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  "body": "\n## Definition\n\nLet $(M, g)$ be a connected [[Hermitian manifold|CK1DCPC3D3]].\nSo $M$ is a complex manifold with integrable [[almost complex structure|9SYWRGVTRS]] $J$.\nThe Riemannian metric $g$ satisfies $g(JX, JY) = g(X, Y)$ for all tangent vectors $X, Y$.\nThe manifold $M$ is a *Hermitian symmetric space* if each point $p \\in M$ has an [[isometry|R9YWQS15AA]] $s_p \\colon M \\to M$ with three properties.\n\n1. $s_p$ is holomorphic.\n2. $s_p \\circ s_p = \\id$.\n3. $p$ is an isolated fixed point of $s_p$.\n\n## Remarks\n\nConditions 2 and 3 force $(ds_p)_p = -\\id$, so $(M, g)$ is a Riemannian [[symmetric space|PAA93R4WYD]] and $s_p$ is its geodesic symmetry.\nThe metric $g$ is then Kähler.\n\nExample: the unit disc $D = \\set{z \\in \\C : \\abs{z} \u003c 1}$ with the metric $\\frac{4\\,\\abs{dz}^2}{(1 - \\abs{z}^2)^2}$.\nAt $0$ take $s_0(z) = -z$: it is holomorphic, an isometry, an involution, and $0$ is its only fixed point.\nThe holomorphic isometries $z \\mapsto e^{i t} \\frac{z - a}{1 - \\bar a z}$ act transitively, and conjugating $s_0$ by them gives $s_p$ at every point $p$.\n\nNon-example: the round sphere $S^4$ is a Riemannian symmetric space, but it has no almost complex structure at all, so it is not a Hermitian symmetric space.\n\nEach simply connected Hermitian symmetric space is a product of a flat part $\\C^k$ and irreducible ones of compact type, such as $\\C P^n$, and of noncompact type, such as $D$.\n",
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