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 way of paradoxes is the way of truth. To test Reality we must see it on the tight-rope. When the Verities become acrobats we can judge them.”
—Oscar Wilde (1854–1900)
“This is one of the paradoxes of the democratic movement—that it loves a crowd and fears the individuals who compose it—that the religion of humanity should have no faith in human beings.”
—Walter Lippmann (1889–1974)