# Properties of $S_2$ and the plane and $[−1,1]^2$

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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• 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 $\mathbb{R}^3$.