Rigidity and characteristic classes of smooth bundles with nonpositively curved fibers

TL;DR

This paper investigates the properties of smooth bundles with nonpositively curved fibers, demonstrating vanishing results for certain characteristic classes and establishing topological rigidity of the vertical tangent bundle under specific conditions.

Contribution

It provides new vanishing theorems for Miller-Morita-Mumford classes and shows topological rigidity of the vertical tangent bundle for bundles with nonpositively curved fibers.

Findings

Vanishing of generalized Miller-Morita-Mumford classes for certain smooth bundles.

Topological rigidity of the vertical tangent bundle under additional conditions.

Results applicable to bundles with nonpositively curved Riemannian fibers.

Abstract

We prove vanishing results for the generalized Miller-Morita-Mumford classes of some smooth bundles whose fiber is a closed manifold that supports a nonpositively curved Riemannian metric. We also find, under some extra conditions, that the vertical tangent bundle is topologically rigid.