The study of relations between the cone structures associated with foliations and their differential geometric properties.

点击翻译按钮获取中文摘要

1)The notion of admissible functions for digraphs is introduced, and given a correspondence between Riemannian metrics of foliated manifolds and labeling of digraphs associated to the foliation. As an application, we give a divergence-like characterization of admissible functions on digraphs. We also show this fact directly via purely graph theoretical view point.2)It is shown that for any vector filed N transverse to the given foliation, there are many functions f so that fZ be a mean curvature vector field of the foliation with respect to some Riemannian metric. A characterization of such vector fields are given by studying the cone structure associated to the foliation.3)It is shown that the result stated in (2)can be extended to the case when the plane field is not integrable. It is also shown that, as an application of this characterization, mean curvature vector fields and functions have a stable property with respect to the variations of plane fields.4)Studying Waring Problem by using circle method, new method of estimation of the integrals over minor arcs is obtained.