Gaussian Curvature
Definition1
Mapping a point $M$ of a surface to the normal vector at $p$
$$ \nu : M \to S^{2} \text{ with } \nu (p) \text{ normal to } M \text{ at } p $$
If it is continuous at every point, $M$ is called an orientable surface.
Description
$\nu$ is called the Gauss map.
Examples
Sphere $S^{2}$
If $\nu (p)$ is called the outward normal vector at $p$, since $\nu (p) = p$, it is continuous. Therefore, the sphere $S^{2}$ is an orientable surface.
Torus $T^{2}$
Is an orientable surface.
Möbius Strip
The Möbius strip is not an orientable surface.
Theorem
All compact surfaces in $\mathbb{R}^{3}$ are orientable surfaces.
Richard S. Millman and George D. Parker, Elements of Differential Geometry (1977), p180 ↩︎