## 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 [M. Bucher, T. Gelander, Milnor-Wood inequalities for manifolds locally isometric to a product of hyperbolic planes, C. R. Acad. Sci. Paris Ser. I 346 (2008) 661-666].

Original language | English (US) |
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Pages (from-to) | 1503-1542 |

Number of pages | 40 |

Journal | Advances in Mathematics |

Volume | 228 |

Issue number | 3 |

DOIs | |

State | Published - Oct 20 2011 |

## Keywords

- Bounded cohomology
- Euler class
- Flat structure

## ASJC Scopus subject areas

- Mathematics(all)