Hypersurfaces in a conformally flat space

2021 ◽  
pp. 2150067
Author(s):  
Sabina Eyasmin
Keyword(s):  
Space Form ◽  
Sufficient Conditions ◽  
Shape Operator ◽  
Flat Space ◽  
Conformally Flat ◽  
Flat Manifold ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Conformally Flat Manifold ◽  
Proper Generalization

The hypersurface of a space is one of the most important objects in a space. Many authors studied the various geometric aspects of hypersurfaces in a space form. The notion of conformal flatness is one of the most primitive concepts in differential geometry. Again, conformally flat space is a proper generalization of a space form. In this paper, we study the geometry of hypersurfaces in a conformally flat manifold. Then we have investigated some sufficient conditions imposed on the shape operator for which the hypersurface satisfies various pseudosymmetric-type conditions imposed on its conformal curvature tensor.

scholarly journals On the quasi-conformal curvature tensor of an almost Kenmotsu manifold with nullity distributions

2018 ◽  
Vol 33 (2) ◽  
pp. 255
Author(s):  
Dibakar Dey ◽  
Pradip Majhi
Keyword(s):  
Vector Field ◽  
Curvature Tensor ◽  
Conformally Flat ◽  
Kenmotsu Manifold ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Kenmotsu Manifolds ◽  
Almost Kenmotsu Manifolds ◽  
Nullity Distribution ◽  
Characteristic Vector Field

The object of the present paper is to characterize quasi-conformally flat and $\xi$-quasi-conformally flat almost Kenmotsu manifolds with  $(k,\mu)$-nullity and $(k,\mu)'$-nullity distributions respectively. Also we characterize almost Kenmotsu manifolds with vanishing extended quasi-conformal curvature tensor and extended $\xi$-quasi-conformally flat almost Kenmotsu manifolds such that the characteristic vector field $\xi$ belongs to the $(k,\mu)$-nullity distribution.

scholarly journals REMARKS ON BIANCHI SUMS AND PONTRJAGIN CLASSES

2014 ◽  
Vol 97 (3) ◽  
pp. 365-382 ◽  
Author(s):  
MOHAMMED LARBI LABBI
Keyword(s):  
Riemannian Curvature ◽  
Conformally Flat ◽  
Flat Manifold ◽  
Equality Case ◽  
Conformally Flat Manifold

AbstractWe use the exterior and composition products of double forms together with the alternating operator to reformulate Pontrjagin classes and all Pontrjagin numbers in terms of the Riemannian curvature. We show that the alternating operator is obtained by a succession of applications of the first Bianchi sum and we prove some useful identities relating the previous four operations on double forms. As an application, we prove that for a $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}k$-conformally flat manifold of dimension $n\geq 4k$, the Pontrjagin classes $P_i$ vanish for any $i\geq k$. Finally, we study the equality case in an inequality of Thorpe between the Euler–Poincaré characteristic and the $k{\rm th}$ Pontrjagin number of a $4k$-dimensional Thorpe manifold.

scholarly journals Some Curvature Properties on Lorentzian Generalized Sasakian-Space-Forms

2019 ◽  
Vol 2019 ◽  
pp. 1-7 ◽  
Author(s):  
Rongsheng Ma ◽  
Donghe Pei
Keyword(s):  
Space Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Conformally Flat ◽  
Sasakian Space Form ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
Projectively Flat ◽  
Generalized Sasakian Space Form ◽  
Necessary And Sufficient

In this paper, we investigate the Lorentzian generalized Sasakian-space-form. We give the necessary and sufficient conditions for the Lorentzian generalized Sasakian-space-form to be projectively flat, conformally flat, conharmonically flat, and Ricci semisymmetric and their relationship between each other. As the application of our theorems, we study the Ricci almost soliton on conformally flat Lorentzian generalized Sasakian-space-form.

scholarly journals Some Curvature Properties of D-conformal Curvature Tensor on LP-Sasakian Manifolds

