Weak metric structures on generalized Riemannian manifolds

TL;DR

This paper explores weak metric structures on generalized Riemannian manifolds, extending classical geometric concepts and proving splitting results under various rank conditions of the structure tensor.

Contribution

It introduces and studies weak metric structures with constant rank, generalizing almost complex and contact structures, and establishes new splitting theorems for these manifolds.

Findings

Manifolds decompose into weighted products of nearly Kähler manifolds under certain conditions.

Splitting results for manifolds with weak $f$-structures when rank$(F)< ext{dim } M$.

Application of weak almost Hermitian structures to fundamental Riemannian geometry results.

Abstract

In the paper, we first study more general models, where $F$ has constant rank and is based on weak metric structures (introduced by the first author and R. Wolak), which generalize almost complex and almost contact metric $f$-contact structures. We consider generalized metric connections (i.e., linear connections preserving $G$) with totally skew-symmetric torsion (0,3)-tensor. For rank$(F)=\dim M$ and non-conformal tensor $A^2$, where $A$ is a skew-symmetric (1,1)-tensor adjoint to $F$, we apply weak almost Hermitian structures to fundamental results (by the second author and S. Ivanov) on generalized Riemannian manifolds and prove that the manifold is a weighted product of several nearly K\"ahler manifolds corresponding to eigen-distributions of $A^2$. For rank$(F)<\dim M$ we apply weak $f$-structures and obtain splitting results for generalized Riemannian manifolds.