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  "body": "\n## Definition\n\nLet $D \\subset \\C^n$ be a bounded domain, that is, a bounded connected open set.\nThe domain $D$ is a *bounded symmetric domain* if for every $p \\in D$ there is a [[biholomorphism|FX92PP3HZR]] $s_p \\colon D \\to D$ such that:\n\n1. $s_p \\circ s_p = \\id_D$;\n2. $p$ is an isolated fixed point of $s_p$.\n\n## Remarks\n\nThe map $s_p$ is called the symmetry at $p$, and it is unique.\nIts derivative at $p$ is $-\\id$.\nA bounded symmetric domain is homogeneous: its automorphism group acts transitively.\nA bounded domain that is homogeneous and has a symmetry at one point is symmetric.\n\nExample: the unit disc $\\Delta = \\set{z \\in \\C : \\abs{z} \u003c 1}$.\nAt $0$ take $s_0(z) = -z$: it is an involution with only the fixed point $0$.\nAt $a \\in \\Delta$ take $s_a = \\varphi_a \\circ s_0 \\circ \\varphi_a$, where $\\varphi_a(z) = (a - z)/(1 - \\bar a z)$ is an automorphism of $\\Delta$ with $\\varphi_a \\circ \\varphi_a = \\id$ and $\\varphi_a(a) = 0$.\nThen $s_a$ is an involution whose only fixed point is $a$.\n\nNon-example: the annulus $A = \\set{z : r \u003c \\abs{z} \u003c 1}$ with $0 \u003c r \u003c 1$.\nIts automorphisms are the rotations $z \\mapsto e^{i\\theta} z$ and the maps $z \\mapsto r e^{i\\theta}/z$.\nSo each orbit of $\\Aut(A)$ is a union of at most two circles, $A$ is not homogeneous, and it cannot be symmetric.\n\nThe unit ball of $\\C^n$ and the polydisc $\\Delta^n$ are bounded symmetric domains, and so is any product of bounded symmetric domains.\n",
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