Isotropic Immersions with Parallel Second Fundamental Form

1983 ◽  
Vol 26 (3) ◽  
pp. 291-296 ◽  
Author(s):  
Sadahiro Maeda
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
Veronese Surface ◽  
Parallel Second Fundamental Form

AbstractThe main purpose of this paper is to give a characterization of a Veronese manifold, as a generalization of a Veronese surface, in terms of isotropic immersions. This is an improvement of Itoh and Ogiue's results.

scholarly journals A characterization of complete Riemannian manifolds minimally immersed in the unit sphere

1993 ◽  
Vol 131 ◽  
pp. 127-133 ◽  
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Form ◽  
Riemannian Manifolds ◽  
Unit Sphere ◽  
Second Fundamental Form ◽  
Clifford Torus ◽  
Veronese Surface ◽  
The Second Fundamental Form ◽  
Complete Riemannian Manifolds

Let Mn be an n-dimensional Riemannian manifold minimally immersed in the unit sphere Sn+p (1) of dimension n + p. When Mn is compact, Chern, do Carmo and Kobayashi [1] proved that if the square ‖h‖2 of length of the second fundamental form h in Mn is not more than , then either Mn is totallygeodesic, or Mn is the Veronese surface in S4 (1) or Mn is the Clifford torus .In this paper, we generalize the results due to Chern, do Carmo and Kobayashi [1] to complete Riemannian manifolds.

scholarly journals CR-hypersurfaces of the six-dimensional sphere

1994 ◽  
Vol 17 (1) ◽  
pp. 197-200
Author(s):  
M. A. Bashir
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
Dimensional Sphere ◽  
Totally Umbilical ◽  
Parallel Second Fundamental Form ◽  
Extrinsic Sphere

We proved that there does not exist a properCR-hypersurface ofS6with parallel second fundamental form. As a result of this we showed thatS6does not admit a properCR-totally umbilical hypersurface. We also proved that an Einstein properCR-hypersurface ofS6is an extrinsic sphere.

Minimal two-spheres in G(2, 4; (C)) with parallel second fundamental form

2017 ◽  
Vol 101 (5-6) ◽  
pp. 899-912
Author(s):  
Wenjuan Zhang ◽  
Xiaoxiang Jiao ◽  
Mingyan Li
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
Parallel Second Fundamental Form
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