Spheres $S^n = SO(n+1)/SO(n)$

The unit sphere with round metric is the prototypical compact symmetric space. Geodesic symmetry at point $p$ is reflection through the great circle perpendicular to $p$.

Involution: $\sigma(A) = I_{n,1} A I_{n,1}$ on $SO(n+1)$

Curvature: constant sectional curvature $+1$

Hyperbolic spaces $\mathbb{H}^n = SO(n,1)^0/SO(n)$

Models of constant negative curvature $-1$. Multiple realizations:

• Upper half-space model: $\{(x_1, \ldots, x_n) : x_n > 0\}$
• Poincaré disk model: $\{x \in \mathbb{R}^n : \|x\| < 1\}$
• Hyperboloid model: $\{x \in \mathbb{R}^{n+1} : x_1^2 - x_2^2 - \cdots - x_{n+1}^2 = -1, x_1 > 0\}$

Dual to $S^n$ with opposite curvature sign.

Complex projective space $\mathbb{CP}^n = SU(n+1)/(S(U(n) \times U(1)))$

Compact Hermitian symmetric space of complex dimension $n$. Points are complex lines through origin in $\mathbb{C}^{n+1}$.

Fubini-Study metric gives sectional curvatures between 1 and 4.

Kähler manifold: admits compatible complex and symplectic structures.

Grassmannians $Gr(k,n) = O(n)/(O(k) \times O(n-k))$

Parametrizes $k$-dimensional subspaces of $\mathbb{R}^n$. Compact symmetric space of dimension $k(n-k)$.

Complex version: $Gr_\mathbb{C}(k,n) = U(n)/(U(k) \times U(n-k))$

Applications: Schubert calculus, intersection theory, quantum cohomology.

Siegel upper half-space $\mathcal{H}_n = Sp(2n, \mathbb{R})/U(n)$

Non-compact Hermitian symmetric space of complex dimension $\frac{n(n+1)}{2}$. Consists of $n \times n$ complex symmetric matrices $Z$ with $\text{Im}(Z) > 0$ (positive definite).

Fundamental in algebraic geometry (moduli of abelian varieties) and number theory (Siegel modular forms).

Exceptional symmetric spaces:

• $E_6/\text{Spin}(10) \times SO(2)$: 32-dimensional, related to exceptional Jordan algebra
• $E_7/E_6 \times SO(2)$: 54-dimensional
• $E_8/\text{Spin}(16)$: 128-dimensional, highest dimensional compact simple symmetric space

These appear in string theory and exceptional holonomy manifolds.