A characterization of complete Riemannian manifolds minimally immersed in the unit sphere
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In this paper, the authors generalize the results due to Chern, do Carmo and Kobayashi [1] to complete Riemannian manifolds and show that if the square of length of the second fundamental form in M n is not more than, then either M n n is totally geodesic, or M n m is the Veronese surface in S 4 (1) or M m n is the Clifford torus.Abstract:
Let M n be an n -dimensional Riemannian manifold minimally immersed in the unit sphere S n+p (1) of dimension n + p . When M n is compact, Chern, do Carmo and Kobayashi [1] proved that if the square ‖ h ‖ 2 of length of the second fundamental form h in M n is not more than , then either M n is totally geodesic, or M n is the Veronese surface in S 4 (1) or M n is the Clifford torus . In this paper, we generalize the results due to Chern, do Carmo and Kobayashi [1] to complete Riemannian manifolds.read more
Citations
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Rigidity theorems of Clifford Torus
TL;DR: In this article, it was shown that if M has (n − 1) principal curvatures with the same sign everywhere, then M is isometric to a Clifford Torus.
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Complete submanifolds with parallel mean curvature vector in hyperbolic spaces
TL;DR: In this article, a better estimate was proved and generalized to higher codimensions of a theorem of Y.B. Shen on complete submanifolds with parallel mean curvature vector in a hyperbolic space.
Journal ArticleDOI
On the Principal Curvatures of Complete Minimal Hypersurfaces in Space Forms
TL;DR: In this article, a general class of complete minimal hypersurfaces in real space forms of constant curvature c, namely those with constant curvatures having the same sign everywhere, is studied.
References
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Book ChapterDOI
Minimal Submanifolds of a Sphere with Second Fundamental Form of Constant Length
TL;DR: In this article, an n-dimensional manifold which is minimally immersed in a unit sphere of dimension n+p is considered. But the manifold is not a sphere, it is a manifold.
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