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Given a compact, connected regular surface $S$ in Euclidean space which has everywhere positive gaussian curvature = an ovaloid. It is a theorem of Hadamard that ovaloids are diffeomorphic to the sphere (discussed in Klingenberg), but the proof of this follows the proof of their orientability = that there exists a continuous, unit normal field defined globally on $S$. Unfortunately, there is no proof of this in doCarmo (as far as I can tell) and the proof in Klingenberg is said to be obvious, but the details are not explained. (He says the second order surface approximating $S$ near any point is an elliptic paraboloid, but it is not explained how to define the unit normal field globally)

Does anyone know a full proof of this theorem? Ovaloids were extensively studied by Blaschke in his book (kreis und kugel), unfortunately I do not speak German. I would be extremely grateful for any references on ovaloids in English (but only classical differential geometry)

Artin Ostal
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    Hint: Every compact and connected surface in $\mathbb{R}^3$ is orientable. You can realize that such surfaces divides the Euclidean space in two disjoints regions, the "inside" of the surface and the "outside" of the surface. – DiegoMath Mar 05 '21 at 23:41
  • The case of $K>0$ really requires less fire power. Follow up on Hirzebruch's suggestion or think about the support hyperplanes to the surface. – Ted Shifrin Mar 06 '21 at 02:20

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Here is a possible proof: you can use the Jordan-Brouwer Separation Theorem (found on page 89 of Guillemin and Pollack's book of Differential Topology) which says

The complement of a compact and connected hypersurface $X$ in $\Bbb{R}^n$ consists of two open sets, the "inside" $D_1$ and the "outside" $D_0$. Moreover, $\overline{D}_1$ is a compact manifold with boundary and $\partial \overline{D}_1=X$.

In particular, being the boundary of a compact manifold, $X$ inherits a boundary orientation. I.e. we define a tangent vector $V_p\in T_pX$ to be outward facing if it is the trajectory of a curve in $D_1$. In particular, this gives a well-defined notion of global outward normal vector field for the surface $X$ (take $n=3$).

Anyway, this implies that $X$ is orientable.

Addendum: It depends on what your notion of "classical" differential geometry is, but I think that the Jordan-Brouwer theorem is a fairly classical theorem of differential topology/geometry.

Alekos Robotis
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