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，其对最大紧子群S(U（p)×U(q)）的限制作用是标准的。本研究的第一个目的是构造这些动作，并将这些动作划分为等变微分同构类。设（g,h_i） （i=1,2）是光滑函数g:RP(1)→R和h_i:U_i→R （U_1∪U_2= RP(1)）对满足微分方程和其他三个条件。然后我们可以构造一个满足前一个条件的光滑SU（p,q）作用于CP（p+q-1）。我们已经构造了这样两对，诱导的SU（p,q）作用是不同的。这次我们找到了新的三对。有必要证明由这些对引起的作用是相同的还是不相同的。其次，我们考虑光滑SL（m，C）×SL（n，C）作用在复投影（m+n-1）空间上，其对最大紧子群SU(m)×SU(n)的限制作用是标准的。我们还研究了（m+n-1）球上的这类作用。定理1存在无穷多个光滑SL（m，C）×SL（n，C）作用于具有三个轨道的CP（m+n-1）上，其中两个轨道紧化，一个轨道开化。设Ψ为平滑的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， Ψ'为SL（m, c）×SL（n, c）行为，其对SU(m)×SU(n)的限制是标准的。如果Ψ(c,d)=Ψ‘（c’，d‘） and d-c=d’-c‘, thenΨ=Ψ’。定理3设α， β:R^2×CP(1)→CP(1)为满足三个条件的函数，则可以构造一个抽象的SL（m，C）×SL（n，C）作用于CP(m+n-1)，其对SU(m)×SU(n)的约束是标准的。总的来说，动作并不流畅。在不久的将来，我们将利用该定理构造具有k个轨道（k>3）的CP（m+n-1）上的光滑SL（m，C）×SL（n，C）作用。

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.