Foliation Theory

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The purpose of this research was to investigate the structures and properties of foliations in a broad sense.Head Investigator T.Nishimori has been studying the qualitative theory of similarity pseudogroups in order to extend the qualitative theory of codimension one foliations for foliations of higher codimension. To prepare a firm base for the trial, he gave a detailed proof for Hector's Uniform Convergence Theorem in an extended form of class C^<1+Lipschitz> category.Investigator T.Suwa extended a theorem of Baum and Bott, connecting the residue of singular holomorphic foliations and their topological invariants, to open manifolds, and extended one of his own theorems to singular varieties of higher dimension. Furthermore, by introducing Nash residue, he gave a partial answer to the rationality Conjecture of Baum and Bott.Invesigator I.Nakai gave a geometric interpretation of the curvature form in terms of fake billiard and proved that a weakly associative n-web is associative if Chern connections of triples of the members are not flat, and then the foliations are defined by members of a pencil (projective linear family of dim 1) of 1-forms. This results completed the classification of weakly associative 4-webs initiated by Poincare, Mayrhofer and Reidemeister for the flat case.Investigator H.Minakawa constructed exceptional homomorphisms of the fundamental group of a closed orientable surface to the diffeomorphism group of a circle whose Euler class satisfies the equality bounding Ghys Inequality (arising from Milnor-Wood Inequality). Furthermore he classified exotic circles in the piecewise liniear homeomorphism group of a circle.