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 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.

scholarly journals A Closed Form for Slant Submanifolds of Generalized Sasakian Space Forms

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1238
Author(s):  
Pablo Alegre ◽  
Joaquín Barrera ◽  
Alfonso Carriazo
Keyword(s):  
Closed Form ◽  
Mean Curvature ◽  
Second Fundamental Form ◽  
Space Forms ◽  
Equality Case ◽  
Slant Submanifolds ◽  
The Second Fundamental Form ◽  
Sasakian Space Forms ◽  
Maslov Form ◽  
The Mean

The Maslov form is a closed form for a Lagrangian submanifold of C m , and it is a conformal form if and only if M satisfies the equality case of a natural inequality between the norm of the mean curvature and the scalar curvature, and it happens if and only if the second fundamental form satisfies a certain relation. In a previous paper we presented a natural inequality between the norm of the mean curvature and the scalar curvature of slant submanifolds of generalized Sasakian space forms, characterizing the equality case by certain expression of the second fundamental form. In this paper, first, we present an adapted form for slant submanifolds of a generalized Sasakian space form, similar to the Maslov form, that is always closed. And, in the equality case, we studied under which circumstances the given closed form is also conformal.

scholarly journals Twisted product CR-submanifolds in a locally conformal Kähler space forms

Filomat ◽  
2020 ◽  
Vol 34 (6) ◽  
pp. 1915-1925
Author(s):  
Vittoria Bonanzinga ◽  
Koji Matsumoto
Keyword(s):  
Fundamental Form ◽  
Mean Curvature ◽  
Second Fundamental Form ◽  
Twisted Product ◽  
Space Forms ◽  
The Second Fundamental Form ◽  
Locally Conformal Kähler ◽  
The Mean ◽  
Locally Conformal ◽  
Cr Submanifolds

Certain twisted product CR-submanifolds in a K?hler manifold and some inequalities of the second fundamental form of these submanifolds are presented ([14]). Then the length of the second fundamental form of a twisted product CR-submanifold in a locally conformal K?hler manifold is considered (2013), ([15]). In this paper, we consider the relation of the mean curvature and the length of the second fundamental form in two twisted product CR-submanifolds in a locally conformal K?hler space forms.

scholarly journals A basic inequality for submanifolds in a cosymplectic space form

2003 ◽  
Vol 2003 (9) ◽  
pp. 539-547 ◽  
Author(s):  
Jeong-Sik Kim ◽  
Jaedong Choi
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Sectional Curvature ◽  
Mean Curvature ◽  
Space Form ◽  
The Other ◽  
Space Forms ◽  
Basic Inequality

For submanifolds tangent to the structure vector field in cosymplectic space forms, we establish a basic inequality between the main intrinsic invariants of the submanifold, namely, its sectional curvature and scalar curvature on one side; and its main extrinsic invariant, namely, squared mean curvature on the other side. Some applications, including inequalities between the intrinsic invariantδMand the squared mean curvature, are given. The equality cases are also discussed.

scholarly journals On totally umbilicalCR-submanifolds of a Kaehler manifold

1993 ◽  
Vol 16 (2) ◽  
pp. 405-408
Author(s):  
M. A. Bashir
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Lens Space ◽  
Holomorphic Sectional Curvature ◽  
Mean Curvature Vector ◽  
Kaehler Manifold ◽  
Totally Umbilical ◽  
The Mean

LetMbe a compact3-dimensional totally umbilicalCR-submanifold of a Kaehler manifold of positive holomorphic sectional curvature. We prove that if the length of the mean curvature vector ofMdoes not vanish, thenMis either diffeomorphic toS3orRP3or a lens spaceLp,q3.

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 Isometric immersions in the hyperbolic space with their image contained in a horoball

2001 ◽  
Vol 43 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Ana Lluch
Keyword(s):  
Hyperbolic Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Complete Riemannian Manifold ◽  
Isometric Immersions ◽  
Special Groups ◽  
The Mean ◽  
Complex Hyperbolic Spaces ◽  
Real Hyperbolic Space ◽  
Sharp Lower Bound

We give a sharp lower bound for the supremum of the norm of the mean curvature of an isometric immersion of a complete Riemannian manifold with scalar curvature bounded from below into a horoball of a complex or real hyperbolic space. We also characterize the horospheres of the real or complex hyperbolic spaces as the only isometrically immersed hypersurfaces which are between two parallel horospheres, have the norm of the mean curvature vector bounded by the above sharp bound and have some special groups of symmetries.

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].

Biharmonic hypersurfaces in pseudo-Riemannian space forms

2016 ◽  
Vol 13 (07) ◽  
pp. 1650094 ◽  
Author(s):  
Dan Yang ◽  
Yu Fu
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Constant Mean Curvature ◽  
Space Form ◽  
Constant Sectional Curvature ◽  
Principal Curvatures ◽  
Space Forms ◽  
Riemannian Space Forms

Let [Formula: see text] be a nondegenerate biharmonic pseudo-Riemannian hypersurface in a pseudo-Riemannian space form [Formula: see text] with constant sectional curvature [Formula: see text]. We show that [Formula: see text] has constant mean curvature provided that it has three distinct principal curvatures and the Weingarten operator can be diagonalizable.

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