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How can I show that there is no isometry between a sphere and a plane?

Wikipedia defines an isometry as follows:

Let $(M,g)$ and $(M',g')$ be two Riemannian manifolds, and let $f:M\to M'$ be a diffeomorphism. Then $f$ is called an isometry if $g'=f^*g'$, where $f^*g'$ denotes the pullback of the rank $(0,2)$ metric tensor $g'$ by $f$.

However, I have no clue how to apply this definition to solve this problem. Any hint would be greatly appreciated!

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There is not even a homeomorphism between a sphere and a plane. One is compact, and the other isn't. –  froggie Nov 2 '12 at 2:01
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Where did you come across this problem? (I ask because it seems odd that you would have to go to Wikipedia to look up the definition of "isometry.") Anyway, the answer to your question is that there cannot exist a diffeomorphism (much less an isometry!) between the sphere and the plane. –  Jesse Madnick Nov 2 '12 at 2:01
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Perhaps a better question would be to take the sphere and remove a point, and show that this is not isometric to the plane. The two spaces are diffeomorphic, and hence it is not an unreasonable question. One way to see they cannot be isometric is that the sphere has positive curvature, while the plane has 0 curvature. Another (easier but less illuminating) way, is to note that the sphere (minus a point) has finite diameter, while the plane does not. –  froggie Nov 2 '12 at 2:06
    
Or that sphere (minus a point) is incomplete while the plane is complete. –  Neal Nov 2 '12 at 20:18

2 Answers 2

up vote 7 down vote accepted

A sphere and a plane are not even homeomorphic and so can't be diffeomorphic or isometric. What is more interesting, is that there isn't a local isometry between the plane and the sphere. This means that there is no map, that, when restricted to a small neighborhood, is an isometry of that small neighborhood onto its image.

To show such a result, you find an invariant that depends only on the metric $g$, calculate that invariant and show that it is different for the two objects involved. Here, the relevant invariant is the curvature. The plane is flat while the sphere has constant positive curvature.

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Imagine you are holding a sheet of paper. This is your plane. Try and fold the sheet if paper into a sphere without creasing or bending the edges. Can you do this? No, isometries preserve curvature. If you try and make a sphere out of a sheet of paper you will have to bend or tear the paper. But we know that a plane and a cylinder are isometric. You can easily wrap the paper around to form a cylinder. There is no bending needed.

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