| Co-Investigator(Kenkyū-buntansha) |
ONO Tomoaki TMCAE Dep. of Gen. Education, Assis. Professor, 一般科, 助教授 (00224270)
SUGIE Michio TMCAE Dep. of Gen. Education, Professor, 一般科, 教授 (90216309)
TOYONARI Toshitaka TMCAE Dep. of Gen. Education, Professor, 一般科, 教授 (20217582)
KADOWAKI Mituteru TMCAE Dep. of Gen. Education, Assis. Professor, 一般科, 助教授 (70300548)
NAKAYA Hideki TMCAE Dep. of Gen. Education, Assis. Professor, 一般科, 助教授 (20271489)
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| Research Abstract |
First we consider smooth SU(p,q) actions on the (2p+2q-1)-sphere and on the complex projective (p+q-1)-space whose restrictedaction to the maximal compact subgroup S(U(p)×U(q)) is standard. We classified the actions up to the equivariant difeomorphism classes.
Theorem 1 There exist infinitely many smooth SU(p,q) actions on the (2p+2q-1)-sphere with three orbits such that the restrictedactions to S(U(p)×U(q)) are standard.
Each SU(p,q) action in this theorem induces the smooth SU(p,q) action on the complex projective (p+q-1)-space. But all inducedactions on the complex projective (p+q-1)-spaceare standard.
Theorem 2 There is a one-to-one correspondence between the set of equivalence classes of smooth SU(p,q) actions on the complex projective (p+q-1)-space whose restricted S(U(p)×U(q)) action is standard and the set of equivalence classes of pairs (g,h_i) (i=1,2), where g : P_1(R)→R, h_i : U_i→R (U_1∪U_2=P_1(R)) are smooth functions satisfying some four conditions.
Corollary 3 There exist two smooth actions on the complex projective (p+q-1)-space with three orbits up to equivariant difeomorphism classes.
Next we consider smooth SL(m,C)×SL(n,C) actions on the (2m+2n-1)-sphere whose restrictedaction to the maximal compact subgroup SU(m)×SU(n) is standard.
Let SL(m,C)×SL(n,C) action Φ definedabove be given, then the subgroup M acts on the submanifold F, where M is isomorphic to R^2 and F is diffeomorphicto S^3. We denote the restrictedM action on F byΦ_M.
Theorem 4 Let Φ, Φ' be SL(m,C)×SL(n,C) actions definedabove. Then Φ=Φ' if and only if Φ_M=Φ'_M.
Corollary 5 Let p be an integer. Then there exist infinitely many smooth SL(m,C)×SL(n,C) actions on the (2m+2n-1)-sphere with (2p-1) orbits whose restrictedSU(m)×SU(n) actions are standard.
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