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  "body": "\n## Definition\n\nLet $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]], and let $f \\colon M \\to N$ be a [[smooth|0V33BNKZ9S]] map.\nThe map $f$ is an *immersion* if for every point $p \\in M$ the [[differential|6P9PXQDSEN]] $df_p \\colon T_pM \\to T_{f(p)}N$ is injective.\n\n## Remarks\n\nEquivalently, the rank of $df_p$ equals $\\dim M$ at every point $p$; in particular $\\dim M \\le \\dim N$.\nAn immersion that is a homeomorphism onto its image, with the subspace topology, is a *smooth embedding*.\nExample: $f \\colon \\R \\to \\R^2$, $f(t) = (\\cos t, \\sin t)$, is an immersion.\nIts derivative $f'(t) = (-\\sin t, \\cos t)$ never vanishes, so $df_t$ is injective for every $t$.\nIt is not an embedding, because it is not injective.\nNon-example: $g(t) = (t^2, t^3)$ is smooth and injective but is not an immersion.\nIts derivative $g'(t) = (2t, 3t^2)$ vanishes at $t = 0$, where the image has a cusp.\nBy the constant rank theorem, near each point an immersion has the local form $(x_1, \\dots, x_m) \\mapsto (x_1, \\dots, x_m, 0, \\dots, 0)$ in suitable charts.\n",
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