Narrow Results

In this article, we prove that the maximum rank r of the Ricci tensor of a Cartan–Hadamard manifold Mⁿ satisfies the inequality 2r - 1 ≥ n - s, where n is the dimension and s is the core number, which measures the flatness of Mⁿ. Examples show that this lower bound is sharp.

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 we prove the logarithmic Sobolev inequality on pinned loop groups over compact Lie groups equipped with -metric (s>1/2). We also compute the Ricci tensor with respect to -metric and investigate its asymptotic behavior as s decreases to 1/2.

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.

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.

In this paper, we investigate homothetic Ricci collineations (HRCs) for non-static plane symmetric spacetimes. The source of the energy–momentum tensor is assumed to be a perfect fluid. Both degenerate as well as non-degenerate cases are considered and the HRC equations are solved in different cases. It is concluded that these spacetimes may possess 6, 7, 8, 10 or 11 HRCs in non-degenerate case, while they admit seven or infinite number of HRCs for degenerate Ricci tensor.

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.

Four-dimensional conformally flat curvature models of signature $(2,2)$ are considered and complete classification of curvature tensor is obtained. Then, some remarks about Ivanov–Petrova and Walker–Ivanov–Petrova properties are stated.

Here, using the projectively invariant pseudo-distance and Schwarzian derivative, it is shown that every connected complete Finsler space of the constant negative Ricci scalar is reversible. In particular, every complete Randers metric of constant negative Ricci (or flag) curvature is Riemannian.

This paper considers the relationship between geometry, symmetry and fundamental interactions — gravity and those mediated by gauge fields. We explore product spacetimes which (a) have the necessary symmetries for gauge interactions and four-dimensional gravity and (b) reduce to an $N$-dimensional isotropic universe in their flat space limit. The key technique is looking at orbits of the operator form of symmetric rank-two tensors under changes of coordinate system. Orbits containing diagonal matrices are seen to correspond to product manifolds. The $GL(N,ℝ)$ symmetry of the decompactified universe acts nonlinearly on such a product spacetime. We explore the resulting Kaluza–Klein theories, in which the internal symmetries act indirectly on space of the extra dimensions, and give two examples: a six-dimensional model in which the gauge symmetry is $U(1)$ and a seven-dimensional model in which it is $SU(2)$. We identify constraints that can be placed on any rank-two symmetric tensor to obtain such spacetimes: relationships between polynomial invariants. The multiplicities of its eigenvalues determine the dimensionalities of the factor spaces and hence the gauge symmetries. If the tensor in question is the Ricci tensor, other than two-dimensional factor spaces all the factor spaces are Einstein manifolds. This situation represents the classical vacuum of the Kaluza–Klein theory.

In this work, generalized weakly ${\mathscr{H}}$-symmetric space-times ${\text{(GWHS)}}_n$ are investigated, where ${\mathscr{H}}$ is any symmetric $(0,2)$ tensor. It is proved that, in a nontrivial ${\text{(GWHS)}}_n$ space-time, the tensor ${\mathscr{H}}$ has a perfect fluid form. Accordingly, sufficient conditions for a nontrivial generalized weakly Ricci symmetric space-time ${\text{(GWRS)}}_n$ to be either an Einstein space-time or a perfect fluid space-time are obtained. Also, conditions for space-times admitting either a generalized weakly symmetric energy-momentum tensor or a generalized weakly symmetric ${𝒵}$ tensor to be Einstein or perfect fluid space-times are provided.

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 examine questions of geometric realizability for algebraic structures which arise naturally in affine and Riemannian geometry.

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.