scholarly journals Willmore spacelike submanifolds in an indefinite space form Nn+p q(c)

2017 ◽  
Vol 102 (116) ◽  
pp. 175-193
Author(s):  
Shichang Shu ◽  
Junfeng Chen
Keyword(s):  
Space Form ◽  
Second Fundamental Form ◽  
Integral Inequalities ◽  
Willmore Functional ◽  
Lorentzian Space ◽  
The Second Fundamental Form ◽  
Spacelike Submanifold ◽  
Spacelike Submanifolds ◽  
The Mean ◽  
Rigidity Theorems

Let Nn+p q(c) be an (n+p)-dimensional connected indefinite space form of index q(1 ? q ? p) and of constant curvature c. Denote by ? : M ? Nn+p q (c) the n-dimensional spacelike submanifold in Nn+p q (c), ? : M ? Nn+p q(c) is called a Willmore spacelike submanifold in Nn+p q(c) if it is a critical submanifold to the Willmore functional W(?) = ?q M ?n dv =?M (S-nH2)n/2 dv, where S and H denote the norm square of the second fundamental form and the mean curvature of M and ?2 = S ? nH2. If q = p, in [14], we proved some integral inequalities of Simons? type and rigidity theorems for n-dimensional Willmore spacelike submanifolds in a Lorentzian space form Nn+p q(c). In this paper, we continue to study this topic and prove some integral inequalities of Simons? type and rigidity theorems for n-dimensional Willmore spacelike submanifolds in an indefinite space form Nn+p q(c) (1 ? q ? p).

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.

The normalized Ricci flow and homology in Lagrangian submanifolds of generalized complex space forms

2020 ◽  
Vol 17 (06) ◽  
pp. 2050094 ◽  
Author(s):  
Fatemah Mofarreh ◽  
Akram Ali ◽  
Wan Ainun Mior Othman
Keyword(s):  
Fundamental Form ◽  
Ricci Flow ◽  
Complex Space ◽  
Space Form ◽  
Second Fundamental Form ◽  
Complex Space Form ◽  
Standard Sphere ◽  
The Second Fundamental Form ◽  
Generalized Complex ◽  
Normalized Ricci Flow

In this paper, we prove that a simply connected Lagrangian submanifold in the generalized complex space form is diffeomorphic to standard sphere [Formula: see text] and the normalized Ricci flow converges to a constant curvature metric, provided its squared norm of the second fundamental form satisfies some upper bound depending only on the squared norm of the mean curvature vector field, the constant sectional curvature, and the dimension of the Lagrangian immersion of the ambient space. Next, we conclude that stable currents do not exist and homology groups vanish in a compact real submanifold of the general complex space form, provided that the second fundamental form satisfies some extrinsic conditions. We show that our results improve some previous results.

A note on rigidity of spacelike self-shrinkers

2020 ◽  
Vol 31 (05) ◽  
pp. 2050035
Author(s):  
Yong Luo ◽  
Hongbing Qiu
Keyword(s):  
Euclidean Space ◽  
Growth Condition ◽  
Fundamental Form ◽  
Mean Curvature ◽  
Integral Method ◽  
Acta Math ◽  
Second Fundamental Form ◽  
The Second Fundamental Form ◽  
The Mean ◽  
A Minor

By using the integral method, we prove a rigidity theorem for spacelike self-shrinkers in pseudo-Euclidean space under a minor growth condition in terms of the mean curvature and the second fundamental form, which generalizes Theorem 1.1 in [H. Q. Liu and Y. L. Xin, Some Results on Space-Like Self-Shrinkers, Acta Math. Sin. (Engl. Ser.) 32(1) (2016) 69–82].

Complete linear Weingarten spacelike submanifolds with higher codimension in the de Sitter space

2019 ◽  
Vol 16 (04) ◽  
pp. 1950050 ◽  
Author(s):  
Jogli Gidel da Silva Araújo ◽  
Henrique Fernandes de Lima ◽  
Fábio Reis dos Santos ◽  
Marco Antonio Lázaro Velásquez
Keyword(s):  
Mean Curvature ◽  
Curvature Vector ◽  
De Sitter Space ◽  
Second Fundamental Form ◽  
Index Formula ◽  
Curvature Function ◽  
De Sitter ◽  
Spacelike Submanifold ◽  
Spacelike Submanifolds ◽  
Complete Riemannian Manifolds

We study complete linear Weingarten spacelike submanifolds with arbitrary high codimension [Formula: see text] in the de Sitter space [Formula: see text] of index [Formula: see text] and whose normalized mean curvature vector is parallel. Under suitable restrictions on the values of the mean curvature function and on the norm of the traceless part of the second fundamental form, we prove that such a spacelike submanifold must be either totally umbilical or isometric to a certain hyperbolic cylinder of [Formula: see text]. Our approach is based on the use of a Simons type formula related to an appropriate Cheng–Yau modified operator jointly with an extension of Hopf’s maximum principle for complete Riemannian manifolds.

