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 (in the spirit of the Erlangen program), 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.

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 (1887-1896).

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 (ℝ) 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 ℝ, 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 (also called metric tensor) 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.

Condensed: the full section is in Wikipedia.

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 ℝ is a subset S of ℝ such that every point of S 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 ℝ which is locally the graph of a smooth function (whether over a region in the yz plane, the xz plane, or the xy plane).

The homeomorphisms appearing in the first definition are known as local parametrizations or local coordinate systems or local charts on S. The equivalence of the first two definitions asserts that, around any point on a regular surface, there always exist local parametrizations of the form (u, v) ↦ (h(u, v), u, v), (u, v) ↦ (u, h(u, v), v), or (u, v) ↦ (u, v, h(u, v)), known as Monge patches. Functions F 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 f : V → U and f ′ : V ′→ U ′ of a regular surface, the composition f ∘ f ′ is necessarily smooth as a map between open subsets of ℝ. 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 ∂_u f and ∂_v f of a local parametrization may fail to be linearly independent. In this case, S 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 {(x, y, z) | x + y + z = 1}; this surface can be covered by six Monge patches (two of each of the three types given above), taking h(u, v) = ± (1 − u − v). It can also be covered by two local parametrizations, using stereographic projection. The set {(x, y, z) : ((x + y) − r) + z = R} is a torus of revolution with radii r and R. It is a regular surface; local parametrizations can be given of the form

\(f(s,t)=\big((R \cos s +r)\cos t, (R \cos s +r) \sin t, R\sin s\big).\)

Condensed: the full section is in Wikipedia.

Let S be a regular surface in ℝ, and let p be an element of S. Using any of the above definitions, one can single out certain vectors in ℝ as being tangent to S at p, and certain vectors in ℝ as being orthogonal (or normal) to S at p.

One sees that the tangent space or tangent plane to S at p, which is defined to consist of all tangent vectors to S at p, is a two-dimensional linear subspace of ℝ; it is often denoted by T_pS.

The normal space to S at p, which is defined to consist of all normal vectors to S at p, is a one-dimensional linear subspace of ℝ which is orthogonal to the tangent space T_pS. As such, at each point p of S, there are two normal vectors of unit length (unit normal vectors). The unit normal vectors at p can be given in terms of local parametrizations, Monge patches, or local defining functions, via the formulas

\(\pm\left.\frac{\frac{\partial f}{\partial u}\times\frac{\partial f}{\partial v}}{\left\|\frac{\partial f}{\partial u}\times\frac{\partial f}{\partial v}\right\|}\right|_{f^{-1}(p)},\qquad \pm\left.\frac{\left(\frac{\partial h}{\partial u},\frac{\partial h}{\partial v},-1\right)}{\sqrt{1+\left(\frac{\partial h}{\partial u}\right)^2+\left(\frac{\partial h}{\partial v}\right)^2}}\right|_{(p_1,p_2)},\qquad \text{or}\qquad \pm\frac{\nabla F(p)}{\big\|\nabla F(p)\big\|},\)

following the same notations as in the previous definitions.

It is also useful to note an "intrinsic" definition of tangent vectors, which is typical of the generalization of regular surface theory to the setting of smooth manifolds. It defines the tangent space as an abstract two-dimensional real vector space, rather than as a linear subspace of ℝ. In this definition, one says that a tangent vector to S at p is an assignment, to each local parametrization f : V → S with p ∈ f(V), of two numbers X and X, such that for any other local parametrization f ′ : V → S with p ∈ f(V) (and with corresponding numbers (X ′) and (X ′)), one has

\(\begin{pmatrix}X^1\\ X^2\end{pmatrix}=A_{f'(p)}\begin{pmatrix}(X')^1\\ (X')^2\end{pmatrix},\)

where A_{f ′(p)} is the Jacobian matrix of the mapping f ∘ f ′, evaluated at the point f ′(p). The collection of tangent vectors to S at p naturally has the structure of a two-dimensional vector space. A tangent vector in this sense corresponds to a tangent vector in the previous sense by considering the vector

\(X^1\frac{\partial f}{\partial u}+X^2\frac{\partial f}{\partial v}.\)

in ℝ. The Jacobian condition on X and X ensures, by the chain rule, that this vector does not depend on f.

\([[X,Y],Z] + [[Y,Z],X] + [[Z,X],Y]=0.\)

Condensed: the full section is in Wikipedia.

