2005 Fiscal Year Final Research Report Summary
Integral geometry and variational problems in homogeneous spaces
| Project/Area Number |
16540051
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | University of Tsukuba |
|---|
Principal Investigator |
TASAKI Hiroyuki University of Tsukuba, Graduate School of Pure and Applied Sciences, Associate Professor, 大学院・数理物質科学研究科, 助教授 (30179684)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
ITOH Mitsuhiro University of Tsukuba, Graduate School of Pure and Applied Sciences, Professor, 大学院・数理物質科学研究科, 教授 (40015912)
YASUKURA Osami University of Fukui, Faculty of Engineering, Professor, 工学部, 教授 (00191122)
HASHIMOTO Hideya Meijo University, Faculty of Science & Technology, Professor, 理工学部, 教授 (60218419)
IKAWA Osamu Fukushima National College of Technology, Department of General Education, Associate Professor, 一般科, 助教授 (60249745)
KOKUBU Masatoshi Tokyo Denki University, Department of Mathematical Sciences, Associate Professor, 工学部, 助教授 (50287439)
|
|---|
| Project Period (FY) |
2004 – 2005
|
| Keywords | homogeneous space / integral geometry / differential geometry / Poincare formula / Crofton formula / reflective submanifold |
|---|
| Research Abstract |
In the last academic year we established a Crofton formula in Riemannian symmetric spaces by the use of reflective submaifolds, which is totally geodesic. In order to get explicite expression of the Crofton formula we need some geometric invariants of submanifolds. In the case where the head investigator gave an explicit expression of Poincare formula of submanifolds in complex space forms, he introduced a notion of multiple Kahler angle. By the use of the multiple Kahler angle he could obtain an explicit Crofton formula of submanifolds in complex space forms. For the purpose that we extend the class of submanifolds we use in Crofton formula in Riemannian symmetric spaces, we extend the notion of reflective submanifolds to that of weakly reflective submanifolds. Weakly reflective submanifolds are special ones of minimal submanifolds. Some arguments show that austere submanifolds are weakly reflective. In order to get austere submanifolds in the spheres we described a condition for a submanifold to be austere among the orbits of the linear isotropy actions of Riemannian symmetric pairs. As a result of this we obtained a classification of austere orbits in the spheres under the linear isotropy actions. We observed that some of them are invariant under an isometry of the sphere which reverses the submanifold with respect to the normal directions. So we called such submanifolds weakly reflective submanifolds and started our research of weakly reflective submanifolds.
|
