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

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