Metrics on complex manifold

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Our results can be divided into three parts ; (I) concerning commutative algebra, (II) concerning differential topology, and (III) concerning probability theory.(I) A. Kato considers modules over a commutative ring R, then he studies modules modulo submodules of a generalized length zero. Fixing a generalized length L, the concepts of L-torsion, L-coannihilator, L-nilpotent, and L-coprimary module can be considered. Especially he gives a L-coprimary decomposition of a module modulo a submodule whose L equals zero.(II) M. Kurata proves the following ; for measure-preserving diffeomorphisms of closed manifolds, there are families of locally invariant manifolds corresponding to Lyapunov exponents less than, where being non-negative.(III) Y. Hasegawa develops a potential theory on a certain hypersurface S of infinite dimensional real sequences space E. M. Yamazato characterizes the class of hitting time distributions of single points of generalized diffusion processes.