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YaJuan ZHANG, . Linear Weingarten hypersurfaces in locally symmetric manifolds[J]. Journal of South China Normal University (Natural Science Edition), 2016, 48(6): 124-128.
Citation: YaJuan ZHANG, . Linear Weingarten hypersurfaces in locally symmetric manifolds[J]. Journal of South China Normal University (Natural Science Edition), 2016, 48(6): 124-128.

Linear Weingarten hypersurfaces in locally symmetric manifolds

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  • Received Date: January 10, 2016
  • Revised Date: March 07, 2016
  • The complete linear Weingarten hypersurfaces with bounded mean curvature in locally symmetric manifold are studied. By supposing a suitable restriction on the norm of which is symmetric tensor in M^{n}, it is proved that such a hypersurface must be either totally umbilical or isometric to the hypersurface with two distinct principle curvatures, one of which is simple.
  • [1]SY Cheng, ST Yau.Hypersurfaces with constant scalar curvature[J].Math. Ann., 1977, 225(3):195-204
    [2]S Shu.Complete hypersurfaces with constant scalar curvature in a hyperbolic space[J].Balkan J. Geom. Appl., 2007, 12(2):107-115
    [3]H Li, YJ Suh, G Wei.Linear Weingarten hypersurfaces in a unit sphere[J].Bull. Korean Math. Soc., 2009, 46(2):321-329
    [4]S Shu.Linear Weingarten hypersurfaces in a real space form[J].Glasgow Math. J., 2010, 52(3):635-648
    [5]R L\'{o}pez.Rotational linear Weingarten surfaces of hyperbolic type[J].Israel J. Math., 2008, 167(1):283-301
    [6]A Barros, J Silva, P Sousa.Rotational linear Weingarten surfaces into the Euclidean sphere[J].Israel J. Math., 2012, 192(2):819-830
    [7]C P Aquino, H F de Lima.On the geometry of linear Weingarten hypersurfaces in the hyperbolic space[J].Monatsh. Math., 2013, 171(3-4):259-268
    [8]C P Aquino, H F de Lima, M A L Vel\'{a}squez.A new characterization of complete linear Weingarten hypersurfaces in real space forms[J].Pacific J. Math., 2013, 261(1):33-43
    [9]C P Aquino, H F de Lima, M A L Vel\'{a}squez.Generalized maximum principles and the characterization of linear Weingarten hypersurfaces in space forms[J].Michigan Math. J., 2014, 63(1):27-40
    [10]C P Aquino, H F de Lima, M A L Vel\'{a}squez.Linear Weingarten hypersurfaces with bounded mean curvature in the hyperbolic space[J].Glasgow Math. J., 2015, 57(3):653-663
    [11]X Chao, P Wang.Linear Weingarten hypersurfaces in locally symmetric manifolds[J].Balkan J.Geom. Appl., 2014, 19(2):50-59
    [12]M Okumura.Hypersurfaces and a pinching problem on the second fundamental tensor[J].Amer. J. Math., 1974, 63(1):207-213

    [1]SY Cheng, ST Yau.Hypersurfaces with constant scalar curvature[J].Math. Ann., 1977, 225(3):195-204
    [2]S Shu.Complete hypersurfaces with constant scalar curvature in a hyperbolic space[J].Balkan J. Geom. Appl., 2007, 12(2):107-115
    [3]H Li, YJ Suh, G Wei.Linear Weingarten hypersurfaces in a unit sphere[J].Bull. Korean Math. Soc., 2009, 46(2):321-329
    [4]S Shu.Linear Weingarten hypersurfaces in a real space form[J].Glasgow Math. J., 2010, 52(3):635-648
    [5]R L\'{o}pez.Rotational linear Weingarten surfaces of hyperbolic type[J].Israel J. Math., 2008, 167(1):283-301
    [6]A Barros, J Silva, P Sousa.Rotational linear Weingarten surfaces into the Euclidean sphere[J].Israel J. Math., 2012, 192(2):819-830
    [7]C P Aquino, H F de Lima.On the geometry of linear Weingarten hypersurfaces in the hyperbolic space[J].Monatsh. Math., 2013, 171(3-4):259-268
    [8]C P Aquino, H F de Lima, M A L Vel\'{a}squez.A new characterization of complete linear Weingarten hypersurfaces in real space forms[J].Pacific J. Math., 2013, 261(1):33-43
    [9]C P Aquino, H F de Lima, M A L Vel\'{a}squez.Generalized maximum principles and the characterization of linear Weingarten hypersurfaces in space forms[J].Michigan Math. J., 2014, 63(1):27-40
    [10]C P Aquino, H F de Lima, M A L Vel\'{a}squez.Linear Weingarten hypersurfaces with bounded mean curvature in the hyperbolic space[J].Glasgow Math. J., 2015, 57(3):653-663
    [11]X Chao, P Wang.Linear Weingarten hypersurfaces in locally symmetric manifolds[J].Balkan J.Geom. Appl., 2014, 19(2):50-59
    [12]M Okumura.Hypersurfaces and a pinching problem on the second fundamental tensor[J].Amer. J. Math., 1974, 63(1):207-213

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