Study of non-compact transformation groups acting smoothly on some smooth manifolds

首先，我们考虑了(2p+2q-1)-球面上的光滑SU(p，q)作用和对极大紧子群S(U(P)×U(Q))的约束作用为标准的复射影(p+q-1)-空间上的SU(p，q)作用。我们将作用归类到等变同构类.定理1在具有三个轨道的(2p+2q-1)-球面上存在无穷多个光滑SU(p，q)作用，使得对S(U(P)×U(Q))的限制作用是标准的.这个定理中的每个SU(p，q)作用都诱导出复射影(p+q-1)-空间上的光滑SU(p，q)作用.但复射影(p+q-1)-空间上的所有诱导作用都是标准的.定理2在限制S(U(P)×U(Q))作用为标准的复射影(p+q-1)-空间上的光滑SU(p，q)作用的等价类集合与(g，h_i)(i=1，2)对的等价类集合之间存在一一对应，其中g：P_1(R)→R，H_i：U_i→R(U_1∪U_2=P_1(R))是满足四个条件的光滑函数。推论3在具有三个轨道直到等变同构类的复射影(p+q-1)-空间上存在两个光滑作用。接下来我们考虑(2m+2n-1)-球面上的光滑SL(m，C)×SL(n，C)作用，其对极大紧子群SU(M)×SU(N)的限制作用是标准的。设上述定义的SL(m，C)×SL(n，C)作用Φ被给定，则子群M作用在子流形F，C上其中M同构于R^2，F微分同构于S^3。我们用Φ_M表示F上的受限M作用。定理4设Φ，Φ‘是上面定义的SL(m，C)×SL(n，C)作用。则Φ=Φ‘当且仅当Φ_M=Φ’_M。推论5设p为整数。则在具有(2p-1)轨道的(2m+2n-1)-球面上存在无穷多个光滑的SL(m，C)×SL(n，C)作用，其受限的SU(M)×SU(N)作用是标准的。

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.