On smooth \(SO_ 0(p,q)\)-actions on \(S^{p+q-1}\) (Q923437)

Earlier papers of the author [Osaka J. Math. 16, 561-579 (1979; Zbl 0425.57015) and Tôhoku Math. J. 33, 145-175 (1981; Zbl 0542.57032)] contain some classification results about real analytic actions of SL(n,\({\mathbb{R}})\) on spheres. The present paper provides some classification results for smooth actions on spheres. More specifically, let SO(p,q) be the group of matrices in \(SL(p+q,{\mathbb{R}})\) which leaves invariant the quadratic form \[ -x^ 2_ 1-...-x^ 2_ p+x^ 2_{p+1}+...+x^ 2_{p+q}, \] and let \(SO_ 0(p,q)\) be the identity component of SO(p,q). The author shows that for \(p\geq 3\), \(q\geq 3\), there is a one-to-one correspondence between the set of smooth actions of \(SO_ 0(p,q)\) on the sphere \(S^{p+q-1}\) such that the restricted action of SO(p)\(\times SO(q)\) is the standard orthogonal action and the set of pairs (\(\phi\),f) satisfies certain conditions, where \(\phi\) is a smooth action of \({\mathbb{R}}\) on \(S^ 1\) and f is a smooth function from \(S^ 1\) into the real projective line. The author uses the method presented by \textit{T. Asoh} [Osaka J. Math. 24, 271-298 (1987; Zbl 0706.57021)], and he notes that it follows from Asoh's considerations that there are infinitely many topologically distinct smooth actions of \(SO_ 0(p,q)\) on \(S^{p+q-1}\) whose restricted SO(p)\(\times SO(q)\)-action is the standard orthogonal action.