Let S be a regular surface in ℝ. Given a local parametrization f : V → S and a unit normal vector field n to f(V), one defines the following objects as real-valued or matrix-valued functions on V. The first fundamental form depends only on f, and not on n. The fourth column records the way in which these functions depend on f, by relating the functions E ′, F ′, G ′, L ′, etc., arising for a different choice of local parametrization, f ′ : V ′ → S, to those arising for f. Here A denotes the Jacobian matrix of f ∘ f ′. The key relation in establishing the formulas of the fourth column is then

\(\begin{pmatrix}\frac{\partial f'}{\partial u}\\ \frac{\partial f'}{\partial v}\end{pmatrix}=A\begin{pmatrix}\frac{\partial f}{\partial u}\\ \frac{\partial f}{\partial v}\end{pmatrix},\)

as follows by the chain rule.

By a direct calculation with the matrix defining the shape operator, it can be checked that the Gaussian curvature is the determinant of the shape operator, the mean curvature is half of the trace of the shape operator, and the principal curvatures are the eigenvalues of the shape operator; moreover the Gaussian curvature is the product of the principal curvatures and the mean curvature is their sum. These observations can also be formulated as definitions of these objects. These observations also make clear that the last three rows of the fourth column follow immediately from the previous row, as similar matrices have identical determinant, trace, and eigenvalues. It is fundamental to note E, G, and EG − F are all necessarily positive. This ensures that the matrix inverse in the definition of the shape operator is well-defined, and that the principal curvatures are real numbers.

Note also that a negation of the choice of unit normal vector field will negate the second fundamental form, the shape operator, the mean curvature, and the principal curvatures, but will leave the Gaussian curvature unchanged. In summary, this has shown that, given a regular surface S, the Gaussian curvature of S can be regarded as a real-valued function on S; relative to a choice of unit normal vector field on all of S, the two principal curvatures and the mean curvature are also real-valued functions on S.

Geometrically, the first and second fundamental forms can be viewed as giving information on how f(u, v) moves around in ℝ as (u, v) moves around in V. In particular, the first fundamental form encodes how quickly f moves, while the second fundamental form encodes the extent to which its motion is in the direction of the normal vector n. In other words, the second fundamental form at a point p encodes the length of the orthogonal projection from S to the tangent plane to S at p; in particular it gives the quadratic function which best approximates this length. This thinking can be made precise by the formulas

\(\begin{aligned} \lim_{(h,k)\to(0,0)}\frac{\big|f(u+h,v+k) - f(u,v)\big|^2-\big(Eh^2+2Fhk+Gk^2\big)}{h^2+k^2} &= 0\\ \lim_{(h,k)\to(0,0)}\frac{\big(f(u+h,v+k) - f(u,v)\big)\cdot n-\frac{1}{2}\big(Lh^2 +2M hk + Nk^2\big)}{h^2+k^2} &= 0, \end{aligned}\)

Condensed: the full section is in Wikipedia.

Let S be a regular surface in ℝ. The Christoffel symbols assign, to each local parametrization f : V → S, eight functions on V, defined by

\(\begin{pmatrix}\Gamma_{11}^1&\Gamma_{12}^1&\Gamma_{21}^1&\Gamma_{22}^1\\ \Gamma_{11}^2&\Gamma_{12}^2&\Gamma_{21}^2&\Gamma_{22}^2\end{pmatrix}=\begin{pmatrix}E&F\\ F&G\end{pmatrix}^{-1}\begin{pmatrix}\frac{1}{2}\frac{\partial E}{\partial u}&\frac{1}{2}\frac{\partial E}{\partial v}&\frac{1}{2}\frac{\partial E}{\partial v} &\frac{\partial F}{\partial v}-\frac{1}{2}\frac{\partial G}{\partial u}\\ \frac{\partial F}{\partial u}-\frac{1}{2}\frac{\partial E}{\partial v}&\frac{1}{2}\frac{\partial G}{\partial u}&\frac{1}{2}\frac{\partial G}{\partial u}&\frac{1}{2}\frac{\partial G}{\partial v}\end{pmatrix}.\)

They can also be defined by the following formulas, in which n is a unit normal vector field along f(V) and L, M, N are the corresponding components of the second fundamental form:

\(\begin{aligned} \frac{\partial^2f}{\partial u^2}&=\Gamma_{11}^1\frac{\partial f}{\partial u}+\Gamma_{11}^2\frac{\partial f}{\partial v}+Ln\\ \frac{\partial^2f}{\partial u\partial v}&=\Gamma_{12}^1\frac{\partial f}{\partial u}+\Gamma_{12}^2\frac{\partial f}{\partial v}+Mn\\ \frac{\partial^2f}{\partial v^2}&=\Gamma_{22}^1\frac{\partial f}{\partial u}+\Gamma_{22}^2\frac{\partial f}{\partial v}+Nn. \end{aligned}\)

The key to this definition is that ⁠∂f/∂u⁠, ⁠∂f/∂v⁠, and n form a basis of ℝ at each point, relative to which each of the three equations uniquely specifies the Christoffel symbols as coordinates of the second partial derivatives of f. The choice of unit normal has no effect on the Christoffel symbols, since if n is exchanged for its negation, then the components of the second fundamental form are also negated, and so the signs of Ln, Mn, Nn are left unchanged.

