Cartan's Classification of Symmetric Spaces

Every simply connected Riemannian symmetric space decomposes uniquely as a Riemannian product: $M = M_0 \times M_1 \times \cdots \times M_k$ where $M_0 = \mathbb{R}^m$ (Euclidean factor) and each $M_i$ is an irreducible symmetric space (compact type, non-compact type, or Euclidean).

Irreducible symmetric spaces are classified by involutions of simple Lie algebras.

Structure Theorem for Symmetric Spaces

Let $M = G/K$ be a symmetric space with Cartan decomposition $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$. Then:

1. Lie algebra structure:

   • $[\mathfrak{k}, \mathfrak{k}] \subseteq \mathfrak{k}$
   • $[\mathfrak{k}, \mathfrak{p}] \subseteq \mathfrak{p}$
   • $[\mathfrak{p}, \mathfrak{p}] \subseteq \mathfrak{k}$

2. Metric structure: The Killing form (or a multiple) restricts to a positive definite metric on $\mathfrak{p}$, giving the Riemannian metric on $M$

3. Curvature: Sectional curvature $K(X,Y) = -\frac{1}{4}\|[X,Y]\|^2$ for orthonormal $X, Y \in \mathfrak{p}$

Duality Theorem

Compact and non-compact irreducible symmetric spaces come in dual pairs. If $M_c = G_c/K$ (compact) corresponds to involution $\sigma$ of $\mathfrak{g}_c = \mathfrak{k} \oplus i\mathfrak{p}$, then $M_{nc} = G_{nc}/K$ (non-compact) corresponds to the same $K$ with $\mathfrak{g}_{nc} = \mathfrak{k} \oplus \mathfrak{p}$.

Properties:

• Same dimension and rank
• Opposite curvature signs
• Isomorphic local geometry (tangent spaces)
• Related representation theories

Rank Rigidity

For symmetric spaces of non-compact type and rank $\geq 2$, the geodesic structure determines the full Riemannian structure (Mostow rigidity). Rank-one spaces admit more flexibility in their metrics.

Iwasawa Decomposition

For symmetric space $M = G/K$ of non-compact type, there exists a decomposition: $G = KAN$ where $A$ is abelian (maximal flat subspace) and $N$ is nilpotent. This provides coordinates on $M$ analogous to polar coordinates, fundamental for harmonic analysis.