Riemannian manifolds admitting a new type of semisymmetric nonmetric connection

Sudhakar K CHAUBEY,Ahmet YILDIZ

doi:10.3906/mat-1902-2

Abstract

We define a new type of semisymmetric nonmetric connection on a Riemannian manifold and establish its existence. It is proved that such connection on a Riemannian manifold is projectively invariant under certain conditions. We also find many basic results of the Riemannian manifolds and study the properties of group manifolds and submanifolds of the Riemannian manifolds with respect to the semisymmetric nonmetric connection. To validate our findings, we construct a nontrivial example of a $3$-dimensional Riemannian manifold equipped with a semisymmetric nonmetric connection.

Highlights

Let M n be an n -dimensional Riemannian manifold and let ∇ denote the Levi-Civita connection corresponding to the Riemannian metric g on M n
Pak [10] considered the Hayden connection ∇ ̃ equipped with the torsion tensor Tdefined as (1.1) and proved that it is a semisymmetric metric connection
We prove the existence of such a connection on an n -dimensional Riemannian manifold in the following theorem

Summary

Introduction

Let M n be an n -dimensional Riemannian manifold and let ∇ denote the Levi-Civita connection corresponding to the Riemannian metric g on M n. In view of (3.7) and Proposition 2.2, we are in a position to state the following proposition: Proposition 3.2 If an n(> 1) -dimensional Riemannian manifold (M n, g) admits a semisymmetric nonmetric connection ∇ ̃ , the Ricci tensor Scorresponding to the connection ∇ ̃ is symmetric if and only if the 1-form π is closed. Theorem 3.3 Let (M n, g) be an n -dimensional Riemannian manifold equipped with a semisymmetric nonmetric connection ∇ ̃ , the following relations hold for all vector fields X , Y , Z , and U on M n :. Corollary 3.7 If an n -dimensional Riemannian manifold (M n, g) , n > 2 , admits a semisymmetric nonmetric connection ∇ ̃ whose curvature tensor vanishes and whose 1 -form π is closed, (n − 2)′L(X, Y, Z, U ) + nR(X, Y, Z, U ) = 0. In view of (3.24) and (3.26), we get the statement of Corollary 3.7

Group manifolds with respect to the semisymmetric nonmetric connection

Example Let