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