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

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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.