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In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs). The h-principle is good for underdetermined PDEs or PDRs, such as occur in the immersion problem, isometric immersion problem, and other areas. The theory was started by Yakov Eliashberg, Mikhail Gromov and Anthony V. Phillips. It was based on earlier results that reduced partial differential relations to homotopy, particularly for immersions. The first evidence of h-principle appeared in the Whitney–Graustein theorem. This was followed by the Nash-Kuiper Isometric embedding theorem and the Smale-Hirsch Immersion theorem. ==Rough idea== Assume we want to find a function ''ƒ'' on R''m'' which satisfies a partial differential equation of degree ''k'', in co-ordinates . One can rewrite it as : where stands for all partial derivatives of ''ƒ'' up to order ''k''. Let us exchange every variable in for new independent variables Then our original equation can be thought as a system of : A solution of : is called a non-holonomic solution, and a solution of the system (which is a solution of our original PDE) is called a holonomic solution. In order to check whether a solution exists, first check if there is a non-holonomic solution (usually it is quite easy and if not then our original equation did not have any solutions). A PDE ''satisfies the h-principle'' if any non-holonomic solution can be deformed into a holonomic one in the class of non-holonomic solutions. Thus in the presence of h-principle, a differential topological problem reduces to an algebraic topological problem. More explicitly this means that apart from the topological obstruction there is no other obstruction to the existence of a holonomic solution. The topological problem of finding a ''non-holonomic solution'' is much easier to handle and can be addressed with the obstruction theory for topological bundles. Many underdetermined partial differential equations satisfy the h-principle. However, the falsity of an h-principle is also an interesting statement, intuitively this means the objects being studied have non-trivial geometry that cannot be reduced to topology. As an example, embedded Lagrangians in a symplectic manifold do not satisfy an h-principle, to prove this one needs to find invariants coming from pseudo-holomorphic curves. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「homotopy principle」の詳細全文を読む スポンサード リンク
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