# Flat $\mathfrak{so}(p,q)$-Connections for Manifolds of Non-Euclidean Signature

## TL;DR

This paper explores flat connections with torsion on manifolds of non-Euclidean signature, extending classical results on spheres to hyperbolic and other pseudo-Riemannian manifolds, and constructs explicit flat f0a7(p,q) connections.

## Contribution

It provides a detailed analysis of torsion and flat connections on non-Euclidean manifolds, including explicit constructions for f0a7(p,q) connections in various signatures.

## Key findings

- Flat connections exist on certain non-Euclidean manifolds.
- Torsion must be covariantly constant for parallelizability.
- Explicit f0a7(p,q) connections are constructed for specific coset manifolds.

## Abstract

The well-known fact that $S^1$, $S^3$ and $S^7$ are parallelizable manifolds admitting flat connections is revisited. The role of torsion in the construction of those flat connections is made explicit, and the possibilities allowed by different metric signatures are examined. A necessary condition for parallelizability in an open region is that the torsion tensor must be covariantly constant. This property can be used to obtain a relation between a torsion-free and flat connections. Our treatment covers Riemannian and pseudo-Riemannian (non-Euclidean signature) hyperbolic manifolds of dimensions three and seven. Apart from the spherical cases mentioned above, the explicit flat $\mathfrak{so}(p,q)$ connections with $p+q=3,7$ are constructed for the coset manifolds $SO(p,q+1)/SO(p,q)$ or $SO(p+1,q)/SO(p,q)$.