Is there always a diffeomorphism between $(0,1)^2$ and any given (not degenerate) triangle?
Diffeomorphism between a triangle and a square?
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real-analysis
differential-topology
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0*Diffeo*-morphism?? I would doubt. At maximum between the open *inner* part of the square and the triangle. – 2012-11-28
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0I don't understand your second sentence. – 2012-11-28
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1@Berci is referring to the fact that [every convex open subsets of the Euclidean plane is diffeomorphic to the unit open ball](http://mathoverflow.net/questions/4468/what-are-the-open-subsets-of-mathbbrn-that-are-diffeomorphic-to-mathbb/4516#4516). – 2012-11-28
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0@Berci Sorry, the second sentence didn't parse. Now it makes sense. – 2012-11-28
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0@WillieWong Thanks. So if I take $(0,1)^2$ then I'm good? (Set of measure zero is nothing to me) – 2012-11-28
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0Question changed, somebody go for the answer. – 2012-11-28
1 Answers
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If by "triangle" you mean the open set bounded by three line segments (the boundaries themselves are not included), then yes, every convex open subset of the Euclidean plane is diffeomorphic to $\mathbb{R}^2$.
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1In fact every simply connected open subset of the Euclidean plane is diffeomorphic to $\mathbb R^2$. – 2012-11-28
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1@Jacob: I see two ways to conclude that, one of which involves the Riemannian mapping theorem. Do you have in mind a easier proof? – 2012-11-28
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1I had the Riemannian mapping theorem in mind. – 2012-11-28