A spacetime with pseudo-projective curvature tensor

2016 ◽  
Vol 57 (6) ◽  
pp. 062501 ◽  
Author(s):  
Sahanous Mallick ◽  
Young Jin Suh ◽  
Uday Chand De
Keyword(s):  
Curvature Tensor ◽  
Projective Curvature ◽  
Projective Curvature Tensor

scholarly journals On projective-symmetric spaces

1964 ◽  
Vol 4 (1) ◽  
pp. 113-121 ◽  
Author(s):  
Bandana Gupta
Keyword(s):  
Symmetric Space ◽  
Curvature Tensor ◽  
Symmetric Spaces ◽  
Covariant Differentiation ◽  
Metric Tensor ◽  
Projective Curvature ◽  
Projective Curvature Tensor ◽  
Image Position ◽  
Riemannian Spaces

This paper deals with a type of Remannian space Vn (n ≧ 2) for which the first covariant dervative of Weyl's projective curvature tensor is everywhere zero, that is where comma denotes covariant differentiation with respect to the metric tensor gij of Vn. Such a space has been called a projective-symmetric space by Gy. Soós [1]. We shall denote such an n-space by ψn. It will be proved in this paper that decomposable Projective-Symmetric spaces are symmetric in the sense of Cartan. In sections 3, 4 and 5 non-decomposable spaces of this kind will be considered in relation to other well-known classes of Riemannian spaces defined by curvature restrictions. In the last section the question of the existence of fields of concurrent directions in a ψ will be discussed.

scholarly journals ON THE $\mathcal{M}$--PROJECTIVE CURVATURE TENSOR OF A $(k, \mu)$-CONTACT METRIC MANIFOLD

2017 ◽  
pp. 117
Author(s):  
D. G. Prakasha ◽  
Kakasab Mirji
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Sasakian Manifold ◽  
Riemannian Curvature ◽  
Contact Metric Manifold ◽  
Contact Metric Manifolds ◽  
Riemannian Curvature Tensor ◽  
Projective Curvature ◽  
Projective Curvature Tensor

The paper deals with the study of $\mathcal{M}$-projective curvature tensor on $(k, \mu)$-contact metric manifolds. We classify non-Sasakian $(k, \mu)$-contact metric manifold satisfying the conditions $R(\xi, X)\cdot \mathcal{M} = 0$ and $\mathcal{M}(\xi, X)\cdot S =0$, where $R$ and $S$ are the Riemannian curvature tensor and the Ricci tensor, respectively. Finally, we prove that a $(k, \mu)$-contact metric manifold with vanishing extended $\mathcal{M}$-projective curvature tensor $\mathcal{M}^{e}$ is a Sasakian manifold.

On M-Projective Curvature Tensor of Lorentzian α-Sasakian Manifolds

2017 ◽  
Vol 18 ◽  
pp. 22-31
Author(s):  
D.G. Prakasha ◽  
Vasant Chavan
Keyword(s):  
Curvature Tensor ◽  
Unit Sphere ◽  
Sasakian Manifold ◽  
Sasakian Manifolds ◽  
Projectively Flat ◽  
Projective Curvature ◽  
Projective Curvature Tensor

In this paper, we study the nature of Lorentzianα-Sasakian manifolds admitting M-projective curvature tensor. We show that M-projectively flat and irrotational M-projective curvature tensor of Lorentzian α-Sasakian manifolds are locally isometric to unit sphere Sn(c) , wherec = α2. Next we study Lorentzianα-Sasakian manifold with conservative M-projective curvature tensor. Finally, we find certain geometrical results if the Lorentzianα-Sasakian manifold satisfying the relation M(X,Y)⋅R=0.

scholarly journals Projective-symmetric spaces

1967 ◽  
Vol 7 (1) ◽  
pp. 48-54 ◽  
Author(s):  
R. F. Reynolds ◽  
A. H. Thompson
Keyword(s):  
Curvature Tensor ◽  
Covariant Derivative ◽  
Symmetric Spaces ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Projective Curvature ◽  
Projective Curvature Tensor ◽  
Riemannian Spaces

Gy. Soos [1] and B. Gupta [2] have discussed the properties of Riemannian spaces Vn (n > 2) in which the first covariant derivative of Weyl's projective curvature tensor is everywhere zero; such spaces they call Protective-Symmetric spaces. In this paper we wish to point out that all Riemannian spaces with this property are symmetric in the sense of Cartan [3]; that is the first covariant derivative of the Riemann curvature tensor of the space vanishes. Further sections are devoted to a discussion of projective-symmetric af fine spaces An with symmetric af fine connexion. Throughout, the geometrical quantities discussed will be as defined by Eisenhart [4] and [5].

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