The generalized Chern conjecture for manifolds that are locally a product of surfaces

TL;DR

This paper proves that for certain manifolds locally modeled on products of symmetric planes, the Euler characteristic obstructs flat structures, confirming a special case of the Chern conjecture using a new Milnor--Wood inequality.

Contribution

It establishes a sharp Milnor--Wood inequality for manifolds locally isometric to products of hyperbolic planes and analyzes flat bundles over these manifolds, confirming the Chern conjecture in this setting.

Findings

Euler characteristic obstructs flat structures in these manifolds

Finitely many flat structures with nonzero Euler number on Hilbert--Blumenthal varieties

No flat structure corresponds to the tangent bundle in these cases

Abstract

We consider closed manifolds that admit a metric locally isometric to a product of symmetric planes. For such manifolds, we prove that the Euler characteristic is an obstruction to the existence of flat structures, confirming an old conjecture proved by Milnor in dimension 2. In particular, the Chern conjecture follows in these cases. The proof goes via a new sharp Milnor--Wood inequality for Riemannian manifolds that are locally a product of hyperbolic planes. Furthermore, we analyze the possible flat vector bundles over such manifolds. Over closed Hilbert--Blumenthal modular varieties, we show that there are finitely many flat structures with nonzero Euler number and none of them corresponds to the tangent bundle. Some of the main results were announced in C.R. Acad. Sci. Paris, Ser. I 346 (2008) 661-666.