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A remark on the weighted Radon transform on the plane
The Gauss-Bonnet-Grotemeyer Theorem in space forms
1. | Department of Mathematics & Statistics, University of New Hampshire, Durham, NH 03824, United States |
2. | Department of Mathematical Sciences, Tsinghua University, 100084, Beijing, China |
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. |
[2] |
I. Bivens, Integral formulas and hyperspheres in a simply connected space form, Proc. Amer. Math. Soc., 88 (1983), 113-118. |
[3] |
B.-Y. Chen, On an integral formula of Gauss-Bonnet-Grotemeyer, Proc. Amer. Math. Soc., 28 (1971), 208-212. |
[4] |
S. Y. Cheng and S.-T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann., 225 (1977), 195-204.
doi: doi:10.1007/BF01425237. |
[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. |
[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. |
[7] |
K. P. Grotemeyer, Über das Normalenbündel differenzierbarer mannigfaltigkeiten, Ann. Acad. Sci. Fenn., Ser. A. I. No. 336/15 (1963), 1-12. |
[8] |
H. Li, Hypersurfaces with constant scalar curvature in space forms, Math. Ann., 305 (1996), 665-672.
doi: doi:10.1007/BF01444243. |
[9] |
H. Li, Global rigidity theorems of hypersurfaces, Ark. Math., 35 (1997), 327-351.
doi: doi:10.1007/BF02559973. |
[10] |
R. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Diff. Geom., 8 (1973), 465-477. |
[11] |
H. Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math., 117 (1993), 211-239. |
[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. |
[13] |
K. Voss, Einige differentialgeometrische kongruenzsätze für geschlossene flächen und hyperflächen, Math. Ann., 131 (1956), 180-218. |
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. |
[2] |
I. Bivens, Integral formulas and hyperspheres in a simply connected space form, Proc. Amer. Math. Soc., 88 (1983), 113-118. |
[3] |
B.-Y. Chen, On an integral formula of Gauss-Bonnet-Grotemeyer, Proc. Amer. Math. Soc., 28 (1971), 208-212. |
[4] |
S. Y. Cheng and S.-T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann., 225 (1977), 195-204.
doi: doi:10.1007/BF01425237. |
[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. |
[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. |
[7] |
K. P. Grotemeyer, Über das Normalenbündel differenzierbarer mannigfaltigkeiten, Ann. Acad. Sci. Fenn., Ser. A. I. No. 336/15 (1963), 1-12. |
[8] |
H. Li, Hypersurfaces with constant scalar curvature in space forms, Math. Ann., 305 (1996), 665-672.
doi: doi:10.1007/BF01444243. |
[9] |
H. Li, Global rigidity theorems of hypersurfaces, Ark. Math., 35 (1997), 327-351.
doi: doi:10.1007/BF02559973. |
[10] |
R. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Diff. Geom., 8 (1973), 465-477. |
[11] |
H. Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math., 117 (1993), 211-239. |
[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. |
[13] |
K. Voss, Einige differentialgeometrische kongruenzsätze für geschlossene flächen und hyperflächen, Math. Ann., 131 (1956), 180-218. |
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