Chapter I: Symmetric spaces and the Lie-functor

Abstract

There are four definitions of a symmetric space M:

• 1 The global one: M = GIH is a homogeneous space, where H is essentially the group of fixed points of an involution σ of a Lie group G.

• 2 The infinitesimal one: M is a real manifold with a torsionfree affine connection $\nabla$ whose curvature is covariantly constant: $\nabla R{\text{ = 0}}$ .

• 3 The mixed one: M is a real manifold with an affine connection $\nabla$ such that the geodesic symmetry s _P with respect to any point p ε M is an automorphism of $\nabla$ .

• 4 The algebraic one, which reflects axiomatically the properties of the map $\mu :M \times M \to M{\mathbf{ }}{\text{given by}}{\mathbf{ }}\mu \left( {x,y} \right): = s_x \left( y \right){\mathbf{ }}{\text{with}}{\mathbf{ }}s_x {\mathbf{ }}{\text{as in}}{\mathbf{ }}\left( {\text{3}} \right)$.