An obstructions to Non-negative curvature on Bundles with Pull-back connections

TL;DR

This paper investigates the geometric conditions necessary for non-negative sectional curvature in specific Riemannian submersion bundles, revealing obstructions in certain exotic sphere constructions.

Contribution

It establishes a necessary condition for non-negative curvature in bundles with pull-back connections and demonstrates that some known exotic sphere bundles have sections of negative curvature.

Findings

Bundles in [5] and [1] have sections of negative curvature

Derived a necessary condition for non-negative sectional curvature

Identified obstructions to non-negative curvature in certain bundles

Abstract

Motivated to study the geometry of the exotic spheres constructed in [5], we derive a necessary condition for non-negative sectional curvature in certain total spaces of Riemannian submersions with totally geodesic fibers. In particular, we prove that the bundles in [5] and [1] have sections of negative curvature.