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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

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), 277-297. 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), 113-118.  Google Scholar

[3]

B.-Y. Chen, On an integral formula of Gauss-Bonnet-Grotemeyer, Proc. Amer. Math. Soc., 28 (1971), 208-212.  Google Scholar

[4]

S. Y. Cheng and S.-T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann., 225 (1977), 195-204. 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), 747-752. 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), 674-684. doi: doi:10.2307/1969203.  Google Scholar

[7]

K. P. Grotemeyer, Über das Normalenbündel differenzierbarer mannigfaltigkeiten, Ann. Acad. Sci. Fenn., Ser. A. I. No. 336/15 (1963), 1-12. Google Scholar

[8]

H. Li, Hypersurfaces with constant scalar curvature in space forms, Math. Ann., 305 (1996), 665-672. doi: doi:10.1007/BF01444243.  Google Scholar

[9]

H. Li, Global rigidity theorems of hypersurfaces, Ark. Math., 35 (1997), 327-351. 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), 465-477.  Google Scholar

[11]

H. Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math., 117 (1993), 211-239.  Google Scholar

[12]

G. Solanes, Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces, Trans. Amer. Math. Soc., 358 (2006), 1105-1115. 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), 180-218. 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), 277-297. 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), 113-118.  Google Scholar

[3]

B.-Y. Chen, On an integral formula of Gauss-Bonnet-Grotemeyer, Proc. Amer. Math. Soc., 28 (1971), 208-212.  Google Scholar

[4]

S. Y. Cheng and S.-T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann., 225 (1977), 195-204. 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), 747-752. 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), 674-684. doi: doi:10.2307/1969203.  Google Scholar

[7]

K. P. Grotemeyer, Über das Normalenbündel differenzierbarer mannigfaltigkeiten, Ann. Acad. Sci. Fenn., Ser. A. I. No. 336/15 (1963), 1-12. Google Scholar

[8]

H. Li, Hypersurfaces with constant scalar curvature in space forms, Math. Ann., 305 (1996), 665-672. doi: doi:10.1007/BF01444243.  Google Scholar

[9]

H. Li, Global rigidity theorems of hypersurfaces, Ark. Math., 35 (1997), 327-351. 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), 465-477.  Google Scholar

[11]

H. Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math., 117 (1993), 211-239.  Google Scholar

[12]

G. Solanes, Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces, Trans. Amer. Math. Soc., 358 (2006), 1105-1115. 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), 180-218. Google Scholar

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