The second definition shows, in the context of local parametrizations, that the Christoffel symbols are geometrically natural. Although the formulas in the first definition appear less natural, they have the importance of showing that the Christoffel symbols can be calculated from the first fundamental form, which is not immediately apparent from the second definition. The equivalence of the definitions can be checked by directly substituting the first definition into the second, and using the definitions of E, F, G.

The Codazzi equations assert that

\(\begin{aligned} \frac{\partial L}{\partial v}-\frac{\partial M}{\partial u}&=L\Gamma_{12}^1 + M(\Gamma_{12}^2-\Gamma_{11}^1) - N\Gamma_{11}^2\\ \frac{\partial M}{\partial v}-\frac{\partial N}{\partial u}&=L\Gamma_{22}^1 + M(\Gamma_{22}^2-\Gamma_{12}^1) - N\Gamma_{12}^2. \end{aligned}\)

These equations can be directly derived from the second definition of Christoffel symbols given above; for instance, the first Codazzi equation is obtained by differentiating the first equation with respect to v, the second equation with respect to u, subtracting the two, and taking the dot product with n. The Gauss equation asserts that

Condensed: the full section is in Wikipedia.

A diffeomorphism \(\varphi\) between open sets \(U\) and \(V\) in a regular surface \(S\) is said to be an isometry if it preserves the metric, i.e. the first fundamental form. Thus for every point \(p\) in \(U\) and tangent vectors \(w_1,\,\, w_2\) at \(p\), there are equalities

\(E(p) w_1\cdot w_1 + 2F(p) w_1\cdot w_2 + G(p) w_2\cdot w_2= E(\varphi(p)) \varphi^\prime(w_1)\cdot \varphi^\prime(w_1) +2F(\varphi(p)) \varphi^\prime(w_1)\cdot \varphi^\prime(w_2) + G (\varphi(p)) \varphi^\prime(w_1)\cdot \varphi^\prime(w_2).\)

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

\((w_1,w_2)_p=(\varphi^\prime(w_1),\varphi^\prime(w_2))_{\varphi(p)}\).

On the other hand, the length of a parametrized curve \(\gamma(t)=(x(t),y(t))\) can be calculated as

\(L(\gamma)=\int_a^b \sqrt{E\dot{x}\cdot \dot{x} +2F \dot{x}\cdot \dot{y} +G\dot{y}\cdot \dot{y} } \, dt\)

and, if the curve lies in \(U\), the rules for change of variables show that

\(L(\varphi\circ \gamma) = L(\gamma).\)

Conversely if \(\varphi\) preserves the lengths of all parametrized in curves then \(\varphi\) is an isometry. Indeed, for suitable choices of \(\gamma\), the tangent vectors \(\dot{x}\) and \(\dot{y}\) give arbitrary tangent vectors \(w_1\) and \(w_2\). The equalities must hold for all choice of tangent vectors \(w_1\) and \(w_2\) as well as \(\varphi^\prime(w_1)\) and \(\varphi^\prime(w_2)\), so that \((\varphi^\prime(w_1),\varphi^\prime(w_2))_{\varphi(p)} = (w_1,w_1)_p\).

A simple example of an isometry is provided by two parametrizations \(f_1\) and \(f_2\) of an open set \(U\) into regular surfaces \(S_1\) and \(S_2\). If \(E_1=E_2\), \(F_1=F_2\) and \(G_1=G_2\), then \(\varphi=f_2\circ f_1^{-1}\) is an isometry of \(f_1(U)\) onto \(f_2(U)\).

The cylinder and the plane give examples of surfaces that are locally isometric but which cannot be extended to an isometry for topological reasons. As another example, the catenoid and helicoid are locally isometric.

