The study of relations between topological properties and differential geometric properties of foliated structures.

1)证明了具有非负Ricci曲率的完备黎曼流形的余维1极小叶理是全测地的，如果它的增长不大于2。进一步证明了米兰达关于欧氏空间中极小图的第二基本形式的平方模的积分的估计。2）得到了正曲率闭黎曼流形上余维q度量叶理的一类“紧叶定理”。作为这个结果的一个推论，得到Berger关于Killing场的结果的一个推广：证明了正曲率闭黎曼流形上的Killing场都有零点或闭轨道; 3）证明了Cheeger常数可以定义在（di-）图上，并且与（di-）图的连通性有关; 4）证明了容许函数的概念，这已经被定义为余维一叶理，也可以定义在有向图，这两个概念之间的“容许函数”通过对应的叶流形与相关的有向图有很强的关系。作为应用，得到了有向图的容许函数的类发散特征。

1) It is shown that codimension-one minimal foliation of a complete Riemannian manifold with non-negative Ricci curvature is totally geodesic if the growth of the foliation is not greater than 2. Further, an another proof of the estimate given by Miranda on the integral of the square norm of the second fundamental form of minimal graphs in Euclidean Spaces is obtained2) A kind of "Compact Leaf Theorem" of codimension-q metric foliations on closed Riemannian manifolds with positive curvature is obtained. As a corollary to this result, an extension of Berger's result on Killing fields is obtained : Any Killing field on a closed Riemannian manifolds with positive curvature has zero points or closed orbits.3) It is shown that Cheeger constant can be defined on (di-)graphs, and is related to connectivities of (di-)graphs.4) It is shown that the notion of admissible functions, which had already been defined for codimension-one foliations, can also be defined on digraphs, and that there is a strong relation between these two notions of "admissible functions" via the correspondence of a foliated manifold with the associated digraph. As an application, a divergence-like characterization of admissible functions of digraphs are obtained.