Narrow Results

This paper intends to obtain concircular vector fields (CVFs) of Kantowski–Sachs and Bianch type-III spacetimes. For this purpose, ten conformal Killing equations and their general solution in the form of conformal Killing vector fields (CKVFs) are derived along with their conformal factors. The obtained conformal Killing vector fields are then placed in Hessian equations to obtain the final form of concircular vector fields. The existence of concircular symmetry imposes restrictions on the metric functions. The conditions imposing restrictions on these metric functions are obtained as a set of integrability conditions. It is shown that Kantowski–Sachs and Bianchi type-III spacetimes admit four-, six-, or fifteen-dimensional concircular vector fields. It is established that for Einstein spaces, every conformal Killing vector field is a concircular vector field. Moreover, it is explored that every concircular vector field obtained here is also a conformal Ricci collineation.

We obtain new invariant Einstein metrics on the compact Lie groups $\mathrm{\text{SO}}(n)$ ($n\ge 12$) which are not naturally reductive. This is achieved by imposing certain symmetry assumptions in the set of all left-invariant metrics on $\mathrm{\text{SO}}(n)$ and by computing the Ricci tensor for such metrics. The Einstein metrics are obtained as solutions of systems polynomial equations, which we manipulate by symbolic computations using Gröbner bases.

Our aim in this paper is to obtain concircular vector fields (CVFs) on the Lorentzian manifold of Bianchi type-I spacetimes. For this purpose, two different sets of coupled partial differential equations comprising ten equations each are obtained. The first ten equations, known as conformal Killing equations are solved completely and components of conformal Killing vector fields (CKVFs) are obtained in different possible cases. These CKVFs are then substituted into second set of ten differential equations to obtain CVFs. It comes out that Bianchi type-I spacetimes admit four-, five-, six-, seven- or 15-dimensional CVFs for particular choices of the metric functions. In many cases, the CKVFs of a particular metric are same as CVFs while there exists few cases where proper CKVFs are not CVFs.

In this paper, Noether symmetry and Killing symmetry analyses of the curved traversable wormholes of $(3+1)$-dimensional spacetime metric in a Riemannian space are discussed. Moreover, a Lie algebra analysis is shown. Using the first and second Cartan’s structure equations, we find connection forms and then the curvature 2-forms are obtained. Finally, the Ricci scalar tensor and the components of Einstein curvature are computed.

Recently we introduced a new definition of metrics on almost commutative algebras. In this paper, we propose a coherent notion of compatible linear connection with respect to any almost commutative tensor and show that to every metric there corresponds a unique torsion-free compatible connection. This connection is called the Levi–Civita connection of the associated metric.

In this paper, we investigate Ricci Inheritance Collineations (RICs) in Kantowski–Sachs spacetimes. RICs are discussed in detail when Ricci tensor is degenerate and nondegenerate. In both the cases, RICs are obtained and it turns out that the dimension of Lie algebra of RICs is finite when Ricci tensor is nondegenerate. In the case when Ricci tensor is degenerate, we get finite as well as infinite dimensional group of RICs.

Let M be a real hypersurface with almost contact metric structure (ϕ, ξ, η, g) in a non-flat complex space form M_n(c). In this paper we investigate real hypersurfaces of M_n(c) whose structure Jacobi operator R_ξ commutes with both the structure tensor ϕ and the Ricci tensor S of M. We characterize Hopf hypersurfaces of M_n(c).

We investigate geometric properties of 3-dimensional real hypersurfaces with Aξ = 0 in a complex 2-dimensional nonflat complex space form from the view-points of their shape operators, Ricci tensors and *-Ricci tensors.