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 (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.