On Fundamental Formulas of Foliations.

In recent years, there have been several studies of foliations from differential geometric aspects. Especially, many differential geometric properties of metric foliations have been studied. In those studies, O'Neill's fundamental equations played a central role. These equations are derived in the study of differential geometric properties of Riemannian submersions, which is defined by O'Neill and is a special class of metric foliations, which we do not treat in this paper. For general foliations, many results are also obtained. However, the approach is done from various view points depending on the mathematicians and, at present, there seems to be no systematic approach. In this paper, we present some fundamental formulas for the differential geometric study of arbitrary foliations. These formulas turns out to be useful especially when curvature conditions are given. We give new proofs of many known results from this view points, and also give some newresults