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

首先考虑复射影(p+q-1)-空间CP(p+q-1)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。设(g，hi)(i=1，2)是光滑函数对g：Rp(1)→R和Hi：U_i→R(U_1∪U_2=Rp(1))满足一定的微分方程和其它三个条件。然后，我们可以在满足前面条件的CP(p+q-1)上构造一个光滑的SU(p，q)作用量。我们已经构造了两个这样的对，使得诱导出的SU(p，q)作用是不同的。这一次我们发现了新的三双。接下来，我们考虑复射影(m+n-1)-空间上的光滑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。定理1在CP(m+n-1)上存在无穷多个光滑的SL(m，C)×SL(n，C)作用，其中两个轨道是紧的，一个轨道是开的.设Ψ是一个光滑的SL(m，C)×SL(n，C)作用，其对SU(M)×SU(N)的限制作用是标准的.定理2设Ψ‘是SL(m，C)×SL(n，C)作用，其对SU(M)×SU(N)的限制是标准的。如果Ψ(c，d)=Ψ‘(c’，d‘)且d-c=d’-c‘，则Ψ=Ψ’.定理3设α，β：r^2×Cp(1)→Cp(1)是满足三个条件的函数，则我们可以在Cp(m+n-1)上构造一个抽象的SL(m，C)×SL(n，C)作用，它对SU(M)×SU(N)的限制是标准的.一般来说，这个作用是不光滑的.在不久的将来，我们将构造光滑的SL(m，C)×SL(n，C)具有k个轨道(k&gt；3)利用这一定理。

First we consider smooth SU(p,q) actions on the complex projective (p+q-1)-space CP(p+q-1) whose restricted action to the maximal compact subgroup S(U(p)×U(q)) is standard. The first aim of this study is to construct such actions and to classify the actions up to the equivariant diffeomorphism classes.Let (g,h_i) (i=1,2) be the pair of smooth functions g:RP(1)→R and h_i:U_i→R (U_1∪U_2= RP(1)) satisfying some differential equations and other three conditions. Then we can construct a smooth SU(p,q) action on CP(p+q-1) satisfying the previous condition. We already constructed such two pairs that the induced SU(p,q) actions are different. This time we found new three pairs. It is necessary to show that the actions induced from the pairs are same or not.Next we consider smooth SL(m,C)×SL(n,C) actions on the complex projective (m+n-1)-space whose restricted action to the maximal compact subgroup SU(m)×SU(n) is standard. We also have studied such actions on the (m+n-1)-sphere.Theorem 1 There exist infinitely many smooth SL(m,C)×SL(n,C) actions on CP(m+n-1) with three orbits, where two orbits are compact and one orbit is open.Let Ψ be a smooth SL(m,C)×SL(n,C) action whose restricted action to SU(m)×SU(n) is standard. Then we can construct a R^2 actionΨ(c,d) on CP(1) for some real numbers c, d.Theorem 2 Let Ψ, Ψ' be SL(m,C)×SL(n,C) actions whose restriction to SU(m)×SU(n) are standard. If Ψ(c,d)=Ψ'(c',d') and d-c=d'-c', thenΨ=Ψ'.Theorem 3 Let α, β :R^2×CP(1)→CP(1) be the functions satisfying some three conditions, then we can construct an abstract SL(m,C)×SL(n,C) action on CP(m+n-1) whose restriction to SU(m)×SU(n) is standard.In general, the action is not smooth.In the near future, we will construct smooth SL(m,C)×SL(n,C) actions on CP(m+n-1) with k orbits (k>3) using this theorem.