Symmetric Spaces

Abstract

In this chapter we study Riemannian manifolds M on which for any point p ∈ M an isometry σ _p : M → M is defined so that σ ² _p = id, σ _p (p) = p and if γ(t) is a geodesic with γ(0) = p, then σ _p (t) = (−t). Such manifolds are called symmetric. The existence of the isometries imposes strong constraints on the structure of the manifolds. For example, M must be homogeneous, which means that for any two points x ₁, x ₂ of M there is an isometry g of M that takes x ₁ to x ₂. In order to construct g, connect x ₁ and x ₂ with a line that is a union of segments of geodesics, and let y _i be the midpoint of the ith segment. We now may put g = σ _yk ◦ ... ◦ σ _y1 (see Figure 1). Sometimes it is required that g belong to the component of the identity in the group of isometries; in this case, if k is odd, we need to add one more symmetry and put g = σ _yk ◦ ... ◦ σ _yl ◦ σ _x1. Another important property is that the group of all isometries of a symmetric space is generated by transformations of the form σ _y . Groups of isometries of symmetric spaces are Lie groups with certain additional properties. There is a tight connection between Lie groups and symmetric spaces; we will consider this connection in this chapter.