Differential geometry of surfaces

Despite measuring different aspects of length and angle, the first and second fundamental forms are not independent from one another, and they satisfy certain constraints called the Gauss-Codazzi equations. A major theorem, often called the fundamental theorem of the differential geometry of surfaces, asserts that whenever two objects satisfy the Gauss-Codazzi constraints, they will arise as the first and second fundamental forms of a regular surface.

First and second fundamental forms, the shape operator, and the curvatureChristoffel symbols, Gauss–Codazzi equations, and the Theorema Egregium

This theorem is baffling. [...] It is the kind of theorem which could have waited dozens of years more before being discovered by another mathematician since, unlike so much of intellectual history, it was absolutely not in the air. [...] To our knowledge there is no simple geometric proof of the theorema egregium today.

In terms of the inner product coming from the first fundamental form, this can be rewritten as

The right-hand side of the three Gauss equations can be expressed using covariant differentiation. For instance, the right-hand side

In 1776 Jean Baptiste Meusnier showed that the differential equation derived by Lagrange was equivalent to the vanishing of the mean curvature of the surface:

Surfaces with (from l. to r.) constant negative, zero and positive Gaussian curvature

(there is no standard agreement whether to use + or − in the definition).

The geodesic curvature measures in a precise way how far a curve on the surface is from being a geodesic.

Contour lines tracking the motion of points on a fixed curve moving along geodesics towards a basepoint

a key result, usually called the Gauss lemma. Geometrically it states that

In isothermal coordinates, first considered by Gauss, the metric is required to be of the special form

The geodesics between two points on the sphere are the great circle arcs with these given endpoints. If the points are not antipodal, there is a unique shortest geodesic between the points. The geodesics can also be described group theoretically: each geodesic through the North pole (0,0,1) is the orbit of the subgroup of rotations about an axis through antipodal points on the equator.

The first model, based on a disk, has the advantage that geodesics are actually line segments (that is, intersections of Euclidean lines with the open unit disk). The last model has the advantage that it gives a construction which is completely parallel to that of the unit sphere in 3-dimensional Euclidean space. Because of their application in complex analysis and geometry, however, the models of Poincaré are the most widely used: they are interchangeable thanks to the Möbius transformations between the disk and the upper half-plane.

Since lines or circles are preserved under Möbius transformations, geodesics are again described by lines or circles orthogonal to the real axis.

Parallel transport of a vector around a geodesic triangle on the sphere. The length of the transported vector and the angle it makes with each side remain constant.