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

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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.