Minimal Submanifolds of a Sphere with Second Fundamental Form of Constant Length

1970 ◽  
pp. 59-75 ◽  
Author(s):  
S. S. Chern ◽  
M. Do Carmo ◽  
S. Kobayashi
Keyword(s):  
Fundamental Form ◽  
Minimal Submanifolds ◽  
Second Fundamental Form ◽  
Constant Length

Totally real submanifolds in a generalized complex space form

2016 ◽  
Vol 10 (02) ◽  
pp. 1750035
Author(s):  
Majid Ali Choudhary
Keyword(s):  
Fundamental Form ◽  
Complex Space ◽  
Space Form ◽  
Second Fundamental Form ◽  
Text Structure ◽  
Complex Space Form ◽  
Real Submanifolds ◽  
Constant Length ◽  
Generalized Complex ◽  
Totally Real

In the present paper, we investigate totally real submanifolds in generalized complex space form. We study the [Formula: see text]-structure in the normal bundle of a totally real submanifold and derive some integral formulas computing the Laplacian of the square of the second fundamental form and using these formulas, we prove a pinching theorem. In fact, the purpose of this note is to generalize results proved in B. Y. Chen and K. Ogiue, On totally real manifolds, Trans. Amer. Math. Soc. 193 (1974) 257–266, S. S. Chern, M. Do Carmo and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, in Functional Analysis and Related Fields (Springer-Verlag, 1970), pp. 57–75 to the case, when the ambient manifold is generalized complex space form.

scholarly journals SELF-SHRINKERS WITH SECOND FUNDAMENTAL FORM OF CONSTANT LENGTH

2017 ◽  
Vol 96 (2) ◽  
pp. 326-332 ◽  
Author(s):  
QIANG GUANG
Keyword(s):  
Fundamental Form ◽  
Simple Proof ◽  
Second Fundamental Form ◽  
Constant Length ◽  
The Second Fundamental Form ◽  
Gap Theorem

We give a new and simple proof of a result of Ding and Xin, which states that any smooth complete self-shrinker in $\mathbb{R}^{3}$ with the second fundamental form of constant length must be a generalised cylinder $\mathbb{S}^{k}\times \mathbb{R}^{2-k}$ for some $k\leq 2$. Moreover, we prove a gap theorem for smooth self-shrinkers in all dimensions.

On the Complete 2-Dimensional λ-Translators with a Second Fundamental form of Constant Length

2020 ◽  
Vol 40 (6) ◽  
pp. 1897-1914
Author(s):  
Xingxiao Li ◽  
Ruina Qiao ◽  
Yangyang Liu
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
Constant Length

EINSTEIN-WEYL DEFORMATIONS AND SUBMANIFOLDS

1996 ◽  
Vol 07 (05) ◽  
pp. 705-719 ◽  
Author(s):  
HENRIK PEDERSEN ◽  
YAT SUN POON ◽  
ANDREW SWANN
Keyword(s):  
Fundamental Form ◽  
Einstein Metrics ◽  
Isometry Group ◽  
Second Fundamental Form ◽  
Local Version ◽  
Positive Ricci Curvature ◽  
Constant Length ◽  
Minimal Hypersurfaces ◽  
Weyl Space ◽  
Weyl Manifold

Motivated by new explicit positive Ricci curvature metrics on the four-sphere which are also Einstein-Weyl, we show that the dimension of the Einstein-Weyl moduli near certain Einstein metrics is bounded by the rank of the isometry group and that any Weyl manifold can be embedded as a hypersurface with prescribed second fundamental form in some Einstein-Weyl space. Closed four-dimensional Einstein-Weyl manifolds are proved to be absolute minima of the L2-norm of the curvature of Weyl manifolds and a local version of the Lafontaine inequality is obtained. The above metrics on the four-sphere are shown to contain minimal hypersurfaces isometric to S1×S2 whose second fundamental form has constant length.

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