Applications of Weak Metric Structures to Non-Symmetrical Gravitational Theory

TL;DR

This paper explores generalized Riemannian manifolds with weak metric structures, studying linear connections satisfying Einstein's conditions, and reveals their geometric properties and decompositions in various special cases.

Contribution

It introduces and analyzes weak metric structures with constant rank, extending almost complex and contact structures, and characterizes Einstein connections in these contexts.

Findings

Manifolds with such connections are weak nearly Kähler or have involutive contact distributions.

Structures are parallel with respect to specific connections.

Manifolds decompose into products of nearly Kähler manifolds under Einstein's conditions.

Abstract

Linear connections satisfying the Einstein metricity condition are important in the study of generalized Riemannian manifolds $(M,G=g+F)$, where the symmetric part $g$ of $G$ is a non-degenerate $(0,2)$-tensor, and $F$ is the skew-symmetric part. Such structures naturally arise in spacetime models in theoretical physics, where $F$ can be defined as an almost complex or almost contact metric (a.c.m.) structure. In the paper, we first study more general models, where $F$ has constant rank and is based on weak metric structures (introduced by the second author and R.~Wolak), which generalize almost complex and a.c.m. structures. We consider linear connections with totally skew-symmetric torsion that satisfy both the Einstein metricity condition and the $A$-torsion condition, where $A$ is a skew-symmetric (1,1)-tensor adjoint to~$F$. In the almost Hermitian case, we prove that the manifold with such a connection is weak nearly K\" ahler, the torsion is completely determined by the exterior derivative of the fundamental 2-form and the Nijenhuis tensor, and the structure tensors are parallel, while in the weak a.c.m. case, the contact distribution is involutive, the Reeb vector field is Levi-Civita parallel, and the structure tensors are also parallel with respect to both connections. For rank$(F)=\dim M$, we apply weak almost Hermitian structures to fundamental results (by the first author and S. Ivanov) on generalized Riemannian manifolds and prove that the manifold equipped with an Einstein's connection is a weighted product of several nearly K\"ahler manifolds. For~rank$(F)<\dim M$ we apply weak almost Hermitian and weak a.c.m. structures and obtain splitting results for generalized Riemannian manifolds equipped with Einstein's connections.