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  "body": "\n## Summary\n\nThe homotopy principle, or h-principle, is a property of a partial differential relation.\nIt says that every formal solution can be deformed through formal solutions into a genuine solution.\nA formal solution assigns to each point a value and candidate derivatives that satisfy the relation pointwise.\nIt does not require the candidates to be the true derivatives of the values.\nWhen the h-principle holds, the existence and classification of solutions reduce to a problem in homotopy theory.\nIt was formulated by Mikhail Gromov, building on results of Smale, Hirsch and Nash.\nIt covers immersions, many symplectic and contact problems, and isometric $C^1$ embeddings.\n\n## Explanations\n\n### L1 human\n\nMany geometric problems ask for a map whose derivative obeys some condition, for example a map whose derivative is never zero.\nThe easy part is to choose, at each point, a value and a matrix that obeys the condition, with no link between the two.\nThe hard part is to find a map whose real derivative obeys it.\nThe h-principle says that whenever the easy problem can be solved, so can the hard one, and up to deformation in the same number of ways.\nA famous consequence is that a sphere can be turned inside out through immersions.\n",
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