Real analytic $\mathrm{SL}(n,\mathbb{R})$-actions on closed manifolds

TL;DR

This paper classifies real-analytic and smooth fixed-point free actions of the special linear group on closed manifolds, extending previous work and analyzing stability properties of these actions.

Contribution

It extends the classification of $ ext{SL}(n, ext{R})$-actions to higher-dimensional manifolds and fixed-point free actions, providing new insights into their stability.

Findings

Classification of real-analytic $ ext{SL}(n, ext{R})$-actions on certain manifolds.

Classification of fixed-point free smooth $ ext{SL}(n, ext{R})$-actions.

Results on the density of structural stability for these actions.

Abstract

We classify real-analytic $\mathrm{SL}(n,\mathbb{R})$-actions on closed manifolds of dimension m for $3\leq n\leq m\leq2n-3$, which extends Fisher--Melnick's work for $\mathrm{SL}(n,\mathbb{R})$-actions on closed n-manifolds. Additionally, we classify smooth $\mathrm{SL}(n,\mathbb{R})$-actions on closed m-manifolds that are fixed-point free. As a corollary, we obtain the density or non-density of structural stability of fixed-point free $\mathrm{SL}(n,\mathbb{R})$-actions.