Graduate School of Science and Engineering Natural Sciences and Mathematics(Science) Graduate School of Science and Engineering(Science) YAMAGUCHI UNIVERSITY

Job title

Professor

Degree

Doctor of Science(Osaka University), Master of Science(Osaka University), Bachelor of Science(Osaka University)

This investigation is on totally geodesic submanifolds of Riemannian symmetric spaces and the Grassmann geometry of submanifolds associated with them. Such typical submanifolds are symmetric submanifolds.
1.Fundamental results on symmetric submanifolds
(1)We clarified the relationship between the construction of symmetric submanifolds and
the theory of Jordan triple system and the associated symmetric R-space, and obtained a summary on the history and transition on these research fields.
(2)We next clarified the details of symmetric submanifolds in the higher-rank irreducible
Riemannian symmetric spaces of noncompact type. This is a collaboration with Berndt, Eschenburg, and Tsukada.
(3)Summing up these results, we published a paper on the classification of symmetric
submanifolds of general Riemannian symmetric spaces in Japanese. This is a collaboration
with Tsukada. This result was announced in a JSPS-DFG seminar held at Kyoto University. A translation of this paper will be also issued in the journal 'Sugaku Expositions' of the American Mathematical Society.
2. Development into another Grassmann geometry
As a development of this research, we also studied the Grassm

2005

This study is on the Grassmann geometry on the Riemannian homogeneous spaces. Our aim is to consider the classification problem of extrinsic homogeneous submanifolds of Riemannian symmetric spaces. For this, in this study, we examine the case where a Riemannian homogeneous space is a 3-dimensional unimodular Lie group with a left invariant metric. The 3-dimensional unimodular Lie groups are classified into six ones; the 3-dimensional vector group, the 3-dimensional Heisenberg group, the groups of rigid motions of the Eucliden 2-plane and the Minkowski 2-plane, the special unitary group SU(2), and the special linear group SL(2,R). Also, for each of them the geometric properties such as the curvatures, the isometry group, and so on, can be expressed concretely. In this study we in particular consider the Grassmann geometry on the spaces SU(2) and SL(2,R), while the cases of the Heisenberg group and the groups of rigid motions of the Eucliden 2-plane and the Minkowski 2-plane are clarify by H. Naitoh, J. Inoguchi, and K. Kuwabara.
The obtained main results are the following.:
for both the spaces SU(2) and SL(2,R),
(1) the classification for all the orbits associated with Gras