| Co-Investigator(Kenkyū-buntansha) |
KATO Takao Faculty of Science, Yamaguchi University Professor, 理学部, 教授 (10016157)
KOMIYA Katsuhiro Faculty of Science, Yamaguchi University Professor, 理学部, 教授 (00034744)
INOUE Toru Faculty of Science, Yamaguchi University Professor, 理学部, 教授 (00034728)
NAKAUCHI Nobumitsu Faculty of Science, Yamaguchi University Associate Professor, 理学部, 助教授 (50180237)
SHIMA Hirohiko Faculty of Science, Yamaguchi University Professor, 理学部, 教授 (70028182)
|
|---|
| Research Abstract |
This investigation is on the submanifold theory of compact simply connected Riemannian symmetric spaces. We study it by using a Grassmann geometry, introduced by R.Harvey and H.B.Lawson in their consideration of calibrated geometry. Particularly we study a Grassmann geometry of orbital type. The main subjects treated here are the following three : (1) the existence problem, (2) the classification, and (3) applications for submanifold theory. For each subject, we have obtained the following results and foreknowledges :
(1) Generally, given a Grassmann geometry of orbital type, the existence problem whether the geometry admits associated submanifolds or not is equivalent to the local solvability of a certain system of 1st order PDE's defined on the isometry group of the ambiant symmetric space. Moreover, under this equivalence, the geometrical property of an associateed submanifold, what is called the 2nd fundamental form, is charac- terized in terms of a solution of the system of 1st order PDE's. Also, for the orbital Grassmann geometries of curves and the ones of real hypersurfaces, the existence problem has been solved affirmatively, and for the orbital Grassmann geometries of strongly curvature-invariant type the geometric structure of associated submanifolds has been clarified.
(2) Among Grassmann geometries of orbital type, there exists an important class, what is called of totally geodesic type. The Grassmann geometries of strongly curvature-invariant type, described above, constitute a subclass of the class of totally geodesic type. We have completed the classification of this subclass, by using such finite diagrams as Dynkin's diagrams. We suppose that this method also is useful even for the cases of general totally geodesic type.
(3) As applications of Grassmann geometry, we have the classification of symmetric submanifolds and the generalization of Gauss mappings. We suppose that these notions are also very useful for the submanifold theory of symmetic spaces.
|
