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 $(g,h_1,h_2)$ of smooth functions to construct smooth $SU(p,q)$-actions on the projective space ${\mathbf{P}}_{\mathrm{ℂ}}^{p+q-1}$ $(p,q\ge 3)$ whose restricted $S\left(U\right(p)\times U(q\left)\right)$-action is the standard action. By using these examples of triples $(g,h_1,h_2)$, we shall construct smooth (resp. analytic) $S{O}_0(p,q)$-actions on $S^{p+q-1}$ $(p,q\ge 3)$ whose restricted $SO\left(p\right)\times SO\left(q\right)$-action is the standard action. Consequently, we shall show that for each positive integer $m$ there exist uncountably infinite smooth conjugacy classes of smooth $S{O}_0(p,q)$-actions on $S^{p+q-1}$ $(p,q\ge 3)$ with $m$ closed and $m+1$ open orbits whose restricted $SO\left(p\right)\times SO\left(q\right)$-action is the standard action. On the other hand, we shall show that there exist exactly countably many analytic conjugacy classes of analytic $S{O}_0(p,q)$-actions on $S^{p+q-1}$ $(p,q\ge 3)$ with one closed and two open orbits whose restricted $SO\left(p\right)\times SO\left(q\right)$-action is the standard action.