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

TL;DR

This paper classifies all analytic actions of the semi-orthogonal group SO^(p,q) on certain high-dimensional manifolds, showing they are all covered by explicitly constructed standard actions.

Contribution

It provides a complete classification of analytic SO^(p,q) actions on closed manifolds of dimension p+q-1, extending Uchida's construction and identifying all such actions.

Findings

All such actions are covered by explicit standard models.

Constructed actions include products of spheres and certain homogeneous spaces.

Main theorem confirms the classification covers all possible actions.

Abstract

This paper provides a classification of analytic actions of the semi-orthogonal group $\text{SO}^\circ(p,q)$, for $p,q \geq 3$, 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 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.