LCS-manifolds and Ricci solitons

This paper is concerned with the study of [Formula: see text]-manifolds and Ricci solitons. It is shown that in a [Formula: see text]-spacetime, the fluid has vanishing vorticity and vanishing shear. It is found that in an [Formula: see text]-manifold, [Formula: see text] is an irrotational vector field, where [Formula: see text] is a non-zero smooth scalar function. It is proved that in a [Formula: see text]-spacetime with generator vector field [Formula: see text] obeying Einstein equation, [Formula: see text] or [Formula: see text] according to [Formula: see text] or [Formula: see text], where [Formula: see text] is a scalar function and [Formula: see text] is the energy momentum tensor. Also, it is shown that if [Formula: see text] is a non-null spacelike (respectively, timelike) vector field on a [Formula: see text]-spacetime with scalar curvature [Formula: see text] and cosmological constant [Formula: see text], then [Formula: see text] if and only if [Formula: see text] (respectively, [Formula: see text]), and [Formula: see text] if and only if [Formula: see text] (respectively, [Formula: see text]), and further [Formula: see text] if and only if [Formula: see text]. The nature of the scalar curvature of an [Formula: see text]-manifold admitting Yamabe soliton is obtained. Also, it is proved that an [Formula: see text]-manifold admitting [Formula: see text]-Ricci soliton is [Formula: see text]-Einstein and its scalar curvature is constant if and only if [Formula: see text] is constant. Further, it is shown that if [Formula: see text] is a scalar function with [Formula: see text] and [Formula: see text] vanishes, then the gradients of [Formula: see text], [Formula: see text], [Formula: see text] are co-directional with the generator [Formula: see text]. In a perfect fluid [Formula: see text]-spacetime admitting [Formula: see text]-Ricci soliton, it is proved that the pressure density [Formula: see text] and energy density [Formula: see text] are constants, and if it agrees Einstein field equation, then we obtain a necessary and sufficient condition for the scalar curvature to be constant. If such a spacetime possesses Ricci collineation, then it must admit an almost [Formula: see text]-Yamabe soliton and the converse holds when the Ricci operator is of constant norm. Also, in a perfect fluid [Formula: see text]-spacetime satisfying Einstein equation, it is shown that if Ricci collineation is admitted with respect to the generator [Formula: see text], then the matter content cannot be perfect fluid, and further [Formula: see text] with gravitational constant [Formula: see text] implies that [Formula: see text] is a Killing vector field. Finally, in an [Formula: see text]-manifold, it is proved that if the [Formula: see text]-curvature tensor is conservative, then scalar potential and the generator vector field are co-directional, and if the manifold possesses pseudosymmetry due to the [Formula: see text]-curvature tensor, then it is an [Formula: see text]-Einstein manifold.

CLASSIFICATIONS OF KANTOWSKI–SACHS, BIANCHI TYPES I AND III SPACETIMES ACCORDING TO RICCI COLLINEATIONS

The Ricci collineation classifications of Kantowski–Sachs, Bianchi types I and III spacetimes are studied according to their degenerate and non-degenerate Ricci tensor. When the Ricci tensor is degenerate, the special cases are classified and it is shown that there are many cases of Ricci collineations (RCs) with infinite number of degrees of freedom, and the group of RCs is ten-dimensional in some spacial cases. Furthermore, it is found that when the Ricci tensor is non-degenerate, the group of RCs is finite-dimensional, and we have only either four which coincides with the isometries or six proper RCs in addition to the four isometries.

RICCI COLLINEATIONS OF THE BIANCHI TYPES I AND III, AND KANTOWSKI–SACHS SPACETIMES

Ricci collineations of the Bianchi types I and III, and Kantowski–Sachs spacetimes are classified according to their Ricci collineation vector (RCV) field of the form (i)–(iv) one component of ξa(xb) is nonzero, (v)–(x) two components of ξa(xb) are nonzero, and (xi)–(xiv) three components of ξa(xb) are nonzero. Their relation with isometries of the spacetimes is established. In case (v), when det (Rab)=0, some metrics are found under the time transformation, in which some of these metrics are known, and the other ones new. Finally, the family of contracted Ricci collineations (CRC) are presented.

TIMELIKE AND SPACELIKE RICCI COLLONEATION VECTORS IN STRING COSMOLOGY

In this paper, we study the consequences of the existence of timelike and spacelike Ricci collineation vectors (RCVs) for string cloud in the context of general relativity. Necessary and sufficient conditions are derived for a space-time with string cloud to admit a timelike RCV parallel to ua and a spacelike RCV parallel to xa. Also, some results are obtained.

Some Robertson-Walker Models with Variable G and λ

Some Robertson-Walker (RW) models admitting a contracted Ricci collineation along the fluid flow vector and having time-varying G and Λ are investigated. The nature of the expansion of the models obtained in the cases k = ±l is found to be interchanged from the corresponding standard FRW models. Estimates of the present values of various cosmological parameters are obtained and found to be well within the observational limits.