Homotopy Principle - Some Paradoxes

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:

    “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)

    “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)