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  "body": "\n## Statement\n\nLet $(M, g)$ be a connected [[Riemannian manifold|7CBT4SM83Y]] with [[Riemannian distance|B4QR306D8G]] $d$.\nThe following are equivalent:\n\n1. the metric space $(M, d)$ is [[complete|0BY52JW6NK]];\n2. every [[closed|0ZCK8DQYN1]] and [[bounded|09QSZVVM26]] subset of $M$ is [[compact|0TVJZYB6SH]];\n3. there is a point $p \\in M$ such that the [[exponential map|Y7Z0AGY1A7]] $\\exp_p$ is defined on all of $T_pM$;\n4. for every point $p \\in M$, $\\exp_p$ is defined on all of $T_pM$;\n5. $M$ is geodesically complete: every [[geodesic|NSYMQMHPDN]] $\\gamma$ can be extended to a geodesic defined on all of $\\R$.\n\nIf these conditions hold, then any two points $p, q \\in M$ are joined by a geodesic of length $d(p, q)$.\n\n## Remarks\n\nThe key step is to show that if $\\exp_p$ is defined on all of $T_pM$, then every $q$ is reached by a minimizing geodesic from $p$.\nChoose a point $x$ nearest to $q$ on a small sphere around $p$, and follow the geodesic from $p$ through $x$.\nOne shows that it decreases the distance to $q$ at unit rate until it reaches $q$.\nFrom this, bounded closed sets are closed subsets of images of compact balls under $\\exp_p$, which gives the other conditions.\nExample: $\\R^n$ and every compact Riemannian manifold satisfy the conditions.\nThe open disc $\\set{x \\in \\R^2 : \\abs{x} \u003c 1}$ with the Euclidean metric fails them: a radial geodesic leaves the disc in finite time, and the sequence $(1 - 1/k, 0)$ is Cauchy with no limit.\nThe last conclusion can hold without completeness, as in the open disc, so it is not equivalent to the conditions.\n\n## Explanations\n\n### L1 human\n\nFor a curved space, several natural meanings of \"having no holes or missing edges\" turn out to be the same.\nOne can never fall off the space by walking straight; Cauchy sequences converge; closed bounded regions are compact.\nAnd when this holds, any two points are joined by a shortest path that is a geodesic.\n\n### L2 human\n\nThe theorem is used constantly in global Riemannian geometry.\nCompleteness is a standing hypothesis in the theorems of Myers, Cartan–Hadamard and others, and one can check it in whichever form is easiest.\nFor example, compact manifolds and homogeneous Riemannian manifolds are complete.\nIt also shows that $\\exp_p$ is surjective for complete $M$, which is the starting point for the study of the [[cut locus|X5WKGC5Q6T]].\nThe theorem fails for pseudo-Riemannian manifolds, where geodesic completeness and connection by geodesics behave differently.\n",
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