Many-sided Research of Foliations

本研究的目的是从多方面的角度来研究叶理。首席研究员西森俊之（NISHIMORI Toshiyuki）一直在研究相似伪群的定性理论，以发展高协维叶状的定性理论。主要的主题是在余维一叶理的定性理论中找到经典定理的高余维类比，并证明了在每个Sacksteder系统中每个带泡轨道的闭包中存在一个收缩不动点。在这项研究中，首席研究员的目标是找到带有气泡的轨道出现的条件。结果证明，对于每一个强半固有轨道，当且仅当它具有有界的乘函数时，它是带泡的。作为这个结果的一个稍微推广的版本，证明了对于每一个强半固有轨道，当且仅当它有一个有界的概乘函数时，它是几乎带泡的。证明的要点是每个强半固有轨道都有一个非空的开放区域。研究者SUWA tatsuo研究了奇异全纯叶的残数，得到了一些结果。研究者以洛伦兹度规为主题，研究流形上的全测地线叶理。他们研究了时间状叶、空间状叶和光状叶的基本例子，并得出了一些结果。

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.