What is the difference between Gaussian and mean curvature?

Gaussian curvature is the product of the two principal curvatures and is intrinsic to the surface; mean curvature is their average and depends on how the surface is embedded in space, governing, for example, minimal surfaces.

What does the Gauss-Bonnet theorem say?

For a closed surface, the integral of the Gaussian curvature equals 2π times the Euler characteristic; total curvature is therefore a topological invariant, unchanged by bending the surface.

Curves and Surfaces

The classical theory of curves and surfaces in three-dimensional space introduces curvature concretely, from the bending and twisting of a curve to the Gaussian curvature of a surface and the global Gauss-Bonnet theorem.

Definition

This is the differential geometry of one- and two-dimensional smooth submanifolds of Euclidean space, describing curves by curvature and torsion and surfaces by their first and second fundamental forms and the curvatures derived from them.

Scope

This topic covers the local theory of space curves via the Frenet-Serret frame (curvature and torsion), regular surfaces and their parametrizations, the first fundamental form measuring intrinsic distances and the second fundamental form measuring bending, and the principal, Gaussian, and mean curvatures. It develops Gauss's Theorema Egregium, geodesics on surfaces, and the Gauss-Bonnet theorem linking total curvature to the Euler characteristic — the classical prototype of the connection between geometry and topology.

Core questions

How do curvature and torsion completely determine a space curve up to rigid motion?
What is the difference between intrinsic geometry (the first fundamental form) and extrinsic bending (the second fundamental form)?
Why is Gaussian curvature intrinsic, as the Theorema Egregium asserts?
How does the Gauss-Bonnet theorem tie total curvature to the topology of a surface?

Key concepts

Frenet-Serret frame, curvature, and torsion of curves
First and second fundamental forms
Principal, Gaussian, and mean curvature
Theorema Egregium and intrinsic geometry
Geodesics and the Gauss-Bonnet theorem

Clinical relevance

The classical theory furnishes the geometric intuition behind general curved spaces, models surfaces in computer graphics, architecture, and materials science, and the Gauss-Bonnet theorem is the historical seed of index theory and characteristic classes.

History

Euler and Monge initiated the study of curves and surfaces; Gauss's Disquisitiones (1827) introduced the intrinsic viewpoint and the Theorema Egregium, and Bonnet's contribution to the Gauss-Bonnet theorem made the global geometry-topology link explicit, anchoring the classical curriculum codified by do Carmo.

Key figures

Carl Friedrich Gauss
Jean Frédéric Frenet
Manfredo do Carmo

Seminal works

docarmo1976
lee2012

Frequently asked questions

What is the difference between Gaussian and mean curvature?: Gaussian curvature is the product of the two principal curvatures and is intrinsic to the surface; mean curvature is their average and depends on how the surface is embedded in space, governing, for example, minimal surfaces.
What does the Gauss-Bonnet theorem say?: For a closed surface, the integral of the Gaussian curvature equals 2π times the Euler characteristic; total curvature is therefore a topological invariant, unchanged by bending the surface.