Editing Homotopy principle (section)

== Simple examples ==

=== Monotone functions ===
Perhaps the simplest partial differential relation is for the derivative to not vanish: <math>f'(x) \neq 0.</math> Properly, this is an ''ordinary'' differential relation, as this is a function in one variable. 

A holonomic solution to this relation is a function whose derivative is nowhere vanishing, i.e. a strictly monotone differentiable function, either increasing or decreasing. The space of such functions consists of two disjoint [[convex set]]s: the increasing ones and the decreasing ones, and has the homotopy type of two points.

A non-holonomic solution to this relation would consist in the data of two functions, a differentiable function f(x), and a [[continuous function]] g(x), with g(x) nowhere vanishing.  A holonomic solution gives rise to a non-holonomic solution by taking g(x) = f'(x).  The space of non-holonomic solutions again consists of two disjoint convex sets, according as g(x) is positive or negative.  

Thus the inclusion of holonomic into non-holonomic solutions satisfies the h-principle. 

[[File:Winding Number Around Point.svg|thumb|The [[Whitney–Graustein theorem]] shows that immersions of the circle in the plane satisfy an h-principle, expressed by [[turning number]].]]
This trivial example has nontrivial generalizations:
extending this to immersions of a circle into itself classifies them by order (or [[winding number]]), by lifting the map to the [[universal covering space]] and applying the above analysis to the resulting monotone map – the [[linear map]] corresponds to multiplying angle: <math>\theta \mapsto n\theta</math> (<math>z \mapsto z^n</math> in complex numbers). Note that here there are no immersions of order 0, as those would need to turn back on themselves. Extending this to circles immersed in the plane – the immersion condition is precisely the condition that the derivative does not vanish – the [[Whitney–Graustein theorem]] classified these by [[turning number]] by considering the homotopy class of the [[Gauss map]] and showing that this satisfies an h-principle; here again order 0 is more complicated.

Smale's classification of immersions of spheres as the homotopy groups of [[Stiefel manifold]]s, and Hirsch's generalization of this to immersions of manifolds being classified as homotopy classes of maps of [[frame bundle]]s are much further-reaching generalizations, and much more involved, but similar in principle – immersion requires the derivative to have rank ''k,'' which requires the partial derivatives in each direction to not vanish and to be linearly independent, and the resulting analog of the Gauss map is a map to the Stiefel manifold, or more generally between frame bundles.

=== A car in the plane ===
As another simple example, consider a car moving in the plane. The position of a car in the plane is determined by three parameters: two coordinates <math>x</math> and <math>y</math> for the location (a good choice is the location of the midpoint between the back wheels) and an angle <math>\alpha</math> which describes the orientation of the car. The motion of the car satisfies the equation

:<math>\dot x \sin\alpha=\dot y\cos \alpha.</math>
since a non-skidding car must move in the direction of its wheels. In [[robotics]] terms, not all paths in the task space are [[holonomic (robotics)|holonomic]].

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) – note that this approximates both the position and the angle of the car arbitrarily closely. This implies that, theoretically, it is possible to parallel park in any space longer than the length of your car. It also implies that, in a contact 3 manifold, any curve is <math>C^0</math>-close to a [[Legendrian knot|Legendrian]] curve.
This last property is stronger than the general h-principle; it is called the <math>C^0</math>-'''dense h-principle'''.

While this example is simple, compare to the [[Nash embedding theorem]], specifically the [[Nash–Kuiper theorem]], which says that any [[short map|short]] smooth (<math>C^\infty</math>) embedding or immersion of <math>M^m</math> in <math>\mathbf{R}^{m+1}</math> or larger can be arbitrarily well approximated by an isometric <math>C^1</math>-embedding (respectively, immersion). This is also a dense h-principle, and can be proven by an essentially similar "wrinkling" – or rather, circling – technique to the car in the plane, though it is much more involved.