Differential geometry of surfaces

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.
Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves. An important role in their study has been played by Lie groups, namely the symmetry groups of the Euclidean plane, the sphere and the hyperbolic plane. These Lie groups can be used to describe surfaces of constant Gaussian curvature; they also provide an essential ingredient in the modern approach to intrinsic differential geometry through connections. On the other hand, extrinsic properties relying on an embedding of a surface in Euclidean space have also been extensively studied. This is well illustrated by the non-linear Euler–Lagrange equations in the calculus of variations: although Euler developed the one variable equations to understand geodesics, defined independently of an embedding, one of Lagrange's main applications of the two variable equations was to minimal surfaces, a concept that can only be defined in terms of an embedding.

History

The volumes of certain quadric surfaces of revolution were calculated by Archimedes. The development of calculus in the seventeenth century provided a more systematic way of computing them. Curvature of general surfaces was first studied by Euler. In 1760 he proved a formula for the curvature of a plane section of a surface and in 1771 he considered surfaces represented in a parametric form. Monge laid down the foundations of their theory in his classical memoir L'application de l'analyse à la géometrie which appeared in 1795. The defining contribution to the theory of surfaces was made by Gauss in two remarkable papers written in 1825 and 1827. This marked a new departure from tradition because for the first time Gauss considered the intrinsic geometry of a surface, the properties which are determined only by the geodesic distances between points on the surface independently of the particular way in which the surface is located in the ambient Euclidean space. The crowning result, the Theorema Egregium of Gauss, established that the Gaussian curvature is an intrinsic invariant, i.e. invariant under local isometries. This point of view was extended to higher-dimensional spaces by Riemann and led to what is known today as Riemannian geometry. The nineteenth century was the golden age for the theory of surfaces, from both the topological and the differential-geometric point of view, with most leading geometers devoting themselves to their study. Darboux collected many results in his four-volume treatise Théorie des surfaces.

Overview

It is intuitively quite familiar to say that the leaf of a plant, the surface of a glass, or the shape of a face, are curved in certain ways, and that all of these shapes, even after ignoring any distinguishing markings, have certain geometric features which distinguish one from another. The differential geometry of surfaces is concerned with a mathematical understanding of such phenomena. The study of this field, which was initiated in its modern form in the 1700s, has led to the development of higher-dimensional and abstract geometry, such as Riemannian geometry and general relativity.
The essential mathematical object is that of a regular surface. Although conventions vary in their precise definition, these form a general class of subsets of three-dimensional Euclidean space which capture part of the familiar notion of "surface." By analyzing the class of curves which lie on such a surface, and the degree to which the surfaces force them to curve in, one can associate to each point of the surface two numbers, called the principal curvatures. Their average is called the mean curvature of the surface, and their product is called the Gaussian curvature.
There are many classic examples of regular surfaces, including:

familiar examples such as planes, cylinders, and spheres
minimal surfaces, which are defined by the property that their mean curvature is zero at every point. The best-known examples are catenoids and helicoids, although many more have been discovered. Minimal surfaces can also be defined by properties to do with surface area, with the consequence that they provide a mathematical model for the shape of soap films when stretched across a wire frame
ruled surfaces, which are surfaces that have at least one straight line running through every point; examples include the cylinder and the hyperboloid of one sheet.

A surprising result of Carl Friedrich Gauss, known as the Theorema Egregium, showed that the Gaussian curvature of a surface, which by its definition has to do with how curves on the surface change directions in three dimensional space, can actually be measured by the lengths of curves lying on the surfaces together with the angles made when two curves on the surface intersect. Terminologically, this says that the Gaussian curvature can be calculated from the first fundamental form of the surface. The second fundamental form, by contrast, is an object which encodes how lengths and angles of curves on the surface are distorted when the curves are pushed off of the surface.
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.
Using the first fundamental form, it is possible to define new objects on a regular surface. Geodesics are curves on the surface which satisfy a certain second-order ordinary differential equation which is specified by the first fundamental form. They are very directly connected to the study of lengths of curves; a geodesic of sufficiently short length will always be the curve of shortest length on the surface which connects its two endpoints. Thus, geodesics are fundamental to the optimization problem of determining the shortest path between two given points on a regular surface.
One can also define parallel transport along any given curve, which gives a prescription for how to deform a tangent vector to the surface at one point of the curve to tangent vectors at all other points of the curve. The prescription is determined by a first-order ordinary differential equation which is specified by the first fundamental form.
The above concepts are essentially all to do with multivariable calculus. The Gauss–Bonnet theorem is a more global result, which relates the Gaussian curvature of a surface together with its topological type. It asserts that the average value of the Gaussian curvature is completely determined by the Euler characteristic of the surface together with its surface area.
Any regular surface is an example both of a Riemannian manifold and Riemann surface. Essentially all of the theory of regular surfaces as discussed here has a generalization in the theory of Riemannian manifolds and their submanifolds.

Regular surfaces in Euclidean space

Definition

It is intuitively clear that a sphere is smooth, while a cone or a pyramid, due to their vertex or edges, are not. The notion of a "regular surface" is a formalization of the notion of a smooth surface. The definition utilizes the local representation of a surface via maps between Euclidean spaces. There is a standard notion of smoothness for such maps; a map between two open subsets of Euclidean space is smooth if its partial derivatives of every order exist at every point of the domain.
A regular surface in Euclidean space is a subset of such that every point of admits any of the following three concepts: local parametrizations, Monge patches, or implicit functions.
The following table gives definitions of such objects; Monge patches is perhaps the most visually intuitive, as it essentially says that a regular surface is a subset of which is locally the graph of a smooth function.

Objects	Definition
Local parametrizations	An open neighborhood for which there is an open subset of and a homeomorphism such that , as a function, is infinitely differentiable for each point of, the two partial derivatives and are linearly independent as elements of.
Monge patches	An open neighborhood for which there is an open subset of and a smooth function such that one of the following holds: .
Implicit functions	An open neighborhood for which there is a smooth function with: at each point of, at least one partial derivative of is nonzero.

The homeomorphisms appearing in the first definition are known as local parametrizations or local coordinate systems or local charts on. The equivalence of the first two definitions asserts that, around any point on a regular surface, there always exist local parametrizations of the form,, or, known as Monge patches. Functions as in the third definition are called local defining functions. The equivalence of all three definitions follows from the implicit function theorem.
Given any two local parametrizations and of a regular surface, the composition is necessarily smooth as a map between open subsets of. This shows that any regular surface naturally has the structure of a smooth manifold, with a smooth atlas being given by the inverses of local parametrizations.
In the classical theory of differential geometry, surfaces are usually studied only in the regular case. It is, however, also common to study non-regular surfaces, in which the two partial derivatives and of a local parametrization may fail to be linearly independent. In this case, may have singularities such as cuspidal edges. Such surfaces are typically studied in singularity theory. Other weakened forms of regular surfaces occur in computer-aided design, where a surface is broken apart into disjoint pieces, with the derivatives of local parametrizations failing to even be continuous along the boundaries.
Simple examples. A simple example of a regular surface is given by the 2-sphere ; this surface can be covered by six Monge patches, taking. It can also be covered by two local parametrizations, using stereographic projection. The set is a torus of revolution with radii and. It is a regular surface; local parametrizations can be given of the form
The hyperboloid on two sheets is a regular surface; it can be covered by two Monge patches, with. The helicoid appears in the theory of minimal surfaces. It is covered by a single local parametrization,.