Einstein connection of nonsymmetric pseudo-Riemannian manifold

TL;DR

This paper explicitly constructs Einstein connections on nonsymmetric pseudo-Riemannian manifolds with applications to almost Hermitian and contact metric manifolds, extending previous solutions and providing concrete examples.

Contribution

It introduces explicit formulas for Einstein connections under specific torsion conditions on nonsymmetric manifolds, generalizing prior results and including new classes of examples.

Findings

Derived explicit Einstein connection formulas for nonsymmetric pseudo-Riemannian manifolds.

Reduced the general solution to known cases like almost Hermitian manifolds.

Provided concrete examples using weighted products of almost Hermitian manifolds.

Abstract

A.Einstein considered a linear connection $\nabla$ with torsion $T$ on a smooth manifold equipped with a nonsymmetric (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, such that $(\nabla_X\,G)(Y,Z)=-G(T(X,Y),Z)$. In this paper, we explicitly present the Einstein connection of a nonsymmetric pseudo-Riemannian manifold with non-degenerate $F$, satisfying the $f^2$-torsion condition $T(f^2X,Y)=T(X,f^2Y)=f^2 T(X,Y)$, where $g(X,fY)=F(X,Y)$, and show that in the almost Hermitian case, it reduces to the M.Prvanovi\'c's (1995) solution. We also explicitly present the Einstein connection of almost contact metric manifolds satisfying the $f^2$-torsion condition, discuss special Einstein connections, and give example in terms of weighted product of almost Hermitian manifolds.