Members of Koike laboratory study Differential Geometry which closely related to Einstein's General Relativity.
Mainly we study Submanifold Theory in Symmetric Spaces.
Main two methods of the research are as follows: 1. Research of submanifolds in symmetric spaces of compact type
〜 by reducing research of submani-
folds in an infinite dimensional
linear space obtained by canceling
holonom of the symmetric space〜 We reasearch submanifolds in a symmetric space
of compact type by reducing to reasearch of sub-
manifolds in an infinite dimensional Hilbert space
through a Riemannian submersion of the Hilbert
space onto the symmetric space. This reducement
of research is valid because symmetric spaces of compact type has holonom
but the Hilbert space
has no holonom.
2. Research of submanifolds in symmetric
spaces of noncompact type
〜 by reducing to research of submani-
folds in an infinite dimensional
complex linear space obtained by
canceling holonom of the complexi-
fication of the symmetric space〜 We reasearch submanifolds in a symmetric space
of compact type by reducing to reasearch of sub-
manifolds in an infinite dimensional anti-Kaehlerian
space through a Riemannian submersion of the anti-Kaehlerian space onto the complexification
of the symmetric space. This reducement of research
is valid in the following two aspects:
(1) A symmetric space of noncompact type has
holonom but the infinite dimensional anti-Kaehlerian
space has no holonom.
(2) Some geometrical structures (of the submanifold) vanishing beyond the
ideal boundary of a symmetric
space of non-compact type appear in the complexified space.
Please click here to see a survey of my
recent research for a proper complex
equifocal submanifold.
(Update : January 17, 2011)
