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  "body": "\n## Summary\n\nA Legendrian submanifold of a [[contact manifold|HBQ8BZ7XB5]] of dimension $2n + 1$ is a submanifold of dimension $n$ that is everywhere tangent to the contact hyperplanes.\nLegendrian submanifolds play in contact geometry the role that [[Lagrangian submanifolds|H8ACPTMX08]] play in symplectic geometry.\nIn a contact $3$-manifold they are Legendrian knots, whose classification up to Legendrian isotopy is a rich subject with invariants such as the Thurston–Bennequin number.\n\n## Explanations\n\n### L1 human\n\nA contact structure gives a plane at each point of a three-dimensional space, and these planes twist so much that no surface can follow them.\nCurves can still follow them: a Legendrian curve is a curve whose velocity always lies in the given plane.\nIn $\\R^3$ with the planes $\\ker(dz - y\\,dx)$, the curve $t \\mapsto (t, f'(t), f(t))$ is Legendrian for every smooth function $f$, because $dz - y\\,dx$ vanishes on its velocity.\nLegendrian knots appear in geometric optics as the fronts of propagating waves, lifted with their direction of travel.\n",
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