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  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $K$ be a real number.\nA *space form* of curvature $K$ is a Riemannian manifold $(M, g)$ of dimension $n$ such that:\n\n1. $M$ is connected.\n2. $M$ is complete as a metric space for the [[Riemannian distance|B4QR306D8G]].\n3. The [[sectional curvature|1X42JAA349]] of every 2-plane at every point of $M$ equals $K$.\n\n## Remarks\n\nExample: the flat torus $\\R^2 / \\Z^2$ with the metric induced from $\\R^2$ is connected and compact, hence complete, and its sectional curvature is $0$ everywhere.\nSo it is a space form of curvature $0$.\n\nNon-example: the open unit disc in $\\R^2$ with the Euclidean metric is connected and flat, but the Cauchy sequence $(1 - 1/k, 0)$ has no limit in the disc.\nIt fails clause 2.\n\nThe simply connected space forms are the sphere of radius $1/\\sqrt{K}$ for $K \u003e 0$, Euclidean space $\\R^n$ for $K = 0$, and hyperbolic space of curvature $K$ for $K \u003c 0$.\nSome authors call only these three models space forms.\n",
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