prove or disprove S^2 is diffeomorphic to M={(x,y,z)|x^4+y^6+z^8 = 1}

Hodol

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Jan 1, 2010, 5:42:59 AM1/1/10

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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?
thanks.

José Carlos Santos

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Jan 1, 2010, 5:49:26 PM1/1/10

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Consider the map from M into S^2 defined by v |-> v/||v||.

Best regards,

Jose Carlos Santos

Hodol

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Jan 3, 2010, 1:53:29 AM1/3/10

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Could you show me more... please?

José Carlos Santos

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Jan 3, 2010, 4:39:28 AM1/3/10

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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?

It is easy to check that any ray _r_ whose initial point is the origin
(that is, (0,0,0)) contains one and only one point r_M of M. Of course,
it also contains one and only one point r_S of S^2. The map that I
mentioned maps r_M into r_S. It is obviously differentiable and, from
what I said above, it must be bijective. Now prove that its inverse is
also differentiable.

José Carlos Santos

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Jan 3, 2010, 4:44:31 AM1/3/10

to

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.

Hodol

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Jan 15, 2010, 7:26:23 AM1/15/10

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I've reach a solution to this problem with your help. :)
Thank you!