Symmetric Spaces - Key Properties

Symmetric spaces exhibit remarkable geometric and analytic properties stemming from their symmetry. These properties make them central objects in differential geometry, representation theory, and mathematical physics.

Theorem

Curvature Properties

For a Riemannian symmetric space $M = G/K$ :

The curvature tensor is parallel: $\nabla R = 0$
Sectional curvatures are determined by Lie brackets: $K(X, Y) = -\frac{1}{4}\|[X,Y]\|^2$ for orthonormal $X, Y \in \mathfrak{p}$
The space is Einstein: $\text{Ric} = \lambda g$ for some constant $\lambda$

These properties severely constrain the geometry—symmetric spaces are the most homogeneous spaces after constant curvature manifolds.

Remark

Geodesics: Every geodesic in a symmetric space $G/K$ through the origin is of the form $\gamma(t) = \exp(tX) \cdot o$ for some $X \in \mathfrak{p}$ . This provides explicit parameterization of all geodesics using the exponential map.

Theorem

Duality of Compact and Non-Compact Types

Irreducible symmetric spaces of compact type and non-compact type come in dual pairs $(M_c, M_{nc})$ where:

$M_c = G_c/K$ (compact)
$M_{nc} = G_{nc}/K$ (non-compact)
$G_c$ and $G_{nc}$ share the same complexification
Same $K$ appears in both
Curvatures have opposite signs

The Lie algebras satisfy $\mathfrak{g}_c = \mathfrak{k} \oplus i\mathfrak{p}$ and $\mathfrak{g}_{nc} = \mathfrak{k} \oplus \mathfrak{p}$ .

Example

The sphere $S^n = SO(n+1)/SO(n)$ (compact, positive curvature) is dual to hyperbolic space $\mathbb{H}^n = SO(n,1)^0/SO(n)$ (non-compact, negative curvature). Both share the same $K = SO(n)$ .

Theorem

Classification by Involutions

Irreducible simply connected symmetric spaces of non-compact type are in bijection with pairs $(\mathfrak{g}, \sigma)$ where:

$\mathfrak{g}$ is a simple real Lie algebra
$\sigma$ is an involutive automorphism
The fixed point set $\mathfrak{k}$ is a maximal compact subalgebra

Classification reduces to classifying involutions of simple Lie algebras.

Rank and root space structure:

Definition

The rank of a symmetric space $M = G/K$ is the dimension of a maximal flat totally geodesic submanifold. Equivalently, it is the dimension of a maximal abelian subspace of $\mathfrak{p}$ .

Example

Rank-one symmetric spaces: spheres, projective spaces (real, complex, quaternionic), hyperbolic spaces
Higher rank: Grassmannians, flag varieties, Hermitian symmetric spaces

Rank-one spaces have constant curvature; higher rank introduces richer geometry.

Theorem

Uniqueness of Geodesics

In a symmetric space of non-compact type, any two points are connected by a unique geodesic. In compact type, minimizing geodesics exist but may not be unique (e.g., antipodal points on spheres).

These properties make symmetric spaces ideal testing grounds for geometric conjectures and provide concrete models for studying curvature effects.