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  "body": "\n## Summary\n\nA submersion is a smooth map between [[smooth manifolds|F45VQ47GPV]] whose [[differential|6P9PXQDSEN]] is surjective at every point.\nLocally it looks like the projection of $\\R^m$ onto $\\R^n$.\nIts fibres are smooth submanifolds, which makes submersions the basic way to cut a manifold into smooth pieces.\nProjections of products and of fibre bundles are typical examples.\n\n## Explanations\n\n### L1 human\n\nA submersion is a map that is locally a projection, like the shadow map $(x, y, z) \\mapsto (x, y)$ that forgets the height.\nNear each point it can move the point in every direction of the target.\nBecause of this, the set of points that land on a given value is a smooth piece of space, with no pinches.\nFor example, $f(x, y) = x^2 + y^2$ is a submersion from $\\R^2 \\setminus \\set{0}$ to $(0, \\infty)$, and its fibres are the circles around the origin.\n",
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