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  "body": "\n## Definition\n\nLet $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]], let $r \\ge 1$, and let $J^r(M, N)$ be the jet bundle of $r$-jets of smooth maps $M \\to N$, with projection $J^r(M, N) \\to M$.\nA *differential relation* of order $r$ is a subset $\\mathcal{R} \\subseteq J^r(M, N)$.\nA *formal solution* of $\\mathcal{R}$ is a continuous section $F \\colon M \\to J^r(M, N)$ with $F(M) \\subseteq \\mathcal{R}$.\nA formal solution is *holonomic* if $F = j^r f$ for a smooth map $f \\colon M \\to N$, where $j^r f$ is the $r$-jet extension of $f$.\nIn that case $f$ is a *genuine solution* of $\\mathcal{R}$.\nThe relation $\\mathcal{R}$ satisfies the *h-principle* if every formal solution is [[homotopic|9YA6S6HD9V]] to a holonomic one.\nThe homotopy must run through formal solutions of $\\mathcal{R}$.\n\n## Remarks\n\nEvery genuine solution gives a formal solution, so the h-principle says that the only obstruction to solving $\\mathcal{R}$ is the homotopy-theoretic obstruction to finding a formal solution.\nStronger versions ask that the inclusion of holonomic solutions into formal solutions be a weak homotopy equivalence; the parametric h-principle is of this kind.\nExample: let $\\dim M \u003c \\dim N$ and let $\\mathcal{R} \\subseteq J^1(M, N)$ be the set of $1$-jets whose linear part is injective.\nGenuine solutions are immersions, formal solutions are fibrewise injective bundle maps $TM \\to TN$, and the Smale–Hirsch theorem shows that $\\mathcal{R}$ satisfies the h-principle.\nNon-example: for $M = S^1$, $N = \\R$ and the analogous relation of injective $1$-jets, the section $F(x) = (0, \\text{the linear map } v \\mapsto v)$, using the standard trivialisation of $TS^1$, is a formal solution.\nBut there is no genuine solution, because a smooth function on the compact circle has a maximum, where its derivative is zero; so the h-principle fails.\n",
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