scholarly journals Some Properties Curvture of Lorentzian Kenmotsu Manifolds

2020 ◽  
Vol 5 (1) ◽  
pp. 283-292
Author(s):  
Ramazan Sari
Keyword(s):  
Sectional Curvature ◽  
Ricci Tensor ◽  
Space Form ◽  
Einstein Manifold ◽  
Holomorphic Sectional Curvature ◽  
Invariant Submanifold ◽  
Kenmotsu Manifolds ◽  
Invariant Submanifolds ◽  
Curvature Tensors ◽  
Kenmotsu Space Form

AbstractIn this paper different curvature tensors on Lorentzian Kenmotsu manifod are studied. We investigate constant ϕ–holomorphic sectional curvature and ℒ-sectional curvature of Lorentzian Kenmotsu manifolds, obtaining conditions for them to be constant of Lorentzian Kenmotsu manifolds in such condition. We calculate the Ricci tensor and scalar curvature for all the cases. Moreover we investigate some properties of semi invariant submanifolds of a Lorentzian Kenmotsu space form. We show that if a semi-invariant submanifold of a Lorentzian Kenmotsu space form M is totally geodesic, then M is an η−Einstein manifold. We consider sectional curvature of semi invariant product of a Lorentzian Kenmotsu manifolds.

scholarly journals On para-Kenmotsu manifolds

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 4971-4980 ◽  
Author(s):  
Simeon Zamkovoy
Keyword(s):  
Sectional Curvature ◽  
Ricci Tensor ◽  
Constant Scalar Curvature ◽  
Holomorphic Sectional Curvature ◽  
Kenmotsu Manifold ◽  
3 Dimensional ◽  
Kenmotsu Manifolds ◽  
Locally Conformally Flat ◽  
Parallel Ricci Tensor ◽  
Negative Sectional Curvature

In this paper we study para-Kenmotsu manifolds. We characterize this manifolds by tensor equations and study their properties. We are devoted to a study of ?-Einstein manifolds. We show that a locally conformally flat para-Kenmotsu manifold is a space of constant negative sectional curvature -1 and we prove that if a para-Kenmotsu manifold is a space of constant ?-para-holomorphic sectional curvature H, then it is a space of constant sectional curvature and H = -1. Finally the object of the present paper is to study a 3-dimensional para-Kenmotsu manifold, satisfying certain curvature conditions. Among other, it is proved that any 3-dimensional para-Kenmotsu manifold with ?-parallel Ricci tensor is of constant scalar curvature and any 3-dimensional para-Kenmotsu manifold satisfying cyclic Ricci tensor is a manifold of constant negative sectional curvature -1.

Non-immersibility of a space form as a totally umbilical hypersurface

1988 ◽  
Vol 109 (1-2) ◽  
pp. 17-21
Author(s):  
Masafumi Okumura ◽  
Hiroshi Takahashi
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Sectional Curvature ◽  
Mean Curvature ◽  
Ricci Tensor ◽  
Constant Mean Curvature ◽  
Space Form ◽  
Einstein Manifold ◽  
Totally Umbilical ◽  
Ambient Manifold

SynopsisSuppose that a space form is immersed into another Riemannian manifold as a totally umbilical hypersurface with constant mean curvature. Then, in the ambient manifold, the lengthof the curvature tensor, that of the Ricci tensor and the scalar curvature must satisfy an inequality. In this paper the authors proved the inequality. As applications of the inequality, some immersibility problems are investigated. For example, it is proved that if a space form is immersed in an Einstein manifold as a totally umbilical hypersurface, then the Einstein manifold has constant sectional curvature along the hypersurface. Moreover, it is proved that a space form cannot be immersed into some Kaehlerian manifolds as a totally umbilical hypersurface with constant mean curvature.

A note on invariant submanifolds of CR-integrable almost Kenmotsu manifolds

Filomat ◽  
2017 ◽  
Vol 31 (19) ◽  
pp. 6211-6218 ◽  
Author(s):  
Young Suh ◽  
Krishanu Mandal ◽  
Uday De
Keyword(s):  
Invariant Submanifold ◽  
Kenmotsu Manifold ◽  
Totally Geodesic ◽  
Almost Kenmotsu Manifold ◽  
Kenmotsu Manifolds ◽  
Almost Kenmotsu Manifolds ◽  
Invariant Submanifolds

The present paper deals with invariant submanifolds of CR-integrable almost Kenmotsu manifolds. Among others it is proved that every invariant submanifold of a CR-integrable (k,?)'-almost Kenmotsu manifold with k < -1 is totally geodesic. Finally, we construct an example of an invariant submanifold of a CR-integrable (k,?)'-almost Kenmotsu manifold which is totally geodesic.

scholarly journals Strongly pseudo-convex CR space forms

2019 ◽  
Vol 6 (1) ◽  
pp. 279-293 ◽  
Author(s):  
Jong Taek Cho
Keyword(s):  
Sectional Curvature ◽  
Hyperbolic Space ◽  
Space Form ◽  
Contact Manifold ◽  
Holomorphic Sectional Curvature ◽  
Sphere Bundle ◽  
Sasakian Space Form ◽  
Space Forms ◽  
Pseudo Convex ◽  
Unit Tangent Sphere Bundle

AbstractFor a contact manifold, we study a strongly pseudo-convex CR space form with constant holomorphic sectional curvature for the Tanaka-Webster connection. We prove that a strongly pseudo-convex CR space form M is weakly locally pseudo-Hermitian symmetric if and only if (i) dim M = 3, (ii) M is a Sasakian space form, or (iii) M is locally isometric to the unit tangent sphere bundle T1(𝔿n+1) of a hyperbolic space 𝔿n+1 of constant curvature −1.

