Differential Geometry of Surfaces - Overview

Overview

Polyhedra in the Euclidean space, such as the boundary of a cube, are among the first surfaces encountered in geometry. It is also possible to define smooth surfaces, in which each point has a neighborhood diffeomorphic to some open set in E2, the Euclidean plane. This elaboration allows calculus to be applied to surfaces to prove many results.

Two smooth surfaces are diffeomorphic if and only if they are homeomorphic. (The analogous result does not hold for higher-dimensional manifolds.) It follows that closed surfaces are classified up to diffeomorphism by their Euler characteristic and orientability.

Smooth surfaces equipped with Riemannian metrics are of foundational importance in differential geometry. A Riemannian metric endows a surface with notions of geodesic, distance, angle, and area. An important class of such surfaces are the developable surfaces: surfaces that can be flattened to a plane An without stretching; examples include the cylinder and the cone.

In addition, there are properties of surfaces which depend on an embedding of the surface into Euclidean space. These surfaces are the subject of extrinsic geometry. They include

• Minimal surfaces are surfaces that minimize the surface area for given boundary conditions; examples include soap films stretched across a wire frame, catenoids and helicoids.
• Ruled surfaces are surfaces that have at least one straight line running through every point; examples include the cylinder and the hyperboloid of one sheet.

Any n-dimensional complex manifold is, at the same time, a real (2n)-dimensional real manifold. Thus any complex one-manifold (also called a Riemann surface) is a smooth oriented surface with an associated complex structure. Every closed surface admits complex structures. Any complex algebraic curve or real algebraic surface is also a smooth surface, possibly with singularities.

Complex structures on a closed oriented surface correspond to conformal equivalence classes of Riemannian metrics on the surface. One version of the uniformization theorem (due to Poincaré) states that any Riemannian metric on an oriented, closed surface is conformally equivalent to an essentially unique metric of constant curvature. This provides a starting point for one of the approaches to Teichmüller theory, which provides a finer classification of Riemann surfaces than the topological one by Euler characteristic alone.

The uniformization theorem states that every smooth Riemannian surface is conformally equivalent to a surface having constant curvature, and the constant may be taken to be 1, 0, or -1. A surface of constant curvature 1 is locally isometric to the sphere, which means that every point on the surface has an open neighborhood that is isometric to an open set on the unit sphere in E3 with its intrinsic Riemannian metric. Likewise, a surface of constant curvature 0 is locally isometric to the Euclidean plane, and a surface of constant curvature -1 is locally isometric to the hyperbolic plane.

Constant curvature surfaces are the two-dimensional realization of what are known as space forms. These are often studied from the point of view of Felix Klein's Erlangen programme, by means of smooth transformation groups. Any connected surface with a three-dimensional group of isometries is a surface of constant curvature.

A complex surface is a complex two-manifold and thus a real four-manifold; it is not a surface in the sense of this article. Neither are algebraic curves or surfaces defined over fields other than the complex numbers.