1
$\begingroup$

Using the definition of a manifold directly prove that the $2$-sphere $S^2 \subset \mathbb{R}^3$ is a $C^\infty$-manifold of dimension $2$.

[Hint: one way (though certainly not the only way) to do this is to try stereographic projections (draw a straight line from the north pole to the $z = −1$ plane and assign the point that intersects the sphere the corresponding point that intersects the $z= 1$ plane,then do something analogous for the south pole and the $z= 1$ plane)

I drew what the hint said and now I am wondering what to do next. How should i go about this?

Definition of a manifold: A subset $M \subseteq \mathbb{R}^k$ is an $m-$ dimensional manifold iff for all $c\in M$ there exists $U,V \in \mathbb{R}^k$ and a diffeomorphism $U \rightarrow \psi ^{\leftarrow \phi} V$ such that $c \in U$ and $\phi(U \cap M)= (\mathbb{R}^m \times {0}\}\cap V$

My drawing looks something like:

https://en.wikipedia.org/wiki/Stereographic_projection#/media/File:Stereographic_projection_in_3D.svg

Where $(0,0,1)\in \mathbb{R}^3$ is the north pole and $(0,0,-1)$ is the south pole.

  • 1
    Ignore the hint if you can think of another way to obtain a smooth atlas. – 2017-02-28
  • 2
    I was going to write an answer, but that hint does almost all of the job, so if you don't understand that then you'll probably not understand any other answer. – 2017-02-28
  • 0
    Explaining the hint would also be helpful. Thanks! – 2017-02-28
  • 0
    I drew what the hint said and now I am wondering what to do next – 2017-02-28
  • 0
    Should my picture look like this? https://en.wikipedia.org/wiki/Stereographic_projection#/media/File:Stereographic_projection_in_3D.svg – 2017-02-28
  • 0
    @AlexM. I am still trying to figure this out using the hint. – 2017-02-28

0 Answers 0