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  "body": "\n## Statement\n\nLet $(M, g)$ be a connected complete [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$ whose [[sectional curvature|1X42JAA349]] satisfies $K \\le 0$ everywhere.\nLet $p \\in M$.\nThen the [[exponential map|Y7Z0AGY1A7]] $\\exp_p \\colon T_p M \\to M$ is a [[covering map|3XZRRE165N]].\nIn particular, the [[universal cover|Y1YPE6PM52]] of $M$ is diffeomorphic to $\\R^n$, and if $M$ is simply connected, then $M$ is diffeomorphic to $\\R^n$.\n\n## Remarks\n\nThe idea of the proof: if $K \\le 0$, then for a [[Jacobi field|9AHMM41PJ3]] $J$ with $J(0) = 0$ the function $\\abs{J(t)}^2$ is convex.\nSo a nonzero such $J$ never vanishes again, and there are no [[conjugate points|ZPXE5VSZ35]].\nHence $\\exp_p$ is a local diffeomorphism.\nPull the metric back to $T_p M$; the lines through $0$ are geodesics defined for all time, so the pulled-back metric is complete by the [[Hopf–Rinow theorem|QCCD2JNFBT]].\nA local isometry from a complete connected Riemannian manifold onto a connected one is a covering map.\n\nExample: for a flat torus $\\R^n / \\Z^n$, the map $\\exp_p$ is the quotient map $\\R^n \\to \\R^n / \\Z^n$ followed by a translation.\nFor a compact hyperbolic surface, $\\exp_p$ is the universal covering by the hyperbolic plane.\n\nA consequence is that all higher homotopy groups of $M$ vanish.\nThe sign condition is needed: on the round sphere $S^n$, the map $\\exp_p$ sends the whole sphere of radius $\\pi$ in $T_p S^n$ to the antipode of $p$.\nSo it is not a covering.\nCompleteness is needed too: the punctured plane $\\R^2 \\setminus \\set{0}$ is flat, but $\\exp_p$ is not defined on all of $T_p M$.\n\n## Explanations\n\n### L1 human\n\nOn a space that curves like a saddle everywhere, geodesics that start at one point spread apart and never meet again.\nSo straight lines from one point unroll the space onto a flat plane, perhaps wrapping it several times around.\nIn particular, such a space with no holes is just a stretched copy of $\\R^n$.\n\n### L2 human\n\nThe theorem shows that curvature restricts topology: a compact manifold with $K \\le 0$ is aspherical, its universal cover is $\\R^n$, and its topology is governed by its fundamental group.\nFor example, no sphere $S^n$ with $n \\ge 2$ carries a metric of nonpositive curvature.\nIt is the starting point of the study of manifolds of negative curvature.\n",
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