scholarly journals A general rigidity theorem for complete submanifolds

1998 ◽  
Vol 150 ◽  
pp. 105-134 ◽  
Author(s):  
Katsuhiro Shiohama ◽  
Hongwei Xu
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Second Fundamental Form ◽  
Geometric Inequalities ◽  
Positive Sectional Curvature ◽  
The Second Fundamental Form ◽  
Complete Submanifolds ◽  
Parallel Mean Curvature ◽  
Rigidity Property ◽  
Simply Connected

Abstract.Making use of 1-forms and geometric inequalities we prove the rigidity property of complete submanifolds Mn with parallel mean curvature normal in a complete and simply connected Riemannian (n+p) -manifold Nn+p with positive sectional curvature. For given integers n, p and for a nonnegative constant H we find a positive number T(n,p) ∈ (0,1) with the property that if the sectional curvature of N is pinched in [T(n,p), 1], and if the squared norm of the second fundamental form is in a certain interval, then Nn+p is isometric to the standard unit (n + p)-sphere. As a consequence, such an M is congruent to one of the five models as seen in our Main Theorem.

scholarly journals Space-like submanifolds with parallel mean curvature in de Sitter spaces

2000 ◽  
Vol 69 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Chongzhen Ouyang ◽  
Zhenqi Li
Keyword(s):  
Fundamental Form ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Second Fundamental Form ◽  
Mean Curvature Vector ◽  
De Sitter ◽  
Parallel Mean Curvature Vector ◽  
Complete Space ◽  
The Second Fundamental Form ◽  
Parallel Mean Curvature

AbstractThis paper investigates complete space-like submainfold with parallel mean curvature vector in the de Sitter space. Some pinching theorems on square of the norm of the second fundamental form are given

scholarly journals Global pinching theorems of submanifolds in spheres

2002 ◽  
Vol 31 (3) ◽  
pp. 183-191
Author(s):  
Kairen Cai
Keyword(s):  
Fundamental Form ◽  
Ricci Curvature ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Curvature Vector ◽  
Second Fundamental Form ◽  
Parallel Mean Curvature Vector ◽  
The Second Fundamental Form ◽  
Parallel Mean Curvature ◽  
Set Up

LetMbe a compact embedded submanifold with parallel mean curvature vector and positive Ricci curvature in the unit sphereS n+p(n≥2 ,p≥1). By using the Sobolev inequalities of P. Li (1980) toLpestimate for the square lengthσof the second fundamental form and the norm of a tensorΦ, related to the second fundamental form, we set up some rigidity theorems. Denote by‖σ‖ptheLpnorm ofσandHthe constant mean curvature ofM. It is shown that there is a constantCdepending only onn,H, andkwhere(n−1) kis the lower bound of Ricci curvature such that if‖σ‖ n/2<C, thenMis a totally umbilic hypersurface in the sphereS n+1.

Characterization of Parallel Isometric Immersions of Space Forms into Space Forms in the Class of Isotropic Immersions

2009 ◽  
Vol 61 (3) ◽  
pp. 641-655
Author(s):  
Sadahiro Maeda ◽  
Seiichi Udagawa
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Space Form ◽  
Curvature Vector ◽  
Second Fundamental Form ◽  
Isometric Immersions ◽  
Space Forms ◽  
The Second Fundamental Form ◽  
The Mean

Abstract.For an isotropic submanifold Mn (n ≧ 3) of a space form of constant sectional curvature c, we show that if the mean curvature vector of Mn is parallel and the sectional curvature K of Mn satisfies some inequality, then the second fundamental form of Mn in is parallel and our manifold Mn is a space form.

scholarly journals Geometric inequalities for CR-warped product submanifolds of locally conformal almost cosymplectic manifolds

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 741-748
Author(s):  
Akram Ali ◽  
Wan Othman ◽  
Sayyadah Qasem
Keyword(s):  
Sectional Curvature ◽  
Fundamental Form ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Integration Theory ◽  
Warping Function ◽  
Geometric Inequalities ◽  
The Second Fundamental Form ◽  
Cosymplectic Manifolds ◽  
Locally Conformal

In this paper, we establish some inequalities for the squared norm of the second fundamental form and the warping function of warped product submanifolds in locally conformal almost cosymplectic manifolds with pointwise ?-sectional curvature. The equality cases are also considered. Moreover, we prove a triviality result for CR-warped product submanifold by using the integration theory on a compact orientate manifold without boundary.

