Some Paradoxes
Here we list a few counter-intuitive results which can be proved by applying the h-principle:
- Cone Eversion. Let us consider functions f on R2 without origin f(x) = |x|. Then there is a continuous one-parameter family of functions such that, and for any, is not zero at any point.
- Any open manifold admits a (non-complete) Riemannian metric of positive (or negative) curvature.
- Smale's paradox can be done using isometric embedding of .
- Nash embedding theorem
Read more about this topic: Homotopy Principle
Famous quotes containing the word paradoxes:
“The paradoxes of today are the prejudices of tomorrow, since the most benighted and the most deplorable prejudices have had their moment of novelty when fashion lent them its fragile grace.”
—Marcel Proust (1871–1922)
“Though your views are in straight antagonism to theirs, assume an identity of sentiment, assume that you are saying precisely that which all think, and in the flow of wit and love roll out your paradoxes in solid column, with not the infirmity of a doubt.”
—Ralph Waldo Emerson (1803–1882)