Homotopy principle

Summary

The homotopy principle, or h-principle, is a property of a partial differential relation. It says that every formal solution can be deformed through formal solutions into a genuine solution. A formal solution assigns to each point a value and candidate derivatives that satisfy the relation pointwise. It does not require the candidates to be the true derivatives of the values. When the h-principle holds, the existence and classification of solutions reduce to a problem in homotopy theory. It was formulated by Mikhail Gromov, building on results of Smale, Hirsch and Nash. It covers immersions, many symplectic and contact problems, and isometric $C^1$ embeddings.