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  "body": "\n## Summary\n\nLet $p$ be a point of a [[Riemannian manifold|7CBT4SM83Y]].\nIts injectivity radius at $p$ is the largest $r$ such that the [[exponential map|Y7Z0AGY1A7]] at $p$ is a diffeomorphism on the $r$-ball in $T_pM$.\nWithin this radius, [[normal coordinates|RRHKMGNKG9]] at $p$ exist and each point is joined to $p$ by a unique shortest geodesic.\nThe injectivity radius of the whole manifold is the infimum over all points.\nLower bounds for it are a key ingredient in compactness and finiteness theorems.\n\n## Explanations\n\n### L1 human\n\nFrom a point $p$, send out geodesics in all directions at the same speed.\nFor a while they spread out without meeting, and every nearby point is reached in exactly one way.\nThe injectivity radius is how far one can go before this breaks down.\nOn the unit sphere all geodesics from the north pole meet again at the south pole, at distance $\\pi$, so the injectivity radius is $\\pi$.\nOn a thin cylinder of circumference $c$, geodesics going around in opposite directions meet after distance $c/2$, so the injectivity radius is $c/2$.\n",
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