On Linear Connection Relating to a Given Curvature

Geometry, 2013

We introduce a metric notion of Ricci curvature forPLmanifolds and study its convergence properties. We also prove a fitting version of the Bonnet-Myers theorem, for surfaces as well as for a large class of higher dimensional manifolds.

Journal of Geometry and Physics, 2007

Geometrical characterizations are given for the tensor R · S, where S is the Ricci tensor of a (semi-)Riemannian manifold (M, g) and R denotes the curvature operator acting on S as a derivation, and of the Ricci Tachibana tensor ∧ g ·S, where the natural metrical operator ∧ g also acts as a derivation on S. As a combination, the Ricci curvatures associated with directions on M, of which the isotropy determines that M is Einstein, are extended to the Ricci curvatures of Deszcz associated with directions and planes on M, and of which the isotropy determines that M is Ricci pseudo-symmetric in the sense of Deszcz.

Universal Journal of Mathematics and Applications, 2020

The aim of the present paper is to study the properties of Riemannian manifolds equipped with a projective semi-symmetric connection.

Journal für die reine und angewandte Mathematik (Crelles Journal), 2022

The prescribed Ricci curvature problem in the context ofG-invariant metrics on a homogeneous spaceM=G/K{M=G/K}is studied. We focus on the metrics at which the mapg↦Rc⁡(g){g\mapsto\operatorname{Rc}(g)}is, locally, as injective and surjective as it can be. Our main result is that such property is generic in the compact case. Our main tool is a formula for the Lichnerowicz Laplacian we prove in terms of the moment map for the variety of algebras.

Israel Journal of Mathematics, 2009

We consider the pseudo-Euclidean space (R n , g), with n ≥ 3 and g ij = δ ij i , i = ±1, where at least one i = 1 and nondiagonal tensors of the form T = ij f ij dx i dx j such that, for i = j, f ij (x i , x j ) depends on x i and x j . We provide necessary and sufficient conditions for such a tensor to admit a metricḡ, conformal to g, that solves the Ricci tensor equation or the Einstein equation. Similar problems are considered for locally conformally flat manifolds. Examples are provided of complete metrics on R n , on the n-dimensional torus T n and on cylinders T k ×R n−k , that solve the Ricci equation or the Einstein equation.

Rendiconti del Circolo Matematico di Palermo

We give coordinate formula and geometric description of the curvature of the tensor product connection of linear connections on vector bundles with the same base manifold. We define the covariant differential of geometric fields of certain types with respect to a pair of a linear connection on a vector bundle and a linear symmetric connection on the base manifold. We prove the generalized Bianchi identity for linear connections and we prove that the antisymmetrization of the second order covariant differential is expressed via the curvature tensors of both connections.

The main purpose of this paper is to study the connections on vector bundle and apply connections to prove the Bianchi identity and Christoffel symbols.

Balkan Journal of Geometry and Its Applications

On a Weyl manifold (M, g, w), we consider • the Levi-Civita connection associated to a metric g ∈ g , the symmetric connection, compatible with the Weyl structure w and the family of linear connections C ={ λ := • + λ(− • | λ ∈ R}. For λ ∈ C, we investigate some properties of the deformation algebra U(M, λ − • Next, we study the case when • and λ determine the same Ricci tensor and the case when the curvature tensors of the connections • and λ are proportional.

Filomat, 2015

This paper discusses the relationships between the metric, the connection and the curvature tensor of 4-dimensional, Ricci flat manifolds which admit a metric. It is shown that these metric and curvature objects are essentially equivalent conditions for such manifolds if one excludes certain very special cases and which occur when the signature is indefinite. In a similar vein, some relevant remarks are made regarding the Weyl conformal tensor.

Differential Geometry and its Applications, 1991

We study a natural generalization of the concepts of torsion and Ricci tensor for a nonlinear connection on a fibred manifolds, with respect to a given fibred soldering form. Our results are achieved by means of the differentials and codifferentials induced by the Frölicher-Nijenhuis graded Lie algebra of tangent valued forms.

2014

The main purpose of this paper is to study the conn ections on vector bundle and apply connections to prove the Bianchi identity and Christoffel symbols.

2011

In this paper we describe the local Ricci and Bianchi identities for an hnormal N-linear connection DΓ(N) on the dual 1-jet space J 1 * (T , M). To reach this aim, we firstly give the expressions of the local distinguished (d-) adapted components of torsion and curvature tensors produced by DΓ(N), and then we analyze their attached local Ricci identities. The derived deflection d-tensor identities are also presented. Finally, we expose the local expressions of the Bianchi identities (in the particular case of an hnormal N-linear connection of Cartan type), which geometrically connect the local torsion and curvature d-tensors of the linear connection DΓ(N).

Annales Polonici Mathematici, 2011

In the first part of our work, some results are given for a Riemannian manifold with semi-symmetric metric connection. In the second part, some special vector fields, such as torse-forming vector fields, recurrent vector fields and concurrent vector fields are examined in this manifold. We obtain some properties of this manifold having the vectors mentioned above.

2014

An n-dimensional Riemannian space V n is called a Riemannian space with cyclic Ricci tensor [2, 3], if the Ricci tensor Rij satisfies the following condition Rij,k +Rjk,i +Rki,j = 0, where Rij the Ricci tensor of V , and the symbol ”,” denotes the covariant derivation with respect to Levi-Civita connection of V . In this paper we would like to treat some results in the subject of geodesic mappings of Riemannian space with cyclic Ricci tensor. Let V n = (M, gij) and V n = (M, gij) be two Riemannian spaces on the underlying manifold M. A mapping V n → V n is called geodesic, if it maps an arbitrary geodesic curve of V n to a geodesic curve of V .[4] At first we investigate the geodesic mappings of a Riemannian space with cyclic Ricci tensor into another Riemannian space with cyclic Ricci tensor. Finally we show that, the Riemannian Einstein space with cyclic Ricci tensor admit only trivial geodesic mapping.

Proceedings of The Estonian Academy of Sciences, 2010

The object of the present paper is to investigate the applications of generalized pseudo Ricci symmetric manifolds admitting a semi-symmetric metric connection to the general relativity and cosmology. Also the existence of a generalized pseudo Ricci symmetric manifold is ensured by an example.