📂Geometry

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.