There is no isometry between a sphere and a plane. 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!
 A: 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. 
A: 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.
