American Institute of Mathematical Sciences

Advanced
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.  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

[1]

Jérôme Bertrand. Prescription of Gauss curvature on compact hyperbolic orbifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1269-1284. doi: 10.3934/dcds.2014.34.1269
[2]

Andrei Agrachev, Ugo Boscain, Mario Sigalotti. A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 801-822. doi: 10.3934/dcds.2008.20.801
[3]

Xumin Jiang. Isometric embedding with nonnegative Gauss curvature under the graph setting. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3463-3477. doi: 10.3934/dcds.2019143
[4]

Cristian Enache. Maximum and minimum principles for a class of Monge-Ampère equations in the plane, with applications to surfaces of constant Gauss curvature. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1347-1359. doi: 10.3934/cpaa.2014.13.1347
[5]

Pei Yean Lee, John B Moore. Gauss-Newton-on-manifold for pose estimation. Journal of Industrial & Management Optimization, 2005, 1 (4) : 565-587. doi: 10.3934/jimo.2005.1.565
[6]

Yuezheng Gong, Jiaquan Gao, Yushun Wang. High order Gauss-Seidel schemes for charged particle dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 573-585. doi: 10.3934/dcdsb.2018034
[7]

Giuseppina di Blasio, Filomena Feo, Maria Rosaria Posteraro. Existence results for nonlinear elliptic equations related to Gauss measure in a limit case. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1497-1506. doi: 10.3934/cpaa.2008.7.1497
[8]

Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Maria Francesca Carfora. Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons. Mathematical Biosciences & Engineering, 2014, 11 (2) : 189-201. doi: 10.3934/mbe.2014.11.189
[9]

Zhong-Qing Wang, Li-Lian Wang. A Legendre-Gauss collocation method for nonlinear delay differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 685-708. doi: 10.3934/dcdsb.2010.13.685
[10]

Mehar Chand, Jyotindra C. Prajapati, Ebenezer Bonyah, Jatinder Kumar Bansal. Fractional calculus and applications of family of extended generalized Gauss hypergeometric functions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 539-560. doi: 10.3934/dcdss.2020030
[11]

Shigeru Sakaguchi. A Liouville-type theorem for some Weingarten hypersurfaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 887-895. doi: 10.3934/dcdss.2011.4.887
[12]

Joel Spruck, Ling Xiao. Convex spacelike hypersurfaces of constant curvature in de Sitter space. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2225-2242. doi: 10.3934/dcdsb.2012.17.2225
[13]

Katsuyuki Ishii, Takahiro Izumi. Remarks on the convergence of an algorithm for curvature-dependent motions of hypersurfaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1103-1125. doi: 10.3934/dcds.2018046
[14]

Qinian Jin, YanYan Li. Starshaped compact hypersurfaces with prescribed $k$-th mean curvature in hyperbolic space. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 367-377. doi: 10.3934/dcds.2006.15.367
[15]

Ning Zhang. A symmetric Gauss-Seidel based method for a class of multi-period mean-variance portfolio selection problems. Journal of Industrial & Management Optimization, 2020, 16 (2) : 991-1008. doi: 10.3934/jimo.2018189
[16]

Rashad M. Asharabi, Jürgen Prestin. Computing eigenpairs of two-parameter Sturm-Liouville systems using the bivariate sinc-Gauss formula. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4143-4158. doi: 10.3934/cpaa.2020185
[17]

Ke Chen, Yiqiu Dong, Michael Hintermüller. A nonlinear multigrid solver with line Gauss-Seidel-semismooth-Newton smoother for the Fenchel pre-dual in total variation based image restoration. Inverse Problems & Imaging, 2011, 5 (2) : 323-339. doi: 10.3934/ipi.2011.5.323
[18]

Ian Blake, V. Kumar Murty, Hamid Usefi. A note on diagonal and Hermitian hypersurfaces. Advances in Mathematics of Communications, 2016, 10 (4) : 753-764. doi: 10.3934/amc.2016039
[19]

Jan Prüss, Gieri Simonett. On the manifold of closed hypersurfaces in $\mathbb{R}^n$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5407-5428. doi: 10.3934/dcds.2013.33.5407
[20]

Antonio Cafure, Guillermo Matera, Melina Privitelli. Singularities of symmetric hypersurfaces and Reed-Solomon codes. Advances in Mathematics of Communications, 2012, 6 (1) : 69-94. doi: 10.3934/amc.2012.6.69

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences