Compact hypersurfaces in a locally symmetric manifold
Keywords:
Locally symmetric, Riemannian manifolds, hypersurfaces, totally umbilical.Abstract
Let M be an n-dimensional compact hypersurface in a locally symmetric manifold Nn+1. Denote by S and H the squared norm of the second fundamental form and the mean curvature of M.
Mathematics Subject Classification (2010): 53B20, 53A10.
References
Alencar, H., do Carmo, M., Hypersurfaces with constant mean curvature in spheres, Proc. of the Amer. Math. Soc., 120(1994), 1223-1229.
Chen, Q., Minimal hypersurfaces of a locally symmetric space, Chinese Science Bulletin, 38(1993), 1057-1059.
Cheng, Q.M., Shu, S.C., Suh, I.J., Compact hypersurfaces in a unit sphere, Proc. Royal Soc. Edinburgh, 135A(2005), 1129-1137.
Cheng, S.Y., Yau, S.T., Hypersurfaces with constant scalar curvature, Math. Ann., 225(1977), 195-204.
Chern, S.S., Do Carmo, M., Kobayashi, S., Minimal submanifolds of a sphere with second fundamental form of constant length, in Functional Analysis and Related Fields (F. Brower, ed.), Springer-Verlag, Berlin, 1970, 59-75.
Fontenele, F., Submanifolds with parallel mean curvature vector in pinched Riemannian manifolds, Pacific J. of Math., 177(1997), 47-70.
Hlineva, S., Belchev, E., On the minimal hypersurfaces of a locally symmetric manifold, Lecture Notes in Math., 1481(1990), 1-4.
Lawson, H.B., Local rigidity theorems for minimal hypersurfaces, Ann. of Math., 89(1969), 187-197.
Li, H., Hypersurfaces with constant scalar curvature in space forms, Math. Ann., 305(1996), 665-672.
Okumura, M., Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math., 96(1974), 207-213.
Omor, H., Isometric immersion of Riemmanian manifolds, J. Math. Soc. Japan, 19(1967), 205-214.
Shiohama, K., Xu, H.W., A general rigidity theorem for complete submanifolds, Nagoya Math. J., 150(1998), 105-134.
Shu, S.C., Liu, S.Y., Complete Hypersurfaces with Constant Mean Curvature in Locally Symmetric Manifold (I), Chinese Ann. of Math., 25A(2004), 99-104.
Shu, S.C., Liu, S.Y., Complete Hypersurfaces with Constant Mean Curvature in Locally Symmetric Manifold (II), Advances in Math. (Chinese), 33(2004), 563-569.
Shui, N.X., Wu, G.Q., Minimal hypersurfaces of a locally symmetric manifold, Chinese Ann. of Math., 16A(1995), 687-691.
Simons, J., Minimal varieties in Riemannian manifolds, Ann. of Math., 88(1968), 62-105.
Xu, H.W., On closed minimal submanifolds in pinched Riemannian manifolds, Trans. Amer. Math. Soc., 347(1995), 1743-1751.
Yau, S.T., Harmonic functions on complete Riemannian manifolds, Comm. Pure and Appl. Math., 28(1975), 201-228.
Zhong, H., Hypersurfaces in a sphere with constant mean curvature, Pro. of the Amer. Math. Soc., 125(1997), 1193-1196.
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