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  "body": "\n## Statement\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] and let $S \\subseteq M$ be an [[embedded submanifold|VKFW27WXEZ]] without boundary.\nLet $NS = TM|_S / TS$ be the normal bundle of $S$.\nIt is the quotient of the restriction to $S$ of the [[tangent bundle|WY6K3P4YCG]] of $M$ by the tangent bundle of $S$.\nIdentify $S$ with the zero section of $NS$.\nThen there is an open neighbourhood $U$ of $S$ in $M$ and a [[diffeomorphism|0KZZ70ZHXP]] $\\Phi \\colon NS \\to U$ such that $\\Phi(s) = s$ for every $s \\in S$.\n\n## Remarks\n\nSuch a $U$, with the diffeomorphism $\\Phi$, is a *tubular neighbourhood* of $S$.\nFor $M = \\R^n$ the proof is direct: identify $NS$ with the bundle of vectors orthogonal to $S$, and consider $E(s, v) = s + v$.\nIts differential along the zero section is an isomorphism.\nBy the inverse function theorem, $E$ is a local diffeomorphism near the zero section.\nAn extra argument shows that $E$ is injective, hence a diffeomorphism onto an open set, on a neighbourhood of the zero section that may shrink along $S$.\nA fibrewise rescaling of $NS$ onto that neighbourhood finishes the proof.\nFor a general $M$ one replaces $s + v$ by the exponential map of a Riemannian metric, or embeds $M$ in some $\\R^N$.\nExample: the circle $S^1 \\times \\set{0}$ in $\\R^3$ has trivial normal bundle, and a solid torus around it is a tubular neighbourhood, diffeomorphic to $S^1 \\times \\R^2$.\nA tubular neighbourhood of the central circle of the open Möbius band is the whole band, which is the total space of the nontrivial line bundle over $S^1$.\nTubular neighbourhoods are unique up to isotopy.\n\n## Explanations\n\n### L1 human\n\nAround a smooth submanifold there is always a thickened copy of it that looks like a bundle of small discs, one disc sticking out at each point.\nAround a closed curve in space this is a thin tube, which is where the name comes from.\nThe theorem lets one study the neighbourhood of a submanifold through linear algebra on its normal bundle.\n\n### L2 human\n\nThe theorem is used to cut and paste manifolds along submanifolds, as in surgery, connected sums along submanifolds and Dehn surgery on knots.\nIt identifies the quotient of $M$ by the complement of a tubular neighbourhood of $S$ with the Thom space of $NS$, which is the key step of the Pontryagin–Thom construction.\nIt also gives a smooth retraction of a neighbourhood of $S$ onto $S$, which is used to approximate continuous maps by smooth ones.\nFor manifolds with boundary, the companion result is the collar neighbourhood theorem.\n",
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  "name": "Tubular neighborhood theorem",
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