scholarly journals Classification of totally umbilical slant submanifolds of a Kenmotsu manifold

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2405-2412 ◽  
Author(s):  
Siraj Uddin ◽  
Zafar Ahsan ◽  
Yaakub Hadi
Keyword(s):  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Slant Submanifold ◽  
Kenmotsu Manifold ◽  
Totally Geodesic ◽  
Totally Umbilical ◽  
Slant Submanifolds ◽  
The Mean

The purpose of this paper is to classify totally umbilical slant submanifolds of a Kenmotsu manifold. We prove that a totally umbilical slant submanifold M of a Kenmotsu manifold ?M is either invariant or anti-invariant or dimM = 1 or the mean curvature vector H of M lies in the invariant normal subbundle. Moreover, we find with an example that every totally umbilical proper slant submanifold is totally geodesic.

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 COMPLETE SPACELIKE SUBMANIFOLDS IN DE SITTER SPACES WITH

2013 ◽  
Vol 87 (3) ◽  
pp. 386-399 ◽  
Author(s):  
JIANCHENG LIU ◽  
JINGJING ZHANG
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Linear Relation ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
De Sitter ◽  
Spacelike Submanifolds ◽  
The Mean

AbstractIn this paper, we give a classification of spacelike submanifolds with parallel normalised mean curvature vector field and linear relation$R= aH+ b$of the normalised scalar curvature$R$and the mean curvature$H$in the de Sitter space${ S}_{p}^{n+ p} (c)$.

scholarly journals Umbilical points on surfaces in RN

1985 ◽  
Vol 100 ◽  
pp. 135-143 ◽  
Author(s):  
Kazuyuki Enomoto
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Normal Connection ◽  
Round Sphere ◽  
Positive Gaussian Curvature ◽  
Connected Surface ◽  
Umbilical Points ◽  
The Mean

Let ϕ: M → RN be an isometric imbedding of a compact, connected surface M into a Euclidean space RN. ψ is said to be umbilical at a point p of M if all principal curvatures are equal for any normal direction. It is known that if the Euler characteristic of M is not zero and N = 3, then ψ is umbilical at some point on M. In this paper we study umbilical points of surfaces of higher codimension. In Theorem 1, we show that if M is homeomorphic to either a 2-sphere or a 2-dimensional projective space and if the normal connection of ψ is flat, then ψ is umbilical at some point on M. In Section 2, we consider a surface M whose Gaussian curvature is positive constant. If the surface is compact and N = 3, Liebmann’s theorem says that it must be a round sphere. However, if N ≥ 4, the surface is not rigid: For any isometric imbedding Φ of R3 into R4 Φ(S2(r)) is a compact surface of constant positive Gaussian curvature 1/r2. We use Theorem 1 to show that if the normal connection of ψ is flat and the length of the mean curvature vector of ψ is constant, then ψ(M) is a round sphere in some R3 ⊂ RN. When N = 4, our conditions on ψ is satisfied if the mean curvature vector is parallel with respect to the normal connection. Our theorem fails if the surface is not compact, while the corresponding theorem holds locally for a surface with parallel mean curvature vector (See Remark (i) in Section 3).

scholarly journals On pinching theorems for compact pseudo-umbilical submanifold

2012 ◽  
Vol 45 (3) ◽  
pp. 645-654
Author(s):  
Jing Mao ◽  
Shaodong Qin
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Mean Curvature ◽  
Sufficient Conditions ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Parallel Mean Curvature Vector ◽  
Totally Umbilical ◽  
Totally Umbilical Submanifold ◽  
Parallel Mean Curvature

AbstractConsider submanifolds in the nested space. For a compact pseudoumbilical submanifold with parallel mean curvature vector of a Riemannian submanifold with constant curvature immersed in a quasi-constant curvature Riemannian manifold, two sufficient conditions are given to let the pseudo-umbilical submanifold become a totally umbilical submanifold.

scholarly journals Pseudo-Reimannian manifolds endowed with an almost paraf-structure

1985 ◽  
Vol 8 (2) ◽  
pp. 257-266 ◽  
Author(s):  
Vladislav V. Goldberg ◽  
Radu Rosca
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Mean Curvature ◽  
Integral Manifold ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Minimal Hypersurfaces ◽  
Reimannian Manifolds ◽  
The Mean ◽  
Orthogonal Distribution

LetM˜(U,Ω˜,η˜,ξ,g˜)be a pseudo-Riemannian manifold of signature(n+1,n). One defines onM˜an almost cosymplectic paraf-structure and proves that a manifoldM˜endowed with such a structure isξ-Ricci flat and is foliated by minimal hypersurfaces normal toξ, which are of Otsuki's type. Further one considers onM˜a2(n−1)-dimensional involutive distributionP⊥and a recurrent vector fieldV˜. It is proved that the maximal integral manifoldM⊥ofP⊥hasVas the mean curvature vector (up to1/2(n−1)). If the complimentary orthogonal distributionPofP⊥is also involutive, then the whole manifoldM˜is foliate. Different other properties regarding the vector fieldV˜are discussed.

Complete spacelike submanifolds in a locally symmetric semi-Riemannian space

2020 ◽  
Vol 31 (04) ◽  
pp. 2050033
Author(s):  
Shicheng Zhang ◽  
Yuntao Zhang
Keyword(s):  
Maximum Principle ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Curvature Vector ◽  
Isoparametric Hypersurface ◽  
Mean Curvature Vector ◽  
Totally Umbilical ◽  
Type Formula ◽  
Spacelike Submanifolds ◽  
Generalized Maximum Principle

In this paper, complete spacelike submanifolds with parallel normalized mean curvature vector are investigated in semi-Riemannian space obeying some standard curvature conditions. In this setting, we obtain a suitable Simons type formula and apply it jointly with the well-known generalized maximum principle of Omori–Yau to show that it must be totally umbilical submanifold or isometric to an isoparametric hypersurface in a submanifold [Formula: see text] of [Formula: see text].

scholarly journals Totally real surfaces inCP2with parallel mean curvature vector

1992 ◽  
Vol 15 (3) ◽  
pp. 589-592
Author(s):  
M. A. Al-Gwaiz ◽  
Sharief Deshmukh
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Real Surface ◽  
Mean Curvature Vector ◽  
Parallel Mean Curvature Vector ◽  
Constant Gaussian Curvature ◽  
Totally Geodesic ◽  
Parallel Mean Curvature ◽  
Totally Real

It has been shown that a totally real surface inCP2with parallel mean curvature vector and constant Gaussian curvature is either flat or totally geodesic.

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