A tangential vector field X on S assigns, to each p in S, a tangent vector X_p to S at p. According to the "intrinsic" definition of tangent vectors given above, a tangential vector field X then assigns, to each local parametrization f : V → S, two real-valued functions X and X on V, so that

\(X_p=X^1\big(f^{-1}(p)\big)\frac{\partial f}{\partial u}\Big|_{f^{-1}(p)}+X^2\big(f^{-1}(p)\big)\frac{\partial f}{\partial v}\Big|_{f^{-1}(p)}\)

for each p in S. One says that X is smooth if the functions X and X are smooth, for any choice of f. According to the other definitions of tangent vectors given above, one may also regard a tangential vector field X on S as a map X : S → ℝ such that X(p) is contained in the tangent space T_pS ⊂ ℝ for each p in S. As is common in the more general situation of smooth manifolds, tangential vector fields can also be defined as certain differential operators on the space of smooth functions on S.

The covariant derivatives (also called "tangential derivatives") of Tullio Levi-Civita and Gregorio Ricci-Curbastro provide a means of differentiating smooth tangential vector fields. Given a tangential vector field X and a tangent vector Y to S at p, the covariant derivative ∇_YX is a certain tangent vector to S at p. Consequently, if X and Y are both tangential vector fields, then ∇_YX can also be regarded as a tangential vector field; iteratively, if X, Y, and Z are tangential vector fields, the one may compute ∇_Z∇_YX, which will be another tangential vector field. There are a few ways to define the covariant derivative; the first below uses the Christoffel symbols and the "intrinsic" definition of tangent vectors, and the second is more manifestly geometric.

Given a tangential vector field X and a tangent vector Y to S at p, one defines ∇_YX to be the tangent vector to p which assigns to a local parametrization f : V → S the two numbers

\((\nabla_YX)^k=D_{(Y^1,Y^2)}X^k\Big|_{f^{-1}(p)}+\sum_{i=1}^2\sum_{j=1}^2\big(\Gamma_{ij}^kX^j\big)\Big|_{f^{-1}(p)}Y^i,\qquad(k=1,2)\)

where D_{(Y, Y)} is the directional derivative. This is often abbreviated in the less cumbersome form (∇_YX) = ∂_Y(X) + YΓ
_ijX, making use of Einstein notation and with the locations of function evaluation being implicitly understood. This follows a standard prescription in Riemannian geometry for obtaining a connection from a Riemannian metric. It is a fundamental fact that the vector

\((\nabla_YX)^1\frac{\partial f}{\partial u}+(\nabla_YX)^2\frac{\partial f}{\partial v}\)

in ℝ is independent of the choice of local parametization f, although this is rather tedious to check.

\(\frac{\partial\Gamma_{11}^2}{\partial v}-\frac{\partial \Gamma_{21}^2}{\partial u}+\Gamma_{21}^2\Gamma_{11}^1+\Gamma_{22}^2\Gamma_{11}^2-\Gamma_{11}^2\Gamma_{21}^1-\Gamma_{12}^2\Gamma_{21}^2\)

\(\nabla_{\frac{\partial f}{\partial v}}\nabla_{\frac{\partial f}{\partial u}}\frac{\partial f}{\partial u}-\nabla_{\frac{\partial f}{\partial u}}\nabla_{\frac{\partial f}{\partial v}}\frac{\partial f}{\partial u}\)

Condensed: the full section is in Wikipedia.

A surface of revolution is obtained by rotating a curve in the xz-plane about the z-axis. Such surfaces include spheres, cylinders, cones, tori, and the catenoid. The general ellipsoids, hyperboloids, and paraboloids are not. Suppose that the curve is parametrized by

\(x= c_1(s),\,\, z=c_2(s)\)

with s drawn from an interval (a, b). If c₁ is never zero, if c₁′ and c₂′ are never both equal to zero, and if c₁ and c₂ are both smooth, then the corresponding surface of revolution

\(S=\Big\{\big(c_1(s)\cos t, c_1(s)\sin t,c_2(s)\big)\colon s\in (a,b)\text{ and }t\in\mathbb{R}\Big\}\)

will be a regular surface in ℝ. A local parametrization f : (a, b) × (0, 2π) → S is given by

\(f(s,t)=\big(c_1(s)\cos t, c_1(s)\sin t,c_2(s)\big).\)

Relative to this parametrization, the geometric data is:

In the special case that the original curve is parametrized by arclength, i.e. (c₁′(s)) + (c₂′(s)) = 1, one can differentiate to find c₁′(s)c₁′′(s) + c₂′(s)c₂′′(s) = 0. On substitution into the Gaussian curvature, one has the simplified

\(K=-\frac{c_1''(s)}{c_1(s)}\qquad\text{and}\qquad H=c_1'(s)c_2''(s)-c_2'(s)c_1''(s)+\frac{c_2'(s)}{c_1(s)}.\)

The simplicity of this formula makes it particularly easy to study the class of rotationally symmetric surfaces with constant Gaussian curvature. By reduction to the alternative case that c₂(s) = s, one can study the rotationally symmetric minimal surfaces, with the result that any such surface is part of a plane or a scaled catenoid.

