Flat Sections and Non-negative Curvature in Pullback Bundles

TL;DR

This paper introduces a geometric obstruction that prevents certain Riemannian submersions with totally geodesic fibers from having non-negative sectional curvature in their total spaces, impacting the understanding of curvature properties in geometric structures.

Contribution

It provides a new geometric criterion to identify when non-negative sectional curvature cannot occur in specific pullback bundle constructions.

Findings

Obstruction applies to various examples of Riemannian submersions.

Certain geometric configurations cannot admit non-negative sectional curvature.

The results clarify limitations in constructing manifolds with non-negative curvature.

Abstract

We give a geometric obstruction to the non-negativity of the sectional curvature in the total spaces of certain Riemannian submersions with totally geodesic fibers; applications of this obstruction to several examples are given.