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From: José Carlos Santos on 3 Jan 2010 04:44 On 03-01-2010 6:53, Hodol wrote: >>> hi all, I have a problem of manifolds as homework, but I don't know >>> how to show. >>> the problem is >> >>> prove or disprove S^2 is diffeomorphic to M where S^2 = {(x,y,z)| >>> x^2+y^2+z^2 = 1} and M = {(x,y,z)|x^4+y^6+z^8 = 1} >> >>> Could somebody help me, please? >> >> Consider the map from M into S^2 defined by v |-> v/||v||. > > Could you show me more... please? Every ray _r_ whose initial point is the origin (that is, (0,0,0)) contains one and only one point r_M of M and one and only one point r_S of S^2. The map from my previous post maps r_M into r_S and it is clearly differentiable. From what I said above, it must also be a bijection. Now prove that its inverse is also differentiable. Best regards, Jose Carlos Santos
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