The Gauss-Bonnet-Grotemeyer Theorem in space forms

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November 2010, 4(4): 655-664. doi: 10.3934/ipi.2010.4.655

The Gauss-Bonnet-Grotemeyer Theorem in space forms

Eric L. Grinberg ^1, and Haizhong Li ^2,

1.	Department of Mathematics & Statistics, University of New Hampshire, Durham, NH 03824, United States
2.	Department of Mathematical Sciences, Tsinghua University, 100084, Beijing, China

Received January 2009 Revised July 2009 Published September 2010

Abstract
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In 1963, K.P.~Grotemeyer proved an interesting variant of the Gauss-Bonnet Theorem. Let $M$ be an oriented closed surface in the Euclidean space $\mathbb R^3$ with Euler characteristic $\chi(M)$, Gauss curvature $G$ and unit normal vector field $\vec n$. Grotemeyer's identity replaces the Gauss-Bonnet integrand $G$ by the normal moment $ ( \vec a \cdot \vec n )^2G$, where $a$ is a fixed unit vector: $ \int_M(\vec a\cdot \vec n)^2 Gdv=\frac{2 \pi}{3}\chi(M) $. We generalize Grotemeyer's result to oriented closed even-dimensional hypersurfaces of dimension $n$ in an $(n+1)$-dimensional space form $N^{n+1}(k)$.

Keywords: hypersurfaces, Grotemeyer., Gauss-Kronecker curvature, Gauss-Bonnet Theorem.

Mathematics Subject Classification: Primary: 53C42; Secondary 53A1.

Citation: Eric L. Grinberg, Haizhong Li. The Gauss-Bonnet-Grotemeyer Theorem in space forms. Inverse Problems & Imaging, 2010, 4 (4) : 655-664. doi: 10.3934/ipi.2010.4.655

References:

[1]	J. L. M. Barbosa and A. G. Colares, Stability of hypersurfaces with constant $r$-mean curvature,, Ann. Global Anal. Geom., 15 (1997), 277. doi: doi:10.1023/A:1006514303828. Google Scholar
[2]	I. Bivens, Integral formulas and hyperspheres in a simply connected space form,, Proc. Amer. Math. Soc., 88 (1983), 113. Google Scholar
[3]	B.-Y. Chen, On an integral formula of Gauss-Bonnet-Grotemeyer,, Proc. Amer. Math. Soc., 28 (1971), 208. Google Scholar
[4]	S. Y. Cheng and S.-T. Yau, Hypersurfaces with constant scalar curvature,, Math. Ann., 225 (1977), 195. doi: doi:10.1007/BF01425237. Google Scholar
[5]	S. S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds,, Ann. of Math. (2), 45 (1944), 747. doi: doi:10.2307/1969302. Google Scholar
[6]	S. S. Chern, On the curvatura integra in a Riemannian manifold,, Ann. of Math. (2), 46 (1945), 674. doi: doi:10.2307/1969203. Google Scholar
[7]	K. P. Grotemeyer, Über das Normalenbündel differenzierbarer mannigfaltigkeiten,, Ann. Acad. Sci. Fenn., (1963), 1. Google Scholar
[8]	H. Li, Hypersurfaces with constant scalar curvature in space forms,, Math. Ann., 305 (1996), 665. doi: doi:10.1007/BF01444243. Google Scholar
[9]	H. Li, Global rigidity theorems of hypersurfaces,, Ark. Math., 35 (1997), 327. doi: doi:10.1007/BF02559973. Google Scholar
[10]	R. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms,, J. Diff. Geom., 8 (1973), 465. Google Scholar
[11]	H. Rosenberg, Hypersurfaces of constant curvature in space forms,, Bull. Sci. Math., 117 (1993), 211. Google Scholar
[12]	G. Solanes, Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces,, Trans. Amer. Math. Soc., 358 (2006), 1105. doi: doi:10.1090/S0002-9947-05-03828-6. Google Scholar
[13]	K. Voss, Einige differentialgeometrische kongruenzsätze für geschlossene flächen und hyperflächen,, Math. Ann., 131 (1956), 180. Google Scholar

show all references

References:

[1]	J. L. M. Barbosa and A. G. Colares, Stability of hypersurfaces with constant $r$-mean curvature,, Ann. Global Anal. Geom., 15 (1997), 277. doi: doi:10.1023/A:1006514303828. Google Scholar
[2]	I. Bivens, Integral formulas and hyperspheres in a simply connected space form,, Proc. Amer. Math. Soc., 88 (1983), 113. Google Scholar
[3]	B.-Y. Chen, On an integral formula of Gauss-Bonnet-Grotemeyer,, Proc. Amer. Math. Soc., 28 (1971), 208. Google Scholar
[4]	S. Y. Cheng and S.-T. Yau, Hypersurfaces with constant scalar curvature,, Math. Ann., 225 (1977), 195. doi: doi:10.1007/BF01425237. Google Scholar
[5]	S. S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds,, Ann. of Math. (2), 45 (1944), 747. doi: doi:10.2307/1969302. Google Scholar
[6]	S. S. Chern, On the curvatura integra in a Riemannian manifold,, Ann. of Math. (2), 46 (1945), 674. doi: doi:10.2307/1969203. Google Scholar
[7]	K. P. Grotemeyer, Über das Normalenbündel differenzierbarer mannigfaltigkeiten,, Ann. Acad. Sci. Fenn., (1963), 1. Google Scholar
[8]	H. Li, Hypersurfaces with constant scalar curvature in space forms,, Math. Ann., 305 (1996), 665. doi: doi:10.1007/BF01444243. Google Scholar
[9]	H. Li, Global rigidity theorems of hypersurfaces,, Ark. Math., 35 (1997), 327. doi: doi:10.1007/BF02559973. Google Scholar
[10]	R. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms,, J. Diff. Geom., 8 (1973), 465. Google Scholar
[11]	H. Rosenberg, Hypersurfaces of constant curvature in space forms,, Bull. Sci. Math., 117 (1993), 211. Google Scholar
[12]	G. Solanes, Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces,, Trans. Amer. Math. Soc., 358 (2006), 1105. doi: doi:10.1090/S0002-9947-05-03828-6. Google Scholar
[13]	K. Voss, Einige differentialgeometrische kongruenzsätze für geschlossene flächen und hyperflächen,, Math. Ann., 131 (1956), 180. Google Scholar

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