Some results on Finsler submanifolds

2016 ◽  
Vol 27 (03) ◽  
pp. 1650021
Author(s):  
B. Y. Wu
Keyword(s):  
Fundamental Form ◽  
Mean Curvature ◽  
Second Fundamental Form ◽  
Moving Frame ◽  
Chern Connection ◽  
Parallel Second Fundamental Form ◽  
The Second Fundamental Form ◽  
Compact Submanifold ◽  
Fundamental Equations ◽  
Simply Connected

In this paper we study the submanifold theory in terms of Chern connection. We introduce the notions of the second fundamental form and mean curvature for Finsler submanifolds, and establish the fundamental equations by means of moving frame for the hypersurface case. We provide the estimation of image radius for compact submanifold, and prove that there exists no compact minimal submanifold in any complete noncompact and simply connected Finsler manifold with nonpositive flag curvature. We also characterize the Minkowski hyperplanes, Minkowski hyperspheres and Minkowski cylinders as the only hypersurfaces in Minkowski space with parallel second fundamental form.

scholarly journals Complete submanifolds with parallel mean curvature in a sphere

1996 ◽  
Vol 38 (3) ◽  
pp. 343-346
Author(s):  
Theodoros Vlachos
Keyword(s):  
Mean Curvature ◽  
Vector Fields ◽  
Positive Root ◽  
Tangent Vector ◽  
Second Fundamental Form ◽  
Dimensional Manifold ◽  
Complete Submanifolds ◽  
Parallel Mean Curvature ◽  
The Mean ◽  
Image Position

Let Mn be an n-dimensional manifold immersed in an (n+p)-dimensional unit sphere Sn+p, with mean curvature H and second fundamental form B. We put φ(X, Y) = B(X, Y)–(X, Y)H where X and Y are tangent vector fields on Mn. Assume that the mean curvature is parallel in the normal bundle of Mn in Sn+p. Following Alencar and do Carmo [1] we denote by BH the square of the positive root of

scholarly journals Positively curved 6-manifolds with simple symmetry groups

2002 ◽  
Vol 74 (4) ◽  
pp. 589-597 ◽  
Author(s):  
FUQUAN FANG
Keyword(s):  
Sectional Curvature ◽  
Isometry Group ◽  
Symmetry Groups ◽  
Positive Sectional Curvature ◽  
Identity Component ◽  
Simply Connected ◽  
Lie Subgroup ◽  
Positively Curved

Let M be a simply connected compact 6-manifold of positive sectional curvature. If the identity component of the isometry group contains a simple Lie subgroup, we prove that M is diffeomorphic to one of the five manifolds listed in Theorem A.

scholarly journals LINEAR WEINGARTEN HYPERSURFACES WITH BOUNDED MEAN CURVATURE IN THE HYPERBOLIC SPACE

2014 ◽  
Vol 57 (3) ◽  
pp. 653-663 ◽  
Author(s):  
CÍCERO P. AQUINO ◽  
HENRIQUE F. DE LIMA ◽  
MARCO ANTONIO L. VELÁSQUEZ
Keyword(s):  
Scalar Curvature ◽  
Hyperbolic Space ◽  
Mean Curvature ◽  
Second Fundamental Form ◽  
Maximum Principles ◽  
Rigidity Result ◽  
The Second Fundamental Form ◽  
Linear Weingarten Hypersurfaces ◽  
Compact Case ◽  
Suitable Restriction

AbstractWe apply appropriate maximum principles in order to obtain characterization results concerning complete linear Weingarten hypersurfaces with bounded mean curvature in the hyperbolic space. By supposing a suitable restriction on the norm of the traceless part of the second fundamental form, we show that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder, when its scalar curvature is positive, or to a spherical cylinder, when its scalar curvature is negative. Related to the compact case, we also establish a rigidity result.

scholarly journals SCALAR CURVATURE OF HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN SPHERES

2011 ◽  
Vol 54 (1) ◽  
pp. 67-75 ◽  
Author(s):  
QIN ZHANG
Keyword(s):  
Scalar Curvature ◽  
Fundamental Form ◽  
Unit Sphere ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Second Fundamental Form ◽  
The Second Fundamental Form ◽  
Closed Hypersurface ◽  
Small Constant

AbstractLet Mn be an n-dimensional closed hypersurface with constant mean curvature H satisfying |H| ≤ ϵ(n) in a unit sphere Sn+1(1), n ≤ 8 and S the square of the length of the second fundamental form of M. There exists a constant δ(n, H) > 0, which depends only on n and H such that if S0 ≤ S ≤ S0 + δ(n, H), then S ≡ S0 and M is isometric to a Clifford hypersurface, where ϵ(n) is a sufficiently small constant depending on n and $S_0=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)}\sqrt{n^2H^4+4(n-1)H^2}$.

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