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  "body": "\n## Statement\n\nLet $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]].\nLet $f \\colon M \\to N$ be a smooth map that is a surjective [[submersion|BMY55EW3RF]] and a [[proper map|15XEBGNGHC]].\nThen $f$ is a locally trivial [[fiber bundle|CX4M8K61AB]].\nThat is, for every $q \\in N$ there are an open neighbourhood $V$ of $q$ and a [[diffeomorphism|0KZZ70ZHXP]] $\\varphi \\colon f^{-1}(V) \\to V \\times f^{-1}(q)$ such that $\\mathrm{pr}_1 \\circ \\varphi = f$ on $f^{-1}(V)$.\n\n## Remarks\n\nThe idea of the proof: choose a complement of the kernel of $df$ in $TM$, for instance the orthogonal complement for a Riemannian metric; this is an [[Ehresmann connection|TD6VEP3SCE]].\nOver a coordinate ball $V$ around $q$, lift the coordinate vector fields of $V$ to horizontal vector fields on $f^{-1}(V)$.\nProperness makes their [[flows|06738GGWWK]] exist as long as the flow downstairs stays in $V$.\nComposing these flows moves the fiber $f^{-1}(q)$ onto every nearby fiber, and this gives $\\varphi$.\n\nExample: if $M$ is compact and $N$ is connected, every submersion $f \\colon M \\to N$ with $M$ nonempty is proper and surjective, so all fibers of $f$ are diffeomorphic.\n\nProperness is needed.\nLet $M = \\R^2 \\setminus \\set{0}$, $N = \\R$ and $f(x, y) = x$.\nThen $f$ is a surjective submersion, but its fiber over $0$ is two disjoint half-lines while its fiber over $1$ is one line.\nThese fibers are not diffeomorphic, so $f$ is not locally trivial near $0$.\nHere $f$ is not proper: the preimage of $[-1, 1]$ is not compact.\n\n## Explanations\n\n### L1 human\n\nA submersion is a smooth map with no critical points, so near each point it looks like a projection.\nThe theorem says that if, in addition, preimages of compact sets are compact, then the map looks like a projection near each whole fiber, not only near each point.\nIn particular all fibers over a connected base are the same manifold.\n\n### L2 human\n\nThe theorem is a main source of fiber bundles in geometry and topology.\nFor instance, a proper family of smooth hypersurfaces with no singular member has diffeomorphic members, which is the start of the study of monodromy and of the Milnor fibration.\nIt is the smooth counterpart of the fact that a proper local homeomorphism between connected manifolds is a covering map.\nFor non-proper maps, local triviality can still hold, but one must control the behaviour at infinity by other means.\n",
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