Topologically speaking, the compact and connected surfaces are classified into three kinds of surfaces:
- a sphere
- a connected sum of tori
- a connected sum of projective planes.
Also, we know that:
- the sphere is an ovaloid
- ovaloids are compact and connected and therefore can be put in one of these categories.
Is there an ovaloid not topologically equivalent to a sphere?
How could I prove that there exist or not an ovaloid that is not topologically equivalent to a sphere?
Best Answer
You can take as a reference these notes (check corollary 5.3.5.1). The corollary states the following:
My understanding is that one requires differentiability to be able to use Gauss curvature.
Now, the ingredients to arrive to this corollary are the following:
a classification of topological surfaces which you cite and which is stated at theorem 5.3.5.1 of the document.
the Gauss-Bonnet theorem which has been mentionned in the comments and appears in the document at theorem 5.3.4.1.
Realize that homeomorphic implies homotopy equivalent.
In fact, the document cites a stronger result by Hadamard in theorem 5.3.6.2:
So as you can see you need quite a bit of machinery to prove the result. However you get to a nice result for ovaloids.