局部为曲面积流形的广义Chern猜想
The generalized Chern conjecture for manifolds that are locally a product of surfaces
摘要：
我们考虑闭流形，它允许度量局部等距于对称平面的乘积。对于这样的流形，我们证明了欧拉示性数是一个障碍的存在，平坦的结构，证实了一个古老的猜想证明米尔诺在2维。特别是，陈省身猜想如下在这些情况下。证明是通过一个新的尖锐的米尔诺尔-伍德不等式的黎曼流形，局部的产品的双曲平面。进一步，我们分析了这类流形上可能的平坦向量丛。在闭的Hilbert-Blumenthal模簇上，我们证明了存在许多具有非零欧拉数的平坦结构，并且没有一个对应于切丛.一些主要结果在C.R.上公布。Acad. Sci. Paris，Ser. I 346（2008）661-666.
英文摘要：
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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