Is there an ovaloid that is not topologically equivalent to a sphere

algebraic-topologygeneral-topologysurfaces

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:

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.

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