Symmetric spaces Sample Clauses

Symmetric spaces. The first (and largest) section of this chapter is dedicated to the general theory of symmetric spaces. We go through the apparatus of Riemannian symmetric pairs and orthogonal symmetric Lie algebras and discuss various aspects of the theory such as type, rank, holonomy, irreducibility, duality, and eventually the classification. The primary reference for this section is [Hel01].‌ 2.1.1. Symmetric and locally symmetric spaces Proposition 2.1.1. Let M be a Riemannian manifold with π0(M ) < , and let I(M ) (a) When endowed with the compact-open topology, I(M ) is a Lie group, and its action on M is smooth. An isometric action of a Lie group G on M is smooth if and only if the corresponding morphism G → I(M ) is smooth. ^∈ ∈ (b) Write i(M ) for Lie(I(M )). Given X i(M ), let X X(M ) be its corresponding fundamental vector field: X^ = exp(tX) · p = d(π ) (X), where πp : I(M ) → M, g '→ g · p. Then X^ lies in the Lie subalgebra K(M ) ⊆ X(M ) of Killing vector fields, and the map2 i(M ) → K(M ), X '→ X^ , is an injective 1Note that the assumption on completeness of M in this paper is redundant. 2This is also known as the infinitesimal generator of the action I(M ) ↷ M . (c) Suppose M is connected and let f, f′ ∈ I(M ) be such that f (p) = f′(p) and dfp = dfp′ for some p ∈ M. Then f = f′. In particular, if an isometry f fixes some point p and dfp = IdTpM, then f = IdM . (d) Let H I(M ) be a Lie subgroup. The action H ↷ M is proper if and only if H is a closed subgroup. In particular, the actions of I(M ) and I0(M ) on M are proper and the stabilizers I(M )p and I0(M )p of any point p are compact (thus π0(I(M )) is also finite). (e) If M is compact, then so is I(M ). We will call the elements of I0(M ) inner isometries. Let M be a connected Riemannian manifold and p ∈ M . Take 0 < r < inj(p), where the latter is the injectivity radius of M at p. The exponential map expp : TpM → M restricts to a diffeomorphism between Br(0) ⊂ TpM and Br(p) ⊆ M . The linear isometry v '→ −v of Br(0) exponentiates to a diffeomorphism exp(tv) exp( tv) of Br(p), called a local geodesic symmetry of M at p. 2.1.2. A connected Riemannian manifold is called a (Riemannian) locally symmetric space if it satisfies the following equivalent conditions: (i) For every p ∈ M , there exists 0 < r < inj(p) such that the local geodesic symmetry of Br(p) is an isometry. (ii) The curvature tensor of M is parallel: ∇R = 0. 2.1.3. A connected Riemannian manifold M is called a (Riemannian) symmetric space...