Topological obstructions to fatness

TL;DR

This paper investigates topological obstructions to the existence of certain Riemannian metrics with positive curvature properties, by linking characteristic numbers to bundle invariants and demonstrating non-existence results.

Contribution

It explicitly computes characteristic invariants in terms of Chern and Pontrjagin numbers, revealing topological obstructions to specific Riemannian metrics.

Findings

Many bundles do not admit metrics with positive vertizontal curvature and totally geodesic fibers.

Explicit formulas relate characteristic numbers to bundle invariants.

Topological obstructions are identified for the existence of certain Riemannian submersions.

Abstract

Alan Weinstein showed that certain characteristic numbers of any Riemannian submersion with totally geodesic fibers and positive vertizontal curvatures are nonzero. In this paper we explicitly compute these invariants in terms of Chern and Pontrjagin numbers of the bundle. This allows us to show that many bundles do not admit such metrics.