scholarly journals Some curvature tensors in N(k)-contact metric manifold

BIBECHANA ◽  
2018 ◽  
Vol 16 ◽  
pp. 55-63
Author(s):  
Riddhi Jung Shah
Keyword(s):  
Ricci Tensor ◽  
Einstein Manifold ◽  
Contact Metric Manifold ◽  
3 Dimensional ◽  
Contact Metric Manifolds ◽  
Curvature Tensors

The purpose of this paper is to study W7and W9-curvature tensors on N(k)-contact metric manifolds. We prove that a N(k)-contact metric manifold satisfying the condition W7( xi,X).W9=0 is eta-Einstein manifold. We also obtain the Ricci tensor S of type (0, 2) for phi-W9flat and divW9=0 conditions on N(k)-contact metric manifolds. Finally, we give an example of 3-dimensional N(k)-contact metric manifold.BIBECHANA 16 (2019) 55-63

On Some Aspects of Geometry of Almost C(λ)-manifolds

2020 ◽  
pp. 19-24
Author(s):  
A.R. Rustanov ◽  
E.A. Polkina ◽  
S.V. Kharitonova
Keyword(s):  
Constant Curvature ◽  
Local Structure ◽  
Ricci Tensor ◽  
Einstein Manifold ◽  
Sufficient Conditions ◽  
Holomorphic Sectional Curvature ◽  
Riemannian Curvature ◽  
Riemannian Curvature Tensor ◽  
Analytic Expressions ◽  
Necessary And Sufficient

In this paper almost C(λ)-manifolds are considered. The local structure of Ricci-flat almost C(λ)-manifolds is obtained. On the space of the adjoint G-structure, necessary and sufficient conditions are obtained under which the al-most C(λ)-manifolds are manifolds of constant curvature and the structure of the Riemannian curvature tensor of an almost C(λ)-manifold of constant curvature is obtained. Relations are obtained that characterize the Einstein almost C(λ)-manifolds. It is proved that a complete almost C(λ)-Einstein manifold is either holomorphically isometrically covered by the product of a real line by a Ricciflat Kähler manifold, or is compact and has a finite fundamental group. For almost C(λ)-manifolds that are -Einstein, analytic expressions for the functions  and  characterizing these manifolds are obtained. It is shown that an almost C(λ)-manifold has an Ф-invariant Ricci tensor. We study also almost C(λ)-manifolds of pointwise constant Ф-holomorphic sectional curvature.

scholarly journals Constancy of -Holomorphic Sectional Curvature for an Indefinite Generalized -Space Form

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Jae Won Lee
Keyword(s):  
Sectional Curvature ◽  
Space Form ◽  
Sasakian Manifold ◽  
Algebraic Characterization ◽  
Holomorphic Sectional Curvature ◽  
Space Forms

Bonome et al., 1997, provided an algebraic characterization for an indefinite Sasakian manifold to reduce to a space of constant -holomorphic sectional curvature. In this present paper, we generalize the same characterization for indefinite -space forms.

On Some Complex Submanifolds in Kaehler Manifolds

1974 ◽  
Vol 26 (6) ◽  
pp. 1442-1449 ◽  
Author(s):  
Masahiro Kon
Keyword(s):  
Sectional Curvature ◽  
Ricci Tensor ◽  
Holomorphic Sectional Curvature ◽  
Kaehler Manifold ◽  
Ricci Operator ◽  
Kaehler Manifolds ◽  
Complex Submanifold ◽  
Complex Submanifolds ◽  
Parallel Ricci Tensor ◽  
Local Result

The purpose of this paper is to give some conditions for complex submanifolds in a Kaehler manifold of constant holomorphic sectional curvature to be Einstein.For a complex hypersurface which is Einstein, Smyth [8] has obtained its classification and Chern [2] has proved the corresponding local result. Moreover, Takahashi [9] and Nomizu-Smyth [3] generalized this to a complex hypersurface with parallel Ricci tensor. We shall consider a condition weaker than the requirement that the Ricci tensor be parallel, that is we shall consider a complex submanifold with commuting curvature and Ricci operator, which condition was treated by Bishop-Goldberg [1].

scholarly journals A note on Chen's basic equality for submanifolds in a Sasakian space form

2003 ◽  
Vol 2003 (11) ◽  
pp. 711-716 ◽  
Author(s):  
Mukut Mani Tripathi ◽  
Jeong-Sik Kim ◽  
Seon-Bu Kim
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Sectional Curvature ◽  
Space Form ◽  
Sasakian Space Form ◽  
Invariant Submanifold ◽  
Basic Equality

It is proved that a Riemannian manifoldMisometrically immersed in a Sasakian space formM˜(c)of constantφ-sectional curvaturec<1, with the structure vector fieldξtangent toM, satisfies Chen's basic equality if and only if it is a3-dimensional minimal invariant submanifold.

scholarly journals CR-submanifolds of a locally conformal Kaehler space form

1994 ◽  
Vol 17 (3) ◽  
pp. 511-514
Author(s):  
M. Hasan Shahid
Keyword(s):  
Scalar Curvature ◽  
Sectional Curvature ◽  
Ricci Curvature ◽  
Space Form ◽  
Holomorphic Sectional Curvature ◽  
Totally Geodesic ◽  
Locally Conformal ◽  
Cr Submanifolds

(Bejancu [1,2]) The purpose of this paper is to continue the study ofCR-submanifolds, and in particular of those of a locally conformal Kaehler space form (Matsumoto [3]). Some results on the holomorphic sectional curvature,D-totally geodesic,D1-totally geodesic andD1-minimalCR-submanifolds of locally conformal Kaehler (1.c.k.)-space fromM¯(c)are obtained. We have also discussed Ricci curvature as well as scalar curvature ofCR-submanifolds ofM¯(c).

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