Riemannian submersions from compact four manifolds

TL;DR

This paper investigates the existence and properties of Riemannian submersions from compact four-dimensional manifolds with positive curvature, providing partial answers to existing conjectures and establishing rigidity results.

Contribution

It proves nonexistence of certain submersions under specific conditions and establishes a rigidity theorem for submersions with totally geodesic fibers from Einstein manifolds.

Findings

Nonexistence of certain Riemannian submersions from positively curved four-manifolds.

Rigidity theorem for submersions with totally geodesic fibers from Einstein four-manifolds.

Partial resolution of a conjecture by Fred Wilhelm.

Abstract

We show that under certain conditions, a nontrivial Riemannian submersion from positively curved four manifolds does not exist. This gives a partial answer to a conjecture due to Fred Wilhelm. We also prove a rigidity theorem for Riemannian submersions with totally geodesic fibers from compact four-dimensional Einstein manifolds.