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Gaussian Curvature 📂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.


  1. Richard S. Millman and George D. Parker, Elements of Differential Geometry (1977), p180 ↩︎