| Co-Investigator(Kenkyū-buntansha) |
OTSU Yukio Kyushu University, Graduate School of Mahematics, Ass. Prof., 大学院・数理学研究院, 助教授 (80233170)
NAGATOMO Yasuyuki Kyushu University, Graduate School of Mahematics, Ass. Prof., 大学院・数理学研究院, 助教授 (10266075)
YAMADA Kotaro Kyushu University, Graduate School of Mahematics, Prof., 大学院・数理学研究院, 教授 (10221657)
UMEHARA Masaaki Osaka Univ., Graduate School of Science, Prof., 大学院・理学研究科, 教授 (90193945)
ISHIKAWA Goo Hokkaido Univ., Graduate School of Science, Ass. Prof., 大学院・理学研究科, 助教授 (50176161)
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| Research Abstract |
I proved the homogeneity of isoparametric hypersurfaces with six principal curvatures with multiplicity two, which I had been tackling for several years. I also got a new proof of Dorfmeister-Neher's theorem which treats the multiplicity one case, in a unified manner.
Investigating the resulted homogeneous hypersurfaces, I got the following As was known in the case of multiplicity one, the hypersurfaces with 6 principal curvatures are given as a fibration over those with 3 principal curvature, where the fibers aret otally geodesic spheres. In the case of multiplicity two, the fiber dimension is six, while in the case of multiplicity one, this is three. Discovery of the fibration structure is an extension of our former results on the degenerate Gauss mapping which was done with G. Ishikawa and M. Kimura.
Moreover, using the fact that the family of isoparametric hypersurfaces fill the ambient space, we get an interesting relation between 13-dimensional sphere and 7-dimensional sphere. Furthermore, using that these hypersurfaces are given as orbits of the exceptional group G_2, we can show that there exists a metric on S^7-CP^2 of which holonomy group is G_2. From this, a real open version of Calabi conjecture will be considered, i.e., when a compact Riemannian manifolds with positive Ricci curvature from which a certain part removed, admits a metric with G_2 holonomy? In this way, hypersurfaces obtained as G_2 orbits suggest us very important and interesting problems.
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