Is there an ovaloid that is not topologically equivalent to a sphere? 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?
 A: You can take as a reference these notes (check corollary 5.3.5.1). The corollary states the following:

If $M$ is a differentiable surface of $\mathbb{R}^3$ connected and
  compact with Gauss curvature $K \ge 0$ and not identically zero then
  $M$ is homeomorphic to a sphere.

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:

If $M$ is an ovalid then the Gauss map $\stackrel{\to}{N}: M \to
 \mathbb{S}^2$ associated with any unitary normal $N$ is a
   dipheomorphism. In particular, $M$ is dipheomorphic to a sphere.

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. 
