scholarly journals A new curvaturelike tensor field in an almost contact Riemannian manifold II

2018 ◽  
Vol 103 (117) ◽  
pp. 113-128 ◽  
Author(s):  
Koji Matsumoto
Keyword(s):  
Riemannian Manifold ◽  
Curvature Tensor ◽  
Tensor Field ◽  
Sasakian Manifold ◽  
Tensor Fields ◽  
Geometrical Properties ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Riemannian Curvature Tensor ◽  
Contact Riemannian Manifold

In the last paper, we introduced a new curvaturlike tensor field in an almost contact Riemannian manifold and we showed some geometrical properties of this tensor field in a Kenmotsu and a Sasakian manifold. In this paper, we define another new curvaturelike tensor field, named (CHR)3-curvature tensor in an almost contact Riemannian manifold which is called a contact holomorphic Riemannian curvature tensor of the second type. Then, using this tensor, we mainly research (CHR)3-curvature tensor in a Sasakian manifold. Then we define the notion of the flatness of a (CHR)3-curvature tensor and we show that a Sasakian manifold with a flat (CHR)3-curvature tensor is flat. Next, we introduce the notion of (CHR)3-?-Einstein in an almost contact Riemannian manifold. In particular, we show that Sasakian (CHR)3- ?-Einstein manifold is ?-Einstain. Moreover, we define the notion of (CHR)3- space form and consider this in a Sasakian manifold. Finally, we consider a conformal transformation of an almost contact Riemannian manifold and we get new invariant tensor fields (not the conformal curvature tensor) under this transformation. Finally, we prove that a conformally (CHR)3-flat Sasakian manifold does not exist.

scholarly journals A (CHR)3-flat trans-Sasakian manifold

2019 ◽  
Vol 12 (2) ◽  
Author(s):  
Koji Matsumoto
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Curvature Tensor ◽  
Ricci Tensor ◽  
Tensor Field ◽  
Einstein Manifold ◽  
Sasakian Manifold ◽  
Hermitian Manifolds ◽  
Riemannian Curvature Tensor ◽  
Contact Riemannian Manifold

In [4] M. Prvanovic considered several curvaturelike tensors defined for Hermitian manifolds. Developing her ideas in [3], we defined in an almost contact Riemannian manifold another new curvaturelike tensor field, which is called a contact holomorphic Riemannian curvature tensor or briefly (CHR)3-curvature tensor. Then, we mainly researched (CHR)3-curvature tensor in a Sasakian manifold. Also we proved, that a conformally (CHR)3-flat Sasakian manifold does not exist. In the present paper, we consider this tensor field in a trans-Sasakian manifold. We calculate the (CHR)3-curvature tensor in a trans-Sasakian manifold. Also, the (CHR)3-Ricci tensor ρ3  and the (CHR)3-scalar curvature τ3  in a trans-Sasakian manifold have been obtained. Moreover, we define the notion of the (CHR)3-flatness in an almost contact Riemannian manifold. Then, we consider this notion in a trans-Sasakian manifold and determine the curvature tensor, the Ricci tensor and the scalar curvature. We proved that a (CHR)3-flat trans-Sasakian manifold is a generalized   ɳ-Einstein manifold. Finally, we obtain the expression of the curvature tensor with respect to the Riemannian metric g of a trans-Sasakian manifold, if the latter is (CHR)3-flat.

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 Quarter-Symmetric Nonmetric Connection on P-Sasakian Manifolds

ISRN Geometry ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Abul Kalam Mondal ◽  
U. C. De
Keyword(s):  
Curvature Tensor ◽  
Sasakian Manifold ◽  
Second Order ◽  
Sasakian Manifolds ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Metric Connection ◽  
Concircular Curvature Tensor ◽  
Order Parallel

The object of the present paper is to study a quarter-symmetric nonmetric connection on a P-Sasakian manifold. In this paper we consider the concircular curvature tensor and conformal curvature tensor on a P-Sasakian manifold with respect to the quarter-symmetric nonmetric connection. Next we consider second-order parallel tensor with respect to the quarter-symmetric non-metric connection. Finally we consider submanifolds of an almost paracontact manifold with respect to a quarter-symmetric non-metric connection.

scholarly journals A new curvature-like tensor in an almost contact Riemannian manifold

2017 ◽  
Vol 9 (3-4) ◽  
Author(s):  
Koji Matsumoto
Keyword(s):  
Riemannian Manifold ◽  
Curvature Tensor ◽  
Space Form ◽  
Einstein Manifold ◽  
Sasakian Manifold ◽  
Special Equation ◽  
Contact Riemannian Manifold

