Classification of analytic $\text{SO}^o(p,q)$-actions on closed $(p+q-1)$-dimensional manifolds II : $p \geq 3$, $q=1,2$
This paper provides a classification of analytic actions of the semi-orthogonal groups $\text{SO}^\circ(p,1)$ and $\text{SO}^\circ(p,2)$, for $p \geq 3$, on closed, connected manifolds of dimension $p$ and $p+1$ respectively. It is a sequel to a previous paper, which treats the case $p,q \geq 3$. Adapting Uchida's methods, we explicitly construct analytic actions of $\text{SO}^\circ(p,1)$ on $\text{S}^{p}$ and $\text{S}^{p-1} \times \text{S}^{1}$, and analytic actions of $\text{SO}^\circ(p,2)$ on $\text{S}^{p+1}$, $\text{S}^{p} \times \text{S}^{1}$ and $\text{S}^{p-1} \times \text{S}^{2}$, and we classify them up to analytic isomorphism, using a result of Hitchin and Schneider's classification of analytic $\text{SL}_2(\mathbb{R})$ actions on surfaces. The main result demonstrates that any analytic $\text{SO}^\circ(p,q)$ action, $q=1,2$, on a closed, connected $(p+q-1)$-dimensional manifold is covered by one of the constructed actions or by an action on $\text{SO}^\circ(p,q) \times_P \text{S}^1$, where $P$ is a maximal parabolic subgroup of $\text{SO}^\circ(p,q)$.