Homotopy principle

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 works of Yakov Eliashberg, Mikhael Gromov and Anthony V. Phillips. It was based on earlier results of Morris W. Hirsch, Nicolaas Kuiper, John Forbes Nash, and Stephen Smale.

Rough idea

Assume we want to find a function $f$ on R^m which satisfies a partial differential equation of degree $k$, in co-ordinates $(u\_1,u\_2,...,u\_m)$. One can rewrite it as

:$Psi(u\_1,u\_2,...,u\_m,\; J^k\_f)=0!,$

where $J^k\_f$ stands for all partial derivatives of $f$ up to order $k$. Let us exchange every variable in $J^k\_f$ for new independent variables $y\_1,y\_2,...,y\_N$.Then our original equation can be thought as a system of:$Psi^\{\}\_\{\}(u\_1,u\_2,...u\_m,y\_1,y\_2,...y\_N)=0!,$and some number of equations of the following type :$y\_j=\{partial\; y\_iover\; partial\; u\_k\}.!,$

A solution of :$Psi^\{\}\_\{\}(u\_1,u\_2,...u\_m,y\_1,y\_2,...y\_N)=0!,$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 if 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.

Therefore, once you prove that an equation satisfies the h-principle, it is really easy to check whether it has solutions. It is surprising that most underdetermined partial differential equations satisfy the h-principle.

The simplest example

The position of a car in the plane is determined by three parameters: two coordinates $x$ and $y$ for the location (a good choice is the location of the midpoint between the back wheels) and an angle $alpha$ which describes the orientation of the car. The motion of the car satisfies the equation

:$dot\; x\; sinalpha=dot\; ycos\; alpha.,$

A non-holonomic solution in this case, roughly speaking, corresponds to a motion of the car by sliding in the plane. In this case the non-holonomic solutions are not only homotopic to holonomic ones but also can be arbitrarily well approximated by the holonomic ones (by going back and forth, like parallel parking in a limited space). This last property is stronger than the general h-principle; it is called the $C^0$-dense h-principle.

Ways to prove the h-principle

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ome paradoxes

Here we list a few counter-intuitive results which can be proved by applying the h-principle:

1. Let us consider functions "f" on R² without origin "f"("x") = |"x"|. Then there is a continuous one-parameter family of functions $f\_t$ such that $f\_0=f$, $f\_1=-f$and for any $t$, $operatorname\{grad\}(f\_t)$ is not zero at any point.

2. Any open manifold admits a (non-complete) Riemannian metric of positive (or negative) curvature.

3. Smale's paradox can be done using $C^1$ isometric embedding of $S^2$.

Related theorems

*Smale's paradox
*Nash embedding theorem