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Differential Geometric Approach to Foliated Structures.

叶状结构的微分几何方法。

基本信息

批准号：
10640055
负责人：
OSHIKIRI Gen-ichi
金额：
$ 2.05万
依托单位：
Iwate University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2000
项目状态：
已结题

项目摘要

(a) We get the following result :Let (M,F,g) be a codimension-q bundle-like foliation on a closed Riemannian manifold of positive curvature. (1) If q is even, then F has a compact leaf. (2) If q is odd, then F has a leaf whose closure is a closed codimsnsion-(q-1) submanifold.As a corollary, we extend Berger's famous result :Any Killing vector field on a closed Riemannian manifold with positive sectional curvature admits a zero point or a closed orbit.(b) We study the dual 1-form to the mean curvature vector of a foliation. We give a characterization of such 1-forms for codimension-one foliations. We also have a simple characterization when the foliation is a bundle foliation, and when the dual 1-form is basic.(c) We get the following result :Let (M,F,g) be a codimension-1 minimal foliation on a complete Riemannian manifold of non-negative Ricci curvature. If the growth of F is not greater than 2, then F is totally geodesic. Further, (M,g) is locally a Riemannina product.As a byproduct, we get a simple proof of Mirand's result on minimal graphs, and foliated version of the result by Alencar and do Carmo on constant mean curvature hypersurfaces.

(a)我们得到了如下结果：设（M，F，g）是正曲率闭黎曼流形上的余维q类球面叶理。(1)如果q是偶数，则F有紧叶。(2)如果q是奇数，则F有一个叶，其闭包是闭余维-（q-1）子流形.作为推论，我们推广了Berger的著名结果：在具有正截面曲率的闭黎曼流形上，任何Killing向量场都有一个零点或一个闭轨. (b)研究叶理平均曲率向量的对偶1-形式。我们给出了余维为1的叶理的1-形式的一个特征。当叶理是丛叶理时，当对偶1-形式是基本的时，我们也有一个简单的刻画。(c)我们得到了如下结果：设（M，F，g）是具有非负Ricci曲率的完备Riemann流形上的余维1极小叶理。如果F的增长不大于2，则F是全测地线。进一步地，（M，g）是局部Riemannina积，作为副产品，我们得到了关于极小图的Mirand结果的一个简单证明，以及Alencar和do Carmo关于常平均曲率超曲面的结果的一个叶状版本.