Milnor-Wood inequalities for manifolds locally isometric to a product of hyperbolic planes

TL;DR

This paper establishes precise Milnor-Wood inequalities for flat vector bundles over certain hyperbolic product manifolds, showing these manifolds cannot have affine structures and characterizing flat bundles by Euler number sign.

Contribution

It provides sharp Milnor-Wood inequalities for these manifolds and confirms Chern--Sullivan's conjecture in this context, offering new insights into their geometric structures.

Findings

Manifolds do not admit affine structures.

Flat vector bundles are classified by Euler number sign.

Sharp bounds for Euler numbers of flat bundles are established.

Abstract

This note describes sharp Milnor--Wood inequalities for the Euler number of flat oriented vector bundles over closed Riemannian manifolds locally isometric to products of hyperbolic planes. One consequence is that such manifolds do not admit an affine structure, confirming Chern--Sullivan's conjecture in this case. The manifolds under consideration are of particular interest, since in contrary to many other locally symmetric spaces they do admit flat vector bundle of the corresponding dimension. When the manifold is irreducible and of higher rank, it is shown that flat oriented vector bundles are determined completely by the sign of the Euler number.