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  "body": "\n## Definition\n\nA *hyperbolic $3$-manifold* is a connected [[3-manifold|R20TB2KEQQ]] $M$ with a [[Riemannian metric|7CBT4SM83Y]] $g$ such that:\n\n1. $(M, g)$ is complete as a metric space, and\n2. every [[sectional curvature|1X42JAA349]] of $g$ is equal to $-1$.\n\n## Remarks\n\nExample: hyperbolic $3$-space $\\mathbb{H}^3 = \\set{(x, y, t) \\in \\R^3 : t \u003e 0}$ with the metric $(dx^2 + dy^2 + dt^2)/t^2$ is complete and has constant sectional curvature $-1$.\nMore generally, if $\\Gamma$ is a discrete group of isometries of $\\mathbb{H}^3$ acting freely, then $\\mathbb{H}^3 / \\Gamma$ with the quotient metric is a hyperbolic $3$-manifold.\n\nNon-example: the open subset $\\mathbb{H}^3 \\setminus \\set{p}$, for a point $p$, has curvature $-1$, but it is not complete, since a geodesic heading to $p$ stops after finite length.\n\nBy the Killing–Hopf theorem, every hyperbolic $3$-manifold is isometric to some $\\mathbb{H}^3 / \\Gamma$ as above.\nIt is orientable exactly when $\\Gamma$ lies in the group of orientation-preserving isometries, which is isomorphic to $PSL(2, \\C)$.\n",
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