In a M. Prvanović’s paper [5], we can find a new curvature-like tensor in an almost Hermitian manifold.In this paper, we define a new curvature-like tensor, named contact holomorphic Riemannian, briefly (CHR), curvature tensor in an almost contactRiemannian manifold. Then, using this tensor, we mainly research (CHR)-curvature tensor in a Kenmotsu and a Sasakian manifold. We introducethe flatness of a (CHR)-curvature tensor and show that a Kenmotsu anda Sasakian manifold with a flat (CHR)-curvature tensor is flat, see Theorems3.1 and 4.1. Next, we introduce the notion of an (CHR)-n-Einstein inan almost contact Riemannian manifold. In particular, in a Sasakian or aKenmotsu manifold, a (CHR)-n-Einstein manifold is n-Einstein, see Theorem5.3. Finally, from this tensor, we introduce a notion of a (CHR)-spaceform in an almost contact Riemannian manifold. In particular, if a Kenmotsuand a Sasakian manifold are (CHR)-space form, then the (CHR)-curvaturetensor satisfies a special equation, see Theorems 6.2 and 7.1.

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

scholarly journals On some classes of generalized quasi Einstein manifolds

Filomat ◽  
2015 ◽  
Vol 29 (3) ◽  
pp. 443-456 ◽  
Author(s):  
Sinem Güler ◽  
Sezgin Demirbağ
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Einstein Manifold ◽  
Sufficient Conditions ◽  
Riemannian Curvature ◽  
Einstein Manifolds ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Riemannian Curvature Tensor ◽  
Concircular Curvature Tensor

In the present paper, we investigate generalized quasi Einstein manifolds satisfying some special curvature conditions R?S = 0,R?S = LSQ(g,S), C?S = 0,?C?S = 0,?W?S = 0 and W2?S = 0 where R, S, C,?C,?W and W2 respectively denote the Riemannian curvature tensor, Ricci tensor, conformal curvature tensor, concircular curvature tensor, quasi conformal curvature tensor and W2-curvature tensor. Later, we find some sufficient conditions for a generalized quasi Einstein manifold to be a quasi Einstein manifold and we show the existence of a nearly quasi Einstein manifolds, by constructing a non trivial example.

scholarly journals On concircular transformations in Riemannian spaces

1986 ◽  
Vol 40 (2) ◽  
pp. 218-225 ◽  
Author(s):  
Luo Chongshan
Keyword(s):  
Symmetric Space ◽  
Curvature Tensor ◽  
Constant Curvature ◽  
Riemannian Space ◽  
Symmetric Spaces ◽  
Riemannian Curvature ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Riemannian Curvature Tensor ◽  
Concircular Transformation

AbstractThis paper introduces a tensor that contains the Riemannian curvature tensor and the conformal curvature tensor as special examples in the Riemannian space (Mn, g), and by using this tensor we define C-semi-symmetric space. In this paper, we have the following main result: if there is a non-trivial concircular transformation between two C-semi-symmetric spaces, then both spaces are of quasi-constant curvature.

scholarly journals QUASI-CONFORMAL CURVATURE TENSOR ON SASAKIAN MANIFOLDS

2017 ◽  
Vol 22 (1) ◽  
pp. 94-98
Author(s):  
Riddhi Jung Shah ◽  
N. V. C. Shukla
Keyword(s):  
Curvature Tensor ◽  
Einstein Manifold ◽  
Sasakian Manifold ◽  
Sasakian Manifolds ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Conformally Recurrent

In this paper we studied some curvature properties of quasi-conformal curvature tensor on Sasakian manifolds. We have proven that a -dimensional Sasakian manifold satisfying the curvature conditions and is an Einstein manifold. We have also obtained some results on quasi-conformally recurrent Sasakian manifold. Finally, Sasakian manifold satisfying the condition was studied. 12n 0 ., S Y XR0 ., W Y XR0 divWJournal of Institute of Science and TechnologyVolume 22, Issue 1, July 2017, Page: 94-98

scholarly journals Some Curvature Conditions on a Para-Sasakian Manifold with Canonical Paracontact Connection

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Bilal Eftal Acet ◽  
Erol Kılıç ◽  
Selcen Yüksel Perktaş
Keyword(s):  
Curvature Tensor ◽  
Einstein Manifold ◽  
Constant Scalar Curvature ◽  
Sasakian Manifold ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Projectively Flat ◽  
Projective Curvature ◽  
Projective Curvature Tensor ◽  
Concircular Curvature Tensor

We study canonical paracontact connection on a para-Sasakian manifold. We prove that a Ricci-flat para-Sasakian manifold with respect to canonical paracontact connection is anη-Einstein manifold. We also investigate some properties of curvature tensor, conformal curvature tensor,W2-curvature tensor, concircular curvature tensor, projective curvature tensor, and pseudo-projective curvature tensor with respect to canonical paracontact connection on a para-Sasakian manifold. It is shown that a concircularly flat para-Sasakian manifold with respect to canonical paracontact connection is of constant scalar curvature. We give some characterizations for pseudo-projectively flat para-Sasakian manifolds.

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