# Non-symmetric Riemannian gravity and Sasaki-Einstein 5-manifolds

## TL;DR

This paper establishes a deep link between non-symmetric Riemannian gravity with skew-symmetric torsion and Sasaki-Einstein 5-manifolds, providing explicit formulas and characterizations in differential geometry.

## Contribution

It characterizes when such connections exist on almost contact metric manifolds and relates them to Sasaki-Einstein 5-manifolds, including explicit curvature formulas.

## Key findings

- Existence of skew torsion connections is equivalent to being D-homothetic to cosymplectic.
- In dimension five, these connections correspond exactly to Sasaki-Einstein 5-manifolds.
- Formulas for curvature and Ricci tensors are derived in terms of SU(2) structures.

## Abstract

We show that a connection with skew-symmetric torsion satisfying the Einstein metricity condition exists on an almost contact metric manifold exactly when it is D-homothetic to a cosymplectic manifold. In dimension five, we get that the existence of a connection with skew torsion satisfying the Einstein metricity condition is equivalent to the existence of a Sasaki-Einstein 5-manifold and vice versa, any Sasaki-Einstein 5-manifold generates a two parametric family of connections with skew torsion satisfying the Einstein metricity condition. Formulas for the curvature and the Ricci tensors of these connections are presented in terms of the Sasaki-Einstein SU(2) structures.