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

TL;DR

This work resolves the analytic action classification problem for the simple Lie group $SO^ o(p,q)$ with $p,q\,\ge 3$ on closed manifolds of dimension $p+q-1$ by identifying a complete set of model actions. It constructs analytic actions on the product and projective model spaces using a basic flow framework (the basic $J_1$-flow and its Uchida-extensions) and a Langlands/ parabolic decomposition to handle the nullcone case through $SO^ o(p,q) imes_{P_{ ext{null}}}S^1$. The main result shows any nontrivial analytic $SO^ o(p,q)$-action is equivariantly covered by one of these models, with four explicit orbit types arising: $SO^ o(p,q)/SO^ o(p-1,q)$, $SO^ o(p,q)/SO^ o(p,q-1)$, $SO^ o(p,q)/G_{ ext{null}}$, or $SO^ o(p,q)/P_{ ext{null}}$. This yields a complete analytic classification for $p,q\ge 3$ and sets the stage for future work addressing the remaining low-rank cases $p,q\, ext{in}\,\{1,2\}$.

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.

Theorems & Definitions (73)

• Theorem 1.1
• Definition 2.1
• Remark 1
• Definition 2.2
• Lemma 2.1
• Remark 2
• Remark 3
• Theorem 2.1
• Theorem 2.2
• Lemma 2.2