ANALYTIC $\text{SO}^\circ(p,q)$ ACTIONS ON CLOSED, CONNECTED $(p+q-1)$-DIMENSIONAL MANIFOLDS

Abstract

This thesis provides a classification of analytic actions of the semiorthogonal group $\text{SO}^\circ(p,q)$, where $p \geq 3$, 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 central result of this thesis 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. For $q \neq 2$, the actions of $\text{SO}^\circ(p,q)$ correspond to a particular class of vector fields on the circle, while for $q = 2$, they correspond to actions of $\text{SO}^\circ(1,2)$ on either the sphere or the torus.