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A rigidity theorem for submanifolds with parallel mean curvature vector field in Sm(1)×R

  • 1.

    School of Mathematics and Statistics, Fuyang Teachers College, Fuyang 236037,China

  • 2.

    School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2016.08.001
  • Received Date: 28 February 2015
  • Accepted Date: 22 September 2015
  • Rev Recd Date: 22 September 2015
  • Publish Date: 30 August 2016
    Abstract

  • Abstract

    Submanifolds with parallel mean curvature vector field in Sm(1)×R were studied. By using the moving-frame method and some algebraic inequalities, a rigidity theorem was obtained, which generalizes the result in the relevant literatures.

    Abstract

    Submanifolds with parallel mean curvature vector field in Sm(1)×R were studied. By using the moving-frame method and some algebraic inequalities, a rigidity theorem was obtained, which generalizes the result in the relevant literatures.
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    • References(8)
  • [1]
    DILLEN F, FASTENAKELS J, VAN DER VEKEN J, et al. Constant angle surfaces in S2×R[J]. Monatsh Math, 2007, 152:89-96.
    [2]
    DANIEL B. Isometric immersions into Sn×R and Hn×R and applications to minimal surfaces[J]. Trans Amer Math Soc, 2009, 361: 6 255-6 282.
    [3]
    BATISTA M. Simons type equation in S2×R and H2×R and applications[J]. Ann Inst Fourier (Grenoble), 2011, 61: 1 299-1 322.
    [4]
    CHEN Q, CUI Q. Normal scalar curvature and a pinching theorem in Sm×R and Hm×R [J].Sci China Math, 2011, 54:1 977-1 984.
    [5]
    CHEN H, CHEN G Y, LI H Z. Some pinching theorems for minimal submanifolds in Sm×R[J]. Sci China Math, 2013, 56: 1 679-1 688.
    [6]
    FETCU D, ONICIUC C, ROSENBERG H. Biharmonic submanifolds with parallel mean curvature in Sm×R[J]. J Geeom Anal, 2013, 23: 2 158-2 176.
    [7]
    LI A M, LI J M. An intrinsic rigidity theorem for minimal submanifolds in a sphere[J]. Arch Math (Basel), 1992, 58(6): 582-594.
    [8]
    ZHANG J F. A rigidity theorem for submanifolds in Sn+p with constant scalar curvature[J]. Journal of Zhejiang University SCIENCE, 2005, 6A(4): 322-328.
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    [1]
    DILLEN F, FASTENAKELS J, VAN DER VEKEN J, et al. Constant angle surfaces in S2×R[J]. Monatsh Math, 2007, 152:89-96.
    [2]
    DANIEL B. Isometric immersions into Sn×R and Hn×R and applications to minimal surfaces[J]. Trans Amer Math Soc, 2009, 361: 6 255-6 282.
    [3]
    BATISTA M. Simons type equation in S2×R and H2×R and applications[J]. Ann Inst Fourier (Grenoble), 2011, 61: 1 299-1 322.
    [4]
    CHEN Q, CUI Q. Normal scalar curvature and a pinching theorem in Sm×R and Hm×R [J].Sci China Math, 2011, 54:1 977-1 984.
    [5]
    CHEN H, CHEN G Y, LI H Z. Some pinching theorems for minimal submanifolds in Sm×R[J]. Sci China Math, 2013, 56: 1 679-1 688.
    [6]
    FETCU D, ONICIUC C, ROSENBERG H. Biharmonic submanifolds with parallel mean curvature in Sm×R[J]. J Geeom Anal, 2013, 23: 2 158-2 176.
    [7]
    LI A M, LI J M. An intrinsic rigidity theorem for minimal submanifolds in a sphere[J]. Arch Math (Basel), 1992, 58(6): 582-594.
    [8]
    ZHANG J F. A rigidity theorem for submanifolds in Sn+p with constant scalar curvature[J]. Journal of Zhejiang University SCIENCE, 2005, 6A(4): 322-328.
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