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

首先考虑了（2 p +2 q-1）-球面和复射影（p+q-1）-空间上的光滑SU（p，q）作用，其对极大紧子群S（U（p）×U（q））的限制作用是标准的.定理1在具有三个轨道的（2 p +2 q-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，定理中的每个SU（p，q）作用诱导复射影（p+q-1）-空间上的光滑SU（p，q）作用。定理2在限制S（U（p）×U（q））作用是标准的复射影（p + q-1）空间上的光滑SU（p，q）作用的等价类集与对（g，h_i）（i= 1，2）的等价类集有一一对应，其中g：P_1（R）→R，h_i：U_i→R（U_1 <$U_2 =P_1（R））是满足某些四个条件的光滑函数.推论3在复射影（p+q-1）-空间上存在两个光滑作用，它们具有三个直到等变自同态类的轨道.其次，我们考虑了（2 m +2n-1）-球面上的SL（m，C）×SL（n，C）光滑作用，它对极大紧子群SU（m）×SU（n）的限制作用是标准的.设SL（m，C）×SL（n，C）光滑作用是（2 m +2n-1）-球面上的光滑作用，C）×SL（n，C）作用Φ，则子群M作用在子流形F上，其中M同构于R^2，F同构于S^3。定理4设Φ，Φ'为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（2 m +2n-1）-球面上存在无穷多个光滑SL（m，C）×SL（n，C）作用量，它们的（2 p-1）个轨道的限制SU（m）×SU（n）作用量是标准的.

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.