Lichnerowicz and Obata theorems for foliations
Lichnerowicz and Obata theorems for foliations
复制标题
叶状结构的 Lichnerowicz 和 Obata 定理
DOI：
10.2140/pjm.2002.206.339
复制
发表时间：
2002
期刊：
影响因子：
--
通讯作者：
Ken Richardson
中科院分区：
文献类型：
--
作者：
Jeffrey M. Lee;Ken Richardson
标准的Lichnerowicz比较定理指出，如果一个封闭的黎曼n流形M的Ricci曲率满足Ric (X， X) > a(n - 1) |X| 2对于每一个固定的> 0的X E TM，那么拉普拉斯最小的正特征值a满足a >和。平等Obata定理指出,当且仅当发生M是等距常数截面曲率的标准n球。在这篇文章中,我们证明如果M是一个封闭的黎曼流形的黎曼叶理余维数问,如果正常里奇曲率满足Ric⊥(X, X)≥(q - 1) | | 2每X在正常包对于一些固定> 0,那么基本的拉普拉斯算子的最小特征值λB满足λB > aq。此外,如果平等时,则叶空间与作用于常截面曲率a的标准q球上的O (q)离散子群的轨道空间是等距的。我们还证明了叶形上的类束度量的一个结果：在任何具有类束度量的黎曼叶形上，存在一个类束度量，其平均曲率是基本的，新度量的基本拉普拉斯量与原度量的基本拉普拉斯量相同。
The standard Lichnerowicz comparison theorem states that if the Ricci curvature of a closed, Riemannian n-manifold M satisfies Ric (X, X) > a(n - 1) |X| 2 for every X E TM for some fixed a > 0, then the smallest positive eigenvalue A of the Laplacian satisfies A > an. The Obata theorem states that equality occurs if and only if M is isometric to the standard n-sphere of constant sectional curvature a. In this paper, we prove that if M is a closed Riemannian manifold with a Riemannian foliation of codimension q, and if the normal Ricci curvature satisfies Ric⊥ (X,X) ≥ a (q - 1) |X| 2 for every X in the normal bundle for some fixed a > 0, then the smallest eigenvalue λ B of the basic Laplacian satisfies λ B > aq. In addition, if equality occurs, then the leaf space is isometric to the space of orbits of a discrete subgroup of O (q) acting on the standard q-sphere of constant sectional curvature a. We also prove a result about bundle-like metrics on foliations: On any Riemannian foliation with bundle-like metric, there exists a bundle-like metric for which the mean curvature is basic and the basic Laplacian for the new metric is the same as that of the original metric.