Homogeneous Weyl connections of non-positive curvature

TL;DR

This paper investigates homogeneous Weyl connections with non-positive curvature, showing that under certain conditions, such connections on unimodular Lie groups are locally product structures, expanding understanding of geometric properties in this setting.

Contribution

It establishes that stretched non-positive homogeneous Weyl connections on unimodular Lie groups are necessarily locally product structures, providing a new classification result.

Findings

Homogeneous Weyl connections with non-positive curvature exist on products like S^1 x M.

Stretched non-positive Weyl connections on unimodular Lie groups are locally product.

The study extends the understanding of curvature constraints in Weyl geometry.

Abstract

We study homogenous Weyl connections with non-positive sectional curvatures. The Cartesian product $\mathbb S^1 \times M$ carries canonical families of Weyl connections with such a property, for any Riemmanian manifold $M$. We prove that if a homogenous Weyl connection on a manifold, modeled on a unimodular Lie group, is non-positive in a stronger sense (streched non-positive), then it must be locally of the product type.