Classification of isometries of spaces of constant curvature and invariant subspaces

TL;DR

This paper classifies isometries of constant curvature spaces by analyzing invariant submanifolds and introduces the Segre symbol as a new invariant to distinguish orbit types.

Contribution

It introduces the Segre symbol to parametrize z-classes of isometries and describes invariant subspace varieties in terms of this invariant.

Findings

Dimensions of invariant submanifold varieties determine isometry orbit types.

Segre symbol encodes the normal form structure of isometries.

Invariant subspace varieties are characterized by the Segre symbol.

Abstract

We study the varieties of invariant totally geodesic submanifolds of isometries of the spherical, Euclidean and hyperbolic spaces in each finite dimension. We show that the dimensions of the connected components of these varieties determine the orbit type (or the z-class) of the isometry. For this purpose, we introduce the Segre symbol of an isometry, a discrete invariant encoding the structure of its normal form, which parametrizes z-classes. We then provide a description of the isomorphism type of the varieties of invariant subspaces in terms of the Segre symbol.