Study of smooth non-compact Lie group actions on compact smooth manifolds

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It is important to study non-compact Lie group actions on compact smooth manifolds.In this note, we classify smooth SU(p, 1)-actions on the (2p+2q-1)-sphere and on the complex projective (p+q-1)-space whose restriction to the maximal compact subgroup S(U (p) X U (q)) is standard.The results are as follows.Let U, V be the set of smooth SU(p, q)-actions on P_<p+q-1> (C), S^<2p+2q-1> whose restricted S(U(p) X U(q))-action is standard, respectively.1. There is a one-to-one correspondence between U and the set of pairs (phi', f' ), where phi' is a smooth N'(p, q)=SU(1,1)-action on P_1(C) whose restriction on S(U (p) X U (q))*N'(p, q) is standard and f : P_1(C)*P_1(C) is a smooth N'(p, q)-equivariant map satisfying one condition.From this fact we see that there exist infinitely many smooth actions on P_<p+q-1>(C) which are mutually distinct.2. There is a one-to-one correspondence between V and the set of pairs (phi, f) , where phi is a smooth N(p, q)=U(1, 1)-action on S^3 whose restriction on S(U(p) X U(q)) *N(p, q) is standard and f : S^3*P_1(C) is a smooth N(p, q)-equivariant map satisfying one condition.3. There exists a mapping rho : V*U which is onto but not one-to-one.