aka Gauss curvature

Determinant of the Differential of the Gauss map, that is the product of the Principal curvatures (video)

A point $p$ on a Surcace is called an

• Elliptic point if $K(p) > 0$
• Hyperbolic point if $K(p) < 0$
• Parabolic point if $K(p) = 0$, it has one of the Principal curvatures non-0
• Planar point
• Umbilical point

Geometric interpretation of Gaussian curvature

For points with positive curvature, the surface, locally, lies on one side of the Tangent plane

For points with negative curvature, the surface, locally, lies on both sides of the Tangent plane

To study this analytically, we need to define the Hessian, and the Height function for the tangent plane around any point

Formal formulation

Note that for Elliptic points, the level curves are locally Ellipses, and for Hyperbolic points they are locally Hyperbolas

There is no such theorem for K(p)=0

Local classification of surfaces by Gauss curvature

A surface with zero Gauss curvature is locally isometric to the Plane (surface)

A surface with constant positive Gauss curvature is locally isometric to a Sphere

A surface with constant negative Gauss curvature is locally isometric to a Pseudosphere

Expression of the Gaussian curvature in Geodesic coordinates

intuition

Gaussian curvature in dicrete geometry

The Gauss-Bonnet theorem relates its integral with the Genus of a surface]]