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

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

2009-02-09

arxiv.org/abs/0902.1215v3

Mathematics

Geometric Topology

36 pages. New title, modified abstract and introduction

Scientific paper

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.

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