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The question:

  1. Is the sphere $S_2$ isometric / diffeomorphic / homeomorphic to the plane?
  2. Is the sphere $S_2$ minus a point isometric / diffeomorphic / homeomorphic to the plane?
  3. Is the sphere $S_2$ isometric / diffeomorphic / homeomorphic to $[−1,1]^2$?
  4. Is the sphere $S_2$ minus a point isometric / diffeomorphic / homeomorphic to $[−1,1]^2$?
  5. Is there any property to the plane and the $[−1,1]^2$ that would make them more discriminable (like properties that depend on the finite diameter of the $[−1,1]^2$)?

I first posted this question mistakenly on MathOverflow; from that thread I would like to add:

  • I wasn't sure about the term "isomorphism" in the area of manifolds. Apparently there is no such thing; I thought it would be any kind of bijection between two manifolds (i.e. "diffeomorphism" minus differentiability). -- Is that correct?

  • From the original thread I got the following answers -- are they correct?:

    1. No, no, no.
    2. No, yes, yes.
    3. No, no, no.
    4. No, no, no.
  • Then why isn't the sphere diffeomorphic to $[-1,1]^2$? What about simple polar coodinate transforms that are mapping to $[0, 2\pi] \times [-\pi, \pi]$?

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up vote 1 down vote accepted
  • The term "isomorphism" in manifold theory depends on what kind of kmanifold you are dealing with. If they are topological manifolds, "isomorphisms" are homeomorphisms. If they are smooth manifolds, "isomorphisms" are diffeomorphisms, and so on. For each category of manifolds and maps, there is an appropriate notion of isomorphism.

  • Yes, you answers are correct. Remark: Whether two manifolds are isometric or not is not a well defined question unless you spesify their metrics. Here I assumed flat manifolds to have their euclidean metric and the sphere's metric to be induced from its embedding as the unit sphere in $\mathbb{R}^3$.

  • Polar coordinates are not in a one-to-one correspondence with points on the sphere. They are singular at the induced poles. Indeed, there does not exist any single chart covering the entire sphere. The easiest way to tell that the sphere and the unit square is not diffeomorphic (or even homeomorphic) is that their (co)homologies don't agree in dimension 2.

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