In the differential geometry of certain F-structures, the importance of concircular curvature tensor is very well known. The relativistic significance of this tensor has been explored here. The spacetimes satisfying Einstein field equations and with vanishing concircular curvature tensor are considered and the existence of Killing and conformal Killing vectors have been established for such spacetimes. Perfect fluid spacetimes with vanishing concircular curvature tensor have also been considered. The divergence of concircular curvature tensor is studied in detail and it is seen, among other results, that if the divergence of the concircular tensor is zero and the Ricci tensor is of Codazzi type then the resulting spacetime is of constant curvature. For a perfect fluid spacetime to possess divergence-free concircular curvature tensor, a necessary and sufficient condition has been obtained in terms of Friedmann-Robertson-Walker model.

https://link.springer.com/content/pdf/10.1007%2Fs10773-009-0121-z.pdf

Concircular Curvature Tensor and Fluid Spacetimes

In $N(k)$-contact metric manifolds and/or $(k,\mu)$-manifolds, gradient Ricci solitons, compact Ricci solitons and Ricci solitons with $V$ pointwise collinear with the structure vector field $\xi $ are studied.

Ricci solitons in contact metric manifolds

We propose a uniform approach for recognizing all black box groups of Lie type which is based on the analysis of the structure of the centralizers of involutions. Our approach can be viewed as a computational version of the classification of the finite simple groups. We present an algorithm which constructs a long root SL2(q)-subgroup in a finite simple group of Lie type of odd characteristic, then we use the Aschbacher's ``Classical Involution Theorem'' as a model in the recognition algorithm and we construct all root SL2(q)-subgroups corresponding to the nodes in the extended Dynkin diagram, that is, we construct the extended Curtis - Phan - Tits system of the finite simple groups of Lie type of odd characteristic. In particular, we construct all subsystem subgroups which can be read from the extended Dynkin diagram. We also present an algorithm which determines whether the p-core (or ``unipotent radical'') Op(G) of a black box group G is trivial or not, where G/Op(G) is a finite simple classical group of odd characteristic p, answering a well-known question of Babai and Shalev

Black Box Groups

The purpose of this paper is to classify α-para Kenmotsu manifolds M3 such that the projection of the image of the concircular curvature tensor L onto the one-dimensional linear subspace of Tp(M3) generated by ${\xi_{p}}$ is zero.

On a Class of α-Para Kenmotsu Manifolds

It is proved that every concircularly recurrent manifold must be necessarily a recurrent manifold.

On pseudo-Riemannian manifolds with recurrent concircular curvature tensor

We classify N(•)-contact metric manifolds which sat- isfy Z(»;X) ¢ Z = 0, Z(»;X) ¢ R = 0 or R(»;X) ¢ Z = 0.

/pdf/on-the-concircular-curvature-tensor-of-a-contact-metric-4ovm6ged76.pdf

D. E. Blair

Papers

On the concircular curvature tensor of a contact metric manifold