Einstein connection of nonsymmetric pseudo-Riemannian manifold

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.