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  "body": "\n## Definition\n\nLet $(M, \\xi)$ be a [[contact manifold|HBQ8BZ7XB5]] of dimension $2n + 1$.\nA *Legendrian submanifold* of $(M, \\xi)$ is a submanifold $L \\subset M$ of dimension $n$ such that $T_x L \\subset \\xi_x$ for every $x \\in L$.\nA *Legendrian knot* is a Legendrian submanifold of a contact $3$-manifold that is diffeomorphic to a circle.\n\n## Remarks\n\nIf $\\xi = \\ker \\alpha$, the condition is that $\\alpha$ restricts to zero on $L$.\nThen $d\\alpha$ also restricts to zero on $L$, so each $T_x L$ is an isotropic subspace of the symplectic vector space $(\\xi_x, d\\alpha)$, of dimension $2n$.\nHence a submanifold tangent to $\\xi$ has dimension at most $n$, and Legendrian submanifolds are the tangent submanifolds of maximal dimension.\n\nExample: in $\\R^3$ with $\\alpha = dz - y\\,dx$, the $x$-axis $\\set{y = z = 0}$ is Legendrian, since along it $dz = 0$ and $y = 0$.\nMore generally, for a smooth $f \\colon \\R \\to \\R$, the curve $\\set{(x, f'(x), f(x))}$ is Legendrian, since $dz - y\\,dx = f'(x)\\,dx - f'(x)\\,dx = 0$ on it.\n\nNon-example: the $z$-axis in the same space is not Legendrian, since $\\alpha(\\partial_z) = 1 \\ne 0$.\n",
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