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  "body": "\n## Statement\n\nLet $n \\ge 2$, let $K$ be a real number, and let $X_K$ be the model space of dimension $n$.\nIt is the sphere of radius $1/\\sqrt{K}$ if $K \u003e 0$, Euclidean space $\\R^n$ if $K = 0$, and hyperbolic space of curvature $K$ if $K \u003c 0$.\nLet $(M, g)$ be a [[space form|1VM1AQT956]] of dimension $n$ and curvature $K$.\nThen:\n\n1. If $M$ is simply connected, then $M$ is isometric to $X_K$.\n2. In general, $M$ is isometric to a quotient $X_K / \\Gamma$ by a group $\\Gamma$ of [[isometries|R9YWQS15AA]] of $X_K$ that acts freely and properly discontinuously.\nThe group $\\Gamma$ is isomorphic to the fundamental group of $M$.\n\n## Remarks\n\nThe idea of the proof: a theorem of Cartan says that two Riemannian manifolds of the same dimension and the same constant [[sectional curvature|1X42JAA349]] are locally isometric.\nFor a complete simply connected $M$, a local isometry from a ball of $X_K$ extends along geodesics to a global isometry; this gives part 1.\nFor general $M$, the [[universal cover|Y1YPE6PM52]] with the pulled-back metric is complete, simply connected and of curvature $K$, so it is $X_K$ by part 1.\nThe deck transformations of the [[covering|3XZRRE165N]] are then isometries.\n\nExample: the real projective space $\\R P^n = S^n / \\set{\\pm \\id}$ with the round metric is a space form of curvature $1$ with $\\Gamma$ of order $2$.\nFlat tori $\\R^n / \\Z^n$ are quotients of $\\R^n$ by a lattice of translations.\n\nCompleteness is needed: an open ball in $\\R^n$ is flat and simply connected, but it is not isometric to $\\R^n$.\n\n## Explanations\n\n### L1 human\n\nThe theorem lists all spaces that curve by the same amount everywhere.\nEach of them is a sphere, a flat space or a hyperbolic space, possibly folded up by a group of symmetries.\nSo to know them all, one only has to know which symmetry groups can be used for folding.\n\n### L2 human\n\nThe theorem turns the classification of space forms into group theory: find the discrete groups of isometries of $S^n$, $\\R^n$ or hyperbolic space that act freely.\nFor $K \u003e 0$ these are finite subgroups of the orthogonal group acting freely on the sphere, which gives for example the lens spaces in dimension $3$.\nFor $K = 0$ and $M$ compact one gets the Bieberbach groups.\nFor $K \u003c 0$ one gets the torsion-free discrete subgroups of the isometry group of hyperbolic space, a very rich family.\n",
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