Einstein connection of a weak almost contact metric manifold

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 $F\ne 0$ is a skew-symmetric tensor associated with electromagnetism, more attractive than ever. 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.