2000 Fiscal Year Final Research Report Summary
Isometric imbedding of Riemannian manifolds
| Project/Area Number |
10640079
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Hiroshima University |
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Principal Investigator |
AGAOKA Yoshio Faculty of Integrated Arts and Sciences, Hiroshima University, Associate Professor, 総合科学部, 助教授 (50192894)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAYAMA Hiromichi Faculty of Integrated Arts and Sciences, Hiroshima University, Associate Professor, 総合科学部, 助教授 (30227970)
YOSHIDA Toshio Faculty of Integrated Arts and Sciences, Hiroshima University, Professor, 総合科学部, 教授 (10033854)
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| Project Period (FY) |
1998 – 2000
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| Keywords | isometric imbedding / symmetric space / Gauss equation / root system / rigidity / plethysm / Grassmann algebra |
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| Research Abstract |
In this research, we obtained the following results on isometric imbeddings of Riemannian manifolds :
1. We determine the value of the intrinsic invariant p(G/K) for many Riemannian symmetric spaces G/K, and obtain the estimates on the dimension of the Euclidean space into which G/K can be locally isometrically immersed. In particular, for the spaces Sp(m)/U(m) and Sp(m), the least dimensional Euclidean spaces are determined.
2. We show that the symmetric space SU(3)/SO(3) and its non-compact dual space admit solutions of the Gauss equation in codimension 5, and also admit almost solutions in codimension 4.
3. We determine the rank of the quadratic map defined by the Gauss equation for the case dim M【less than or equal】9. This result shows the existence an obstruction of local isometric imbeddings for the case M^9⊂R^<23>.
4. We give a new formulation of the Gauss equation in the exterior algebra, and state the relation to the original equation.
5. It is quite important to know the GL(V)-irreducible decomposition of the polynomial ring on the space of curvature like tensors. This is a sort of "plethysm" appeared in the representation theory. We give some decomposition formulas of special plethysms.
6. We show that the least dimensional Euclidean space into which the quaternion projective plane can be locally isometrically immersed is R^<14>.
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