A Note on Equivalence Classes of SO0(p,q)-actions on Sp+q-1 whose Restricted SO(p)×SO(q)-action is the Standard Action

Abstract

In [5] we gave examples of triples (g,h1,h2) of smooth functions to construct smooth SU(p,q)-actions on the projective space Pℂp+q-1 (p,q≥3) whose restricted S(U(p)×U(q))-action is the standard action. By using these examples of triples (g,h1,h2), we shall construct smooth (resp. analytic) SO0(p,q)-actions on Sp+q-1 (p,q≥3) whose restricted SO(p)×SO(q)-action is the standard action. Consequently, we shall show that for each positive integer m there exist uncountably infinite smooth conjugacy classes of smooth SO0(p,q)-actions on Sp+q-1 (p,q≥3) with m closed and m+1 open orbits whose restricted SO(p)×SO(q)-action is the standard action. On the other hand, we shall show that there exist exactly countably many analytic conjugacy classes of analytic SO0(p,q)-actions on Sp+q-1 (p,q≥3) with one closed and two open orbits whose restricted SO(p)×SO(q)-action is the standard action.