2015 ◽  
Vol 19 (1) ◽  
pp. 30-34
Author(s):  
Riddhi Jung Shah
Keyword(s):  
Curvature Tensor ◽  
Science And Technology ◽  
Sasakian Manifold ◽  
Conformally Flat ◽  
Sasakian Manifolds ◽  
Curvature Condition ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor

This paper deals with the study of geometry of Lorentzian para-Sasakian manifolds. We investigate some properties of D-conformally flat, D-conformally semi-symmetric, Xi-D-conformally flat and Phi-D-conformally flat curvature conditions on Lorentzian para-Sasakian manifolds. Also it is proved that in each curvature condition an LP-Sasakian manifold (Mn,g)(n>3) is an eta-Einstein manifold.Journal of Institute of Science and Technology, 2014, 19(1): 30-34

scholarly journals QUASI-CONFORMAL CURVATURE TENSOR OF GENERALIZED SASAKIAN-SPACE-FORMS

2020 ◽  
Vol 35 (1) ◽  
pp. 089
Author(s):  
Braj B. Chaturvedi ◽  
Brijesh K. Gupta
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Space Form ◽  
Riemannian Curvature ◽  
Space Forms ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Sasakian Space Forms ◽  
Riemannian Curvature Tensor

The present paper deals the study of generalised Sasakian-space-forms with the conditions Cq(ξ,X).S = 0, Cq(ξ,X).R = 0 and Cq(ξ,X).Cq = 0, where R, S and Cq denote Riemannian curvature tensor, Ricci tensor and quasi-conformal curvature tensor of the space-form, respectively and at last, we have given some examples to improve our results.

scholarly journals On contact conformal curvature tensor in LP-Sasakian manifolds

BIBECHANA ◽  
2017 ◽  
Vol 15 ◽  
pp. 24-29
Author(s):  
Riddhi Jung Shah
Keyword(s):  
Curvature Tensor ◽  
Conformally Flat ◽  
Sasakian Manifolds ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor

The purpose of the present paper is to study the contact conformal curvature tensor in LP-Sasakian manifolds. Some properties of contact conformally flat, ξ -contact conformally flat and contact conformally semi-symmetric LP-Sasakian manifolds are obtained.BIBECHANA 15 (2018) 24-29

scholarly journals Hypersurfaces in a conformally flat space with curvature collineation

1991 ◽  
Vol 14 (3) ◽  
pp. 595-604
Author(s):  
K. L. Duggal ◽  
R. Sharma
Keyword(s):  
Fundamental Problem ◽  
Energy Condition ◽  
Shape Operator ◽  
Flat Space ◽  
Space Time ◽  
Conformally Flat ◽  
Direct Relation ◽  
Shape Operators

We classify the shape operators of Einstein and pseudo Einstein hypersurfaces in a conformally flat space with a symmetry called curvature collineation. We solve the fundamental problem of finding all possible forms of non-diagonalizable shape operators. A physical example of space-time with matter is presented to show that the energy condition has direct relation with the diagonalizability of shape operator.

scholarly journals On contact conformal curvature tensor in trans-Sasakian manifolds

BIBECHANA ◽  
2014 ◽  
Vol 12 ◽  
pp. 80-88
Author(s):  
Riddhi Jung Shah
Keyword(s):  
Curvature Tensor ◽  
Sasakian Manifold ◽  
Conformally Flat ◽  
Sasakian Manifolds ◽  
3 Dimensional ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
In Trans

The purpose of this paper is to study some results on contact conformal curvature tensor in trans-Sasakian manifolds. Contact conformally flat trans-Sasakian manifold, ζ-contact conformally flat trans-Sasakian manifold and curvature conditions C0(ζ.X).S = 0 and C0(ζ.X).C0 = 0 are studied with some interesting results. Finally, we study an example of 3-dimensional trans-Sasakian manifold. DOI: http://dx.doi.org/10.3126/bibechana.v12i0.11783  BIBECHANA 12 (2015) 80-88

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