Nonnegatively Curved Manifolds with Finite Fundamental Groups Admit Metrics with Positive Ricci Curvature
Nonnegatively Curved Manifolds with Finite Fundamental Groups Admit Metrics with Positive Ricci Curvature
复制标题
具有有限基本群的非负曲流形承认具有正里奇曲率的度量
DOI：
--
复制
发表时间：
2007
期刊：
影响因子：
--
通讯作者：
Burkhard Wilking
中科院分区：
文献类型：
--
作者：
C. Böhm;Burkhard Wilking
在本文中，我们解决了非负截面曲率的完整黎曼度量是否可以变形为正里奇曲率度量的问题。这个问题在最近各种具有正里奇曲率的度量的新构造中隐含地出现了。 Grove 和 Ziller [GZ] 表明，任何具有有限基本群的紧致同质流形都承认具有正 Ricci 曲率的不变度量。两个非规则轨道都具有余维二的情况尤其具有弹性。通过 Grove 和 Ziller 的早期工作，我们知道这些流形允许不变的非负弯曲度量。然而，在某些情况下，这些度量的 Ricci 曲率在任何点都不是正的，因此它们不能应用 Aubin [A] 和 Ehrlich [E] 的变形定理：当且仅当 Ricci 曲率在某个点为正时，非负 Ricci 曲率的度量与具有正 Ricci 曲率的度量共形等价。 Schwachhöfer 和 Tuschmann 关于商空间 [ST] 的工作中也出现了类似的问题。我们的主要结果是： 定理A。设(Mn, g) 是具有有限基本群和非负截面曲率的紧黎曼流形。那么 Mn 承认一个具有正里奇曲率的度量。
In this paper we address the question whether a complete Riemannian metric of nonnegative sectional curvature can be deformed to a metric of positive Ricci curvature. This problem came up implicitly in various recent new constructions for metrics with positive Ricci curvature. Grove and Ziller [GZ] showed that any compact cohomogeneity one manifold with finite fundamental group admits invariant metrics with positive Ricci curvature. The case that both non-regular orbits have codimension two is especially resilient. By earlier work of Grove and Ziller it has been known that these manifolds admit invariant nonnegatively curved metrics. However, in certain cases the Ricci curvature of these metrics is not positive at any point and hence they cannot apply the deformation theorem of Aubin [A] and Ehrlich [E]: a metric of nonnegative Ricci curvature is conformally equivalent to a metric with positive Ricci curvature if and only if the Ricci curvature is positive at some point. Similar problems arise in the work of Schwachhöfer and Tuschmann on quotient spaces [ST]. Our main result is: Theorem A. Let (Mn, g) be a compact Riemannian manifold with finite fundamental group and nonnegative sectional curvature. Then Mn admits a metric with positive Ricci curvature.