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  "body": "\n## Definition\n\nLet $(M, g)$ be a connected [[Riemannian manifold|7CBT4SM83Y]].\nThe manifold $M$ is a *(Riemannian globally) symmetric space* if each point $p \\in M$ has an [[isometry|R9YWQS15AA]] $s_p \\colon M \\to M$ with two properties.\nThe map $s_p$ satisfies $s_p(p) = p$, and its [[differential|6P9PXQDSEN]] $(ds_p)_p \\colon T_p M \\to T_p M$ equals $-\\id$.\nThe map $s_p$ is the *geodesic symmetry* at $p$.\n\n## Remarks\n\nExample: the unit sphere $S^n \\subset \\R^{n+1}$ with the induced metric.\nFor $p \\in S^n$ put $s_p(x) = 2 \\ip{x}{p} p - x$.\nIt 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 $v \\mapsto -v$.\n\nNon-example: a closed surface of genus $2$ with a metric of constant curvature $-1$.\nIts isometry group is finite, so it does not act transitively on the surface.\nA 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.\nIt is *locally* symmetric: its curvature tensor is parallel.\n\nA symmetric space is complete, and $s_p$ is unique: it maps $\\exp_p(v)$ to $\\exp_p(-v)$.\nLet $G$ be the identity component of the isometry group and $K$ the stabilizer of a point; then $M = G/K$.\nConjugation by $s_p$ gives an involution $\\sigma$ of $G$, and its fixed point group satisfies $(G^\\sigma)_0 \\subseteq K \\subseteq G^\\sigma$.\n",
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