Many-sided Research of Foliations

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The purpose of this research was to study foliations from many sided points of view.The head investigator (NISHIMORI Toshiyuki) had been studying the qualitative theory of similarity pseudogroup in order to develop the qualitative theory of foliations of higher codimension. The main theme was to find a higher codimensional analogy of classical theorems in the qualitative theory of codimension-one foliations, and proved that there is a fixed point of a contraction in the closure of each orbits with bubbles in each Sacksteder system. In this research, the aim of the head investigator was to find the condition under which orbits with bubbles appear. As a results, it was proved that, for each strongly semiproper orbit, it is with bubbles if and only if it has a bounded multiplicative function. As a somewhat generalized version of this result, it was proved that, for each strongly semiproper orbit, it is almost with bubbles if and only if it has a bounded almost multiplicative function. The point of the proof was each strongly semiproper orbit has a non-empty open territory.The investigator SUWA tatsuo studied the residues of singular holomorphic foliations and obtained some results. The investigators took totally geodesic foliations on manifolds with Lorentzian metric as the theme. They studied fundamental examples of timelike leaves, spacelike leaves and lightlike leaves and some results.