Differential Geometric Approach to Foliated Structures.

(A)我们得到如下结果：设(M，F，g)是闭的正曲率黎曼流形上的余维-Q丛样叶。(1)如果Q是偶数，则F有紧叶。(2)如果q是奇数，则F有一个闭码-(q-1)子流形的叶.作为推论，我们推广了Berger的著名结果：具有正截面曲率的闭黎曼流形上的任何Kill向量场都允许零点或闭轨.(B)我们研究了叶层的平均曲率向量的对偶1-形式.我们给出了余维一叶的这种1-形式的刻画。得到了如下结果：设(M，F，g)是非负Ricci曲率完备黎曼流形上的余维-1极小叶化.如果F的增长量不大于2，则F是全测地线。进一步，(M，g)是局部Riemannina积.作为副产品，我们得到了极小图上Mirand结果的一个简单证明，并得到了Alencar和Do Carmo在常平均曲率超曲面上结果的分叶版本.

(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.