Classification of analytic ${\text{SO}}^{\circ }(p,q)$-actions on closed $(p+q-1)$-dimensional manifolds I : $p,q\ge 3$

This paper provides a classification of analytic actions of the semi-orthogonal group ${\text{SO}}^{\circ }(p,q)$, for $p,q\ge 3$, on closed, connected $(p+q-1)$-dimensional manifolds. Adapting Uchida’s construction of ${\text{SO}}^{\circ }(p,q)$ actions on ${\text{S}}^{p+q-1}$, we explicitly construct analytic actions of ${\text{SO}}^{\circ }(p,q)$ on ${\text{S}}^p\times {\text{S}}^{q-1}$ and ${\text{S}}^{p-1}\times {\text{S}}^q$, as well as actions on ${\text{SO}}^{\circ }(p,q){\times }_P{\text{S}}^1$, where $P$ is a maximal parabolic subgroup of ${\text{SO}}^{\circ }(p,q)$. The main result demonstrates that any analytic ${\text{SO}}^{\circ }(p,q)$ action on a closed, connected $(p+q-1)$-dimensional manifold is covered by one of the constructed actions.