Quarter-symmetric non-metric connection

TL;DR

This paper introduces a new quarter-symmetric non-metric connection on generalized Riemannian manifolds, exploring its properties, torsion relations, curvature tensors, and conditions for dual symmetry, with implications for symplectic structures.

Contribution

It defines and analyzes a novel quarter-symmetric non-metric connection, detailing its torsion, curvature tensors, and dual symmetry conditions on generalized Riemannian manifolds.

Findings

The exterior derivative of the skew-symmetric part F is invariant under the connection.

Manifolds with F are symplectic if F is closed under the connection.

Curvature tensors of the connection and its dual are explicitly characterized.

Abstract

The paper will study a new quarter-symmetric non-metric connection on a generalized Riemannian manifold. It will determine the relations that the torsion tensor satisfies. The exterior derivative of the skew-symmetric part $F$ of basic tensor $G$ with respect to the Levi-Civita connection coincides with that of skew-symmetric part $F$ with respect to quarter-symmetric non-metric connection, which implies that the even-dimensional manifold endowed with $F$ is symplectic manifold if and only if it is closed with respect to quarter-symmetric non-metric connection. The linearly independent curvature tensors of this connection and its dual connection are determined and the properties of these tensors are discussed. Finally, the condition is given that the connection should be dual symmetric.