scholarly journals Self-Expanders of the Mean Curvature Flow

2021 ◽  
Author(s):  
Knut Smoczyk
Keyword(s):  
Scalar Curvature ◽  
Fundamental Form ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Vector ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
The Second Fundamental Form ◽  
The Mean ◽  
Mean Convex

AbstractWe study self-expanding solutions $M^{m}\subset \mathbb {R}^{n}$ M m ⊂ ℝ n of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are products of self-expanding curves and flat subspaces, if and only if the function |A|2/|H|2 attains a local maximum, where A denotes the second fundamental form and H the mean curvature vector of M. If the principal normal ξ = H/|H| is parallel in the normal bundle, then a similar result holds in higher codimension for the function |Aξ|2/|H|2, where Aξ is the second fundamental form with respect to ξ. As a corollary we obtain that complete mean convex self-expanders attain strictly positive scalar curvature, if they are smoothly asymptotic to cones of non-negative scalar curvature. In particular, in dimension 2 any mean convex self-expander that is asymptotic to a cone must be strictly convex.

scholarly journals Surfaces with Mean Curvature Vector Parallel in the Normal Bundle

1972 ◽  
Vol 47 ◽  
pp. 161-167 ◽  
Author(s):  
Bang-Yen Chen ◽  
Gerald D. Ludden
Keyword(s):  
Mean Curvature ◽  
Tangent Space ◽  
Curvature Vector ◽  
Second Fundamental Form ◽  
Normal Space ◽  
Mean Curvature Vector ◽  
Bilinear Mapping ◽  
The Second Fundamental Form ◽  
Connected Surface ◽  
The Mean

Let M be a connected surface immersed in a Euclidean m-space Em. Let h be the second fundamental form of this immersion it is a certain symmetric bilinear mapping for X ∈ M, where Tx is the tangent space and the normal space of M at x. Let H be the mean curvature vector of M in Em. If there exists a real λ such that for all tangent vectors X, Y in Tx, then ilf is said to be pseudo-umbilical at x.

Horizontal Newton operators and high-order Minkowski formula

2020 ◽  
pp. 2050004
Author(s):  
Chiara Guidi ◽  
Vittorio Martino
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Real Space ◽  
Second Fundamental Form ◽  
Space Forms ◽  
Nonlinear Operators ◽  
The Second Fundamental Form ◽  
Real Space Forms ◽  
Newton Transformations ◽  
Minkowski Formula

In this paper, we study the horizontal Newton transformations, which are nonlinear operators related to the natural splitting of the second fundamental form for hypersurfaces in a complex space form. These operators allow to prove the classical Minkowski formulas in the case of real space forms: unlike the real case, the horizontal ones are not divergence-free. Here, we consider the highest order of nonlinearity and we will show how a Minkowski-type formula can be obtained in this case.

ANCIENT SOLUTIONS OF CODIMENSION TWO SURFACES WITH CURVATURE PINCHING

2020 ◽  
Vol 102 (1) ◽  
pp. 162-171
Author(s):  
ZHENGCHAO JI
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Geometric Flows ◽  
Codimension Two ◽  
The Second Fundamental Form ◽  
Ancient Solutions ◽  
The Mean ◽  
Curvature Pinching

We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find conditions on the second fundamental form that characterise the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension two surfaces.

scholarly journals On contact CR-submanifolds of sasakian manifolds

1983 ◽  
Vol 6 (2) ◽  
pp. 313-326
Author(s):  
Koji Matsumoto
Keyword(s):  
Space Form ◽  
Sasakian Manifold ◽  
Second Fundamental Form ◽  
Sasakian Space Form ◽  
Sasakian Manifolds ◽  
Fundamental Properties ◽  
The Second Fundamental Form ◽  
Ambient Manifold ◽  
The One ◽  
Kaehlerian Manifold

Recently, K.Yano and M.Kon [5] have introduced the notion of a contactCR-submanifold of a Sasakian manifold which is closely similar to the one of aCR-submanifold of a Kaehlerian manifold defined by A. Bejancu [1].In this paper, we shall obtain some fundamental properties of contactCR-submanifolds of a Sasakian manifold. Next, we shall calculate the length of the second fundamental form of a contactCR-product of a Sasakian space form (THEOREM 7.4). At last, we shall prove that a totally umbilical contactCR-submanifold satisfying certain conditions is totally geodesic in the ambient manifold (THEOREM 8.1).

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