1998 Fiscal Year Final Research Report Summary
Study of smooth non-compact Lie group actions on compact smooth manifolds
| Project/Area Number |
09640140
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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 | Tokyo Metropolitan College of Aeronautical Engineering |
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Principal Investigator |
MUKOYAMA Kazuo Tokyo Metro.College.of A.E., General Edu., Prof., 一般科, 教授 (60219847)
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| Co-Investigator(Kenkyū-buntansha) |
KADOWAKI Mituteru Tokyo Metro.C.of A.E., General Edu., Lecturer, 一般科, 講師 (70300548)
ONO Tomoaki Tokyo Metro.College.of A.E., General Edu., Assis.Prof., 一般科, 助教授 (00224270)
SUGIE Michio Tokyo Metro.College.of A.E., General Edu., Prof., 一般科, 教授 (90216309)
MIYAUCHI Mutsuo Tokyo Metro.College.of A.E., General Edu., Prof., 一般科, 教授 (00219726)
TOYONARI Toshitaka Tokyo Metro.College.of A.E., General Edu., Prof., 一般科, 教授 (20217582)
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| Project Period (FY) |
1997 – 1998
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| Keywords | smooth action / non-compact Lie group / orbit space |
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| Research Abstract |
It is important to study non-compact Lie group actions on compact smooth manifolds.
In this note, we classify smooth SU(p, 1)-actions on the (2p+2q-1)-sphere and on the complex projective (p+q-1)-space whose restriction to the maximal compact subgroup S(U (p) X U (q)) is standard.
The results are as follows.
Let U, V be the set of smooth SU(p, q)-actions on P_<p+q-1> (C), S^<2p+2q-1> whose restricted S(U(p) X U(q))-action is standard, respectively.
1. There is a one-to-one correspondence between U and the set of pairs (phi', f' ), where phi' is a smooth N'(p, q)=SU(1,1)-action on P_1(C) whose restriction on S(U (p) X U (q))*N'(p, q) is standard and f : P_1(C)*P_1(C) is a smooth N'(p, q)-equivariant map satisfying one condition.
From this fact we see that there exist infinitely many smooth actions on P_<p+q-1>(C) which are mutually distinct.
2. There is a one-to-one correspondence between V and the set of pairs (phi, f) , where phi is a smooth N(p, q)=U(1, 1)-action on S^3 whose restriction on S(U(p) X U(q)) *N(p, q) is standard and f : S^3*P_1(C) is a smooth N(p, q)-equivariant map satisfying one condition.
3. There exists a mapping rho : V*U which is onto but not one-to-one.
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