanifolds in a Riemannian manifold*, University of Kansas, Department of Mathematics Technical Report 19 (New Series), Univ. of Kansas, Lawrence, Kan., 1968. MR**0248648****[3]**Chuan-Chih Hsiung,*Some integral formulas for closed hypersurfaces*, Math. Scand.**2**(1954), 286–294. MR**0068236****[4]**Chuan-chih Hsiung and Jamal K. Shahin,*Affine differential geometry of closed hypersurfaces*, Proc. London Math. Soc. (3)**17**(1967), 715–735. MR**0218986****[5]**Robert C. Reilly,*Extrinsic rigidity theorems for compact submanifolds of the sphere.*, J. Differential Geometry**4**(1970), 487–497. MR**0290296**

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DOI:
https://doi.org/10.1090/S0002-9939-1972-0296867-3

Keywords:
Combined operation of the vector product,
combined operation of the exterior product,
principal curvatures,
*r*th mean curvature,
Stokes theorem

Article copyright:
© Copyright 1972
American Mathematical Society