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On the quasi Gauss map for a compact sub-manifold in Euclidean space

Cite this: JUSTC, 2020, 50(3): 360-362
https://doi.org/10.3969/j.issn.0253-2778.2020.03.014
Funds: Supported by the Special Fund of Guiyang Science and Technology Bureau (GYU-KYZ [2019-2020]).
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  • Corresponding author:

    WANG Qi (Corresponding author), PhD/Prof. E-mail: wangqihn@126.com

  • Received Date: December 29, 2019
  • Revised Date: March 27, 2020
  • Accepted Date: March 27, 2020
  • Published Date: March 30, 2020
    Abstract
  • Let  σ be the quasi Gauss map of a compact and oriented  n-dimensional isometric immersion sub-manifold  Mn in the  (n+p)-dimensional Euclid space  Rn+p. Denote by ξ the unit mean curvature vector field to  Mn and denote by  Hi the  i-mean curvature along the direction  ξ.Assume that  Hi>0,  i=1,2,…,r for some integer  r  (1≤r≤n-1) and  Hr is a constant. By applying an integral formula recently given by themselves, it is proven that if the image σ(Mn) lies within an open  n-dimension unit semi sphere  Sn+ then  Mn must be totally quasi umbilical. This result generalizes a relevant theorem on hypersurfaces in Euclid space.
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    [1]
    WANG Q. On the Gauss image of compact and convex hyper-surfaces with a constant higher order mean curvature in Euclidean space [J].Journal of Fuzhou University (Natural Science Edition), 2018, 46(3): 307-310.
    [2]
    WANG Q, ZHOU Z J. Integral formulas for compact sub-manifolds in Euclid space [J].J.of Math.(PRC), 2020, 40(4): 415-420..
    [3]
    WANG Q. Totally umbilical hyper-surfaces of the hyperbolical space and Gauss image [J].Journal of Zhejiang University (Science Edition), 2016, 43(5): 537-538,549.
    [4]
    WANG Q. Totally umbilical property of hyper-surfaces in a positive curvature space form and higher order mean curvature [J]. Acta Mathematica Sinica, Chinese Series, 2014, 57(1): 47-50.)

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