Einstein connection of nonsymmetric pseudo-Riemannian manifold, II

TL;DR

This paper explores Einstein connections on nonsymmetric pseudo-Riemannian manifolds, explicitly constructing such connections using almost contact structures and providing specific examples.

Contribution

It introduces explicit formulas for Einstein connections on nonsymmetric manifolds with a weak almost contact structure, extending Einstein's original ideas.

Findings

Explicit Einstein connection formulas using almost contact structures

Discussion of special Einstein connections

Example involving weighted product of almost Hermitian manifold and real line

Abstract

Advances in modern physics since Einstein have made the nonsymmetric metric (0,2)-tensor $G=g+F$, where $g$ is a pseudo-Riemannian metric associated with gravity, and $F\ne0$ is a skew-symmetric tensor associated with electromagnetism, more attractive than ever. A. Einstein considered a linear connection $\nabla$ with torsion $T$ such that $(\nabla_X\,G)(Y,Z)=G(T(Y,X),Z)$. In this paper, we explicitly present the Einstein connection of $G=g+F$ using a weak almost contact structure $(f,\xi,\eta)$ with $g(X,fY)=F(X,Y)$ with a natural condition (trivial in the almost contact case). We discuss special Einstein connections, and give an example in terms of the weighted product of almost Hermitian manifold and a real line.