From: José Carlos Santos on
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