A Note on Locally Metric 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.

Mathematical Proceedings of the Cambridge Philosophical Society, 2005

The concept of generalised (in the sense of Colombeau) connection on a principal fibre bundle is introduced. This definition is then used to extend results concerning the geometry of principal fibre bundles to those that only have a generalised connection. Some applications to singular solutions of Yang-Mills theory are given.

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.

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.

In this paper we define a class of torsion-free connections on the total space of the (co-)tangent bundle over a base-manifold with a connection and for which tangent spaces to the fibers are parallel. Each tangent space to a fiber is flat for these connections and the canonical projection from the (co-)tangent bundle to the base manifold is totally geodesic. In particular cases the connection is metric with signature (n,n) or symplectic and admits a single parallel totally isotropic tangent n-plane.

European Journal of Pure and Applied Mathematics, 2011

In this study, we consider a manifold equipped with semi symmetric metric connection whose the torsion tensor satisfies a special condition. We investigate some properties of the Ricci tensor and the curvature tensor of this manifold. We obtain a necessary and sufficient condition for the mixed generalized quasi-constant curvature of this manifold. Finally, we prove that if the manifold mentioned above is conformally flat, then it is a mixed generalized quasi-Einstein manifold and we prove that if the sectional curvature of a Riemannian manifold with a semi symmetric metric connection whose the special torsion tensor is independent from orientation chosen, then this manifold is of a mixed generalized quasi constant curvature.

Kyungpook mathematical journal, 2016

In this paper we study some properties of submanifolds of a Riemannian manifold equipped with a generalized connection ▽. We also consider almost Hermitian manifolds that admits a special case of this generalized connection and derive some results about the behavior of this manifolds.

2016

Abstract: 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.

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.

1986

Considering two natural principal bundle structures on T 2 M , one prove that a Finsler metric defines flat connections in these bundles.

2011

Journal of Geometry and Physics, 2021

We consider a family of α-connections defined by a pair of generalized dual quasistatistical connections (∇,∇ *) on the generalized tangent bundle (T M ⊕ T * M,ȟ) and determine their curvature, Ricci curvature and scalar curvature. Moreover, we provide the necessary and sufficient condition for∇ * to be an equiaffine connection and we prove that if h is symmetric and ∇h = 0, then (T M ⊕ T * M,ȟ,∇ (α) ,∇ (−α)) is a conjugate Ricci-symmetric manifold. Also, we characterize the integrability of a generalized almost product, of a generalized almost complex and of a generalized metallic structure w.r.t. the bracket defined by the α-connection. Finally we study α-connections defined by the twin metric of a pseudo-Riemannian manifold, (M, g), with a non-degenerate g-symmetric (1, 1)-tensor field J such that d ∇ J = 0, where ∇ is the Levi-Civita connection of g.