Editing Homotopy principle (section)

==Some paradoxes==

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

*Cone eversion.<ref>D. Fuchs, S. Tabachnikov, ''Mathematical Omnibus: Thirty Lectures on Classic Mathematics''</ref> Consider functions ''f'' on '''R'''<sup>2</sup> without origin ''f''(''x'')&nbsp;=&nbsp;|''x''|. Then there is a continuous one-parameter family of functions <math>f_t</math> such that <math>f_0=f</math>, <math>f_1=-f</math> and for any <math>t</math>, <math>\operatorname{grad}(f_t)</math> is not zero at any point.
*Any open manifold admits a (non-complete) Riemannian metric of positive (or negative) curvature.
*[[Sphere eversion]] without creasing or tearing can be done using <math>C^1</math> immersions of <math>S^2</math>.
*The [[Nash-Kuiper theorem|Nash-Kuiper C<sup>1</sup> isometric embedding theorem]], in particular implies that there is a <math>C^1</math> isometric immersion of the round <math>S^2</math> into an arbitrarily small ball of <math>\mathbb R^3</math>. This immersion cannot be <math>C^2</math> because a small oscillating sphere would provide a large lower bound for the principal curvatures, and therefore for the [[Gauss curvature]] of the immersed sphere, but on the other hand if the immersion is <math>C^2</math> this has to be equal to 1 everywhere, the Gauss curvature of the standard <math>S^2</math>, by Gauss' [[Theorema Egregium]].