Example 2.3.3. The manifold image in image will now be considered. We know that this sphere can only be homeomorphic to the plane image by employing two charts of its atlas and two differentiable functions ϕ1, ϕ2 given below:

image

[see pp. 81-82]. Thus, we cannot find a single diffeomorphism. Hence, the sphere cannot be diffeomorphic to the plane. On the other hand, ...

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