On some families of linear connections

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.

2007

For a torsionless connection on the tangent bundle of a manifold M the Weyl curvature W is the part of the curvature in kernel of the Ricci contraction. We give a coordinate free proof of Weyl's result that the Weyl curvature vanishes if and only if the manifold is (locally) diffeomorphic to a real projective space with the connection, when transported to the projective space, in the projective class of the Levi-Civita connection of the Fubini-Study metric. Associated to a connection on an even-dimensional M is an almost complex structure on J(M) the bundle of all complex structures on the tangent spaces of M, c.f. [O'Brian-Rawnsley]. We show that this structure is a projective invariant, and when integrable can be obtained from a torsionless connection which must then have W=0. We also show that two torsionless connections define the same almost complex structure if and only if they are projectively equivalent.

International Journal of Pure and Apllied Mathematics, 2016

We study the existence and the behavior of a linear connection from a curvature given in a n-dimensional riemannian manifold M. For a polynomial section of the dual space of T M on R n , in particular, we find that there is a polynomial linear connection on R n. We prove that if the nullity space of the Ricci tensor is equal to that of the curvature, then the Ricci tensor and the curvature coincide.

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.

Journal of Mathematical Physics, 2005

On a (pseudo-) Riemannian manifold of dimension n 3, the space of tensors which transform covariantly under Weyl rescalings of the metric is built. This construction is related to a Weyl-covariant operator D whose commutator [D, D] gives the conformally invariant Weyl tensor plus the Cotton tensor. So-called generalized connections and their transformation laws under diffeomorphisms and Weyl rescalings are also derived. These results are obtained by application of BRST techniques.

International Journal of Geometric Methods in Modern Physics, 2012

The aim of this paper is to study from the point of view of linear connections the data (M, D, g, W ) with M a smooth (n + p) dimensional real manifold, (D, g) a n-dimensional semi-Riemannian distribution on M, G the conformal structure generated by g and W a Weyl substructure: a map W :

arXiv (Cornell University), 2011

A Riemannian manifold M with an integrable almost product structure P is called a Riemannian product manifold. Our investigations are on the manifolds (M, P, g) of the largest class of Riemannian product manifolds, which is closed with respect to the group of conformal transformations of the metric g. This class is an analogue of the class of locally conformal Kähler manifolds in almost Hermitian geometry. In the present paper we study a natural connection D on (M, P, g) (i.e. DP = Dg = 0). We find necessary and sufficient conditions the curvature tensor of D to have properties similar to the Kähler tensor in Hermitian geometry. We pay attention to the case when D has a parallel torsion. We establish that the Weyl tensors for the connection D and the Levi-Civita connection coincide as well as the invariance of the curvature tensor of D with respect to the usual conformal transformation. We consider the case when D is a flat connection. We construct an example of the considered manifold by a Lie group where D is a flat connection with non-parallel torsion.

Facta Universitatis, Series: Mathematics and Informatics, 2021

The differential geometry of the tangent bundle is an effective domain of differential geometry which reveals many new problems in the study of modern differential geometry. The generalization of connection on any manifold to its tangent bundle is an application of differential geometry. Recently a new type of semi-symmetric non-metric connection on a Riemannian manifold has been studied and a relationship between Levi-Civita connection and semi-symmetric non-metric connection has been established. The various properties of a Riemannian manifold with relation to such connection have also been discussed. The present paper aims to study the tangent bundle of a new type of semi-symmetric non-metric connection on a Riemannian manifold. The necessary and sufficient conditions for projectively invariant curvature tensors corresponding to such connection are proved and show many basic results on the Riemannian manifold in the tangent bundle. Furthermore, the properties of group manifolds of the Riemannian manifolds with respect to the semi-symmetric non-metric connection in the tangent bundle have been studied. Moreover, theorems on the symmetry property of Ricci tensor and Ricci soliton in the tangent bundle are established.

International Mathematical Forum, 2009

In this paper, the existence of the projective quarter symmetric metric connection is proved in Riemannian manifolds. In particular two cases, this connection reduces to a semi-symmetric metric connection and to a projective semi-symmetric connection. Furthermore, we study a scalar curvature of Riemannian manifolds with keeping the covariant derivative of tensor W l ikj .