H-principle

In mathematics, the homotopy principle (h-principle) is a very general way to solve partial differential equations (PDE), and more generally partial differential relations (PDR). The h-principle is good for underdetermined PDE or PDR such as immersion problem, isometric immersions problem and so on.

Some paradoxes

Here we list few paradoxical results which can be proved by applying the

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h-principle:

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1. Let us consider functions f on R2 without origin f(x) = |x|. Then there is a continuous one parameter family of functions f_t such that f_0=f, f_1=-f

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and for any t we have that operatorname{grad}(f_t) is not zero at any point.

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2. Any open manifold admits a (non-complete) Riemannian metric of positive (or negative) curvature.

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3. Smale's paradox can be done using C^1 isometric embedding of S^2.

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