SYMMETRIC SPACES WITH DISSECTING INVOLUTIONS

Abstract

An involutive diffeomorphism σ 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, σ), where M is an irreducible connected symmetric space, not necessarily Riemannian, and σ is a dissecting involutive automorphism. In particular, we show that the only irreducible, connected and simply connected Riemannian symmetric spaces with dissecting isometric involutions are $$ {\mathbbm{S}}^n $$ and ℍn, where the corresponding fixed point spaces are $$ {\mathbbm{S}}^{n-1} $$ and ℍn − 1, respectively.