Symmetric space

Remarks

Example: the unit sphere $S^n \subset \R^{n+1}$ with the induced metric. For $p \in S^n$ put $s_p(x) = 2 \ip{x}{p} p - x$. It is the restriction of a linear orthogonal map, so it is an isometry of $S^n$; it fixes $p$; and on $T_p S^n = p^\perp$ it acts as $v \mapsto -v$.

Non-example: a closed surface of genus $2$ with a metric of constant curvature $-1$. Its isometry group is finite, so it does not act transitively on the surface. A symmetric space is homogeneous, since $s_q$, for $q$ the midpoint of a shortest geodesic from $p$ to $p'$, swaps $p$ and $p'$; so the surface is not symmetric. It is locally symmetric: its curvature tensor is parallel.

A symmetric space is complete, and $s_p$ is unique: it maps $\exp_p(v)$ to $\exp_p(-v)$. Let $G$ be the identity component of the isometry group and $K$ the stabilizer of a point; then $M = G/K$. Conjugation by $s_p$ gives an involution $\sigma$ of $G$, and its fixed point group satisfies $(G^\sigma)_0 \subseteq K \subseteq G^\sigma$.