Study on the geometry of symmetric spaces and their totally geodesic submanifolds

点击翻译按钮获取中文摘要

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 geometryAs a development of this research, we understood the study the Grassmann geometries on Lie groups with left invariant metric. So we studied the Grassmann geometries on the 3-dimensional nilpotent Lie group called Heisenberg group and two 3-dimensional unimodular Lie groups, and obtained the classification of their Grassmann geometries and the details about the associated surface theories. As a result, we found that the Grassmann geometry is closely related to the structure of Lie group. The study for Heisenberg case is a collaboration with Inoguchi and Kuwabara.3.A view for future studyA problem remaining in this research is the complete classification of general totally geodesic submanifolds of Riemannian symmetric spaces. Aso, the Grassmann geometry on Lie groups should be developed much.