# Symmetric spaces with dissecting involutions

## TL;DR

This paper classifies all irreducible symmetric spaces with dissecting involutions, revealing that only spheres and hyperbolic spaces admit such involutions with fixed point spaces of one dimension lower.

## Contribution

It provides a complete classification of dissecting involutions on irreducible symmetric spaces, extending understanding beyond Riemannian cases.

## Key findings

- Only spheres and hyperbolic spaces have dissecting involutions among irreducible 1-connected Riemannian symmetric spaces.
- Dissecting involutions have fixed point spaces of codimension one, namely spheres and hyperbolic spaces.
- The classification links dissecting involutions to geometric structures relevant in quantum field theory.

## Abstract

An involutive diffeomorphism $\sigma$ of a connected smooth manifold $M$ is called dissecting if the complement of its fixed point set is not connected. Dissecting involutions on a complete Riemannian manifold are closely related to constructive quantum field theory through the work of Dimock and Jaffe/Ritter on the construction of reflection positive Hilbert spaces. In this article we classify all pairs $(M,\sigma)$, where $M$ is an irreducible symmetric space, not necessarily Riemannian, and $\sigma$ is a dissecting involutive automorphism. In particular, we show that the only irreducible $1$-connected Riemannian symmetric spaces are $S^n$ and $H^n$ with dissecting isometric involutions whose fixed point spaces are $S^{n-1}$ and $H^{n-1}$, respectively.