TY - JOUR

T1 - The generalized Chern conjecture for manifolds that are locally a product of surfaces

AU - Bucher, Michelle

AU - Gelander, Tsachik

N1 - Funding Information:
✩ Michelle Bucher acknowledges support from the Swedish Research Council (VR) grant 621-2007-6250 and from the Swiss National Science Foundation grant PP00P2-128309/1. Tsachik Gelander acknowledges the financial support from the European Research Council (ERC) grant agreement 203418, and a partial support from the Israeli Science Foundation and the Gustafsson Foundation. * Corresponding author. E-mail addresses: [email protected] (M. Bucher), [email protected] (T. Gelander).

PY - 2011/10/20

Y1 - 2011/10/20

N2 - 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].

AB - 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].

KW - Bounded cohomology

KW - Euler class

KW - Flat structure

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U2 - 10.1016/j.aim.2011.06.022

DO - 10.1016/j.aim.2011.06.022

M3 - Article

AN - SCOPUS:80051571475

SN - 0001-8708

VL - 228

SP - 1503

EP - 1542

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 3

ER -