Each constant-t curve on S can be parametrized as a geodesic; a constant-s curve on S can be parametrized as a geodesic if and only if c₁′(s) is equal to zero. Generally, geodesics on S are governed by Clairaut's relation.

Consider the quadric surface defined by

\({x^2\over a} + {y^2\over b} +{z^2\over c}=1.\)

This surface admits a parametrization

\(x=\sqrt{a(a-u)(a-v)\over (a-b)(a-c)},\,\, y=\sqrt{b(b-u)(b-v)\over (b-a) (b-c)}, \,\, z=\sqrt{c(c-u)(c-v)\over (c-b)(c-a)}.\)

The Gaussian curvature and mean curvature are given by

\(K={abc\over u^2 v^2} ,\,\,K_m=-(u+v)\sqrt{abc\over u^3v^3}.\)

A ruled surface is one which can be generated by the motion of a straight line in E. Choosing a directrix on the surface, i.e. a smooth unit speed curve c(t) orthogonal to the straight lines, and then choosing u(t) to be unit vectors along the curve in the direction of the lines, the velocity vector v = c_t and u satisfy

\(u\cdot v=0, \,\,\|u\|=1,\,\,\|v\|=1.\)

The surface consists of points

\(c(t) + s\cdot u(t)\)

as s and t vary.

Then, if

\(a=\|u_t\|, \,\, b=u_t\cdot v, \,\, \alpha=-\frac{b}{a^2}, \,\, \beta=\frac{\sqrt{a^2-b^2}}{a^2},\)

the Gaussian and mean curvature are given by

\(K=-{\beta^2\over ((s-\alpha)^2 +\beta^2)^2} ,\,\, K_m=-{r[(s-\alpha)^2 +\beta^2)] +\beta_t(s-\alpha) + \beta\alpha_t\over [(s-\alpha)^2 +\beta^2]^{\frac32}}.\)

The Gaussian curvature of the ruled surface vanishes if and only if u_t and v are proportional, This condition is equivalent to the surface being the envelope of the planes along the curve containing the tangent vector v and the orthogonal vector u, i.e. to the surface being developable along the curve. More generally a surface in E has vanishing Gaussian curvature near a point if and only if it is developable near that point. (An equivalent condition is given below in terms of the metric.)

In 1760 Lagrange extended Euler's results on the calculus of variations involving integrals in one variable to two variables. He had in mind the following problem:

Such a surface is called a minimal surface.

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:

Minimal surfaces have a simple interpretation in real life: they are the shape a soap film will assume if a wire frame shaped like the curve is dipped into a soap solution and then carefully lifted out. The question as to whether a minimal surface with given boundary exists is called Plateau's problem after the Belgian physicist Joseph Plateau who carried out experiments on soap films in the mid-nineteenth century. In 1930 Jesse Douglas and Tibor Radó gave an affirmative answer to Plateau's problem (Douglas was awarded a Fields Medal for this work in 1936).

Many explicit examples of minimal surface are known explicitly, such as the catenoid, the helicoid, the Scherk surface and the Enneper surface. There has been extensive research in this area, summarised in Osserman (2002). In particular a result of Osserman shows that if a minimal surface is non-planar, then its image under the Gauss map is dense in S.

If a surface has constant Gaussian curvature, it is called a surface of constant curvature.

• The unit sphere in E has constant Gaussian curvature +1.
• The Euclidean plane and the cylinder both have constant Gaussian curvature 0.
• A unit pseudosphere has constant Gaussian curvature -1 (apart from its equator, that is singular). Pseudosphere can be obtained by rotating a tractrix around its asymptote. In 1868 Eugenio Beltrami showed that the geometry of the pseudosphere was directly related to that of the more abstract hyperbolic plane, discovered independently by Lobachevsky (1830) and Bolyai (1832). Already in 1840, F. Minding, a student of Gauss, had obtained trigonometric formulas for the pseudosphere identical to those for the hyperbolic plane. The intrinsic geometry of this surface is now better understood in terms of the Poincaré metric on the upper half plane or the unit disc, and has been described by other models such as the Klein model or the hyperboloid model, obtained by considering the two-sheeted hyperboloid q(x, y, z) = −1 in three-dimensional Minkowski space, where q(x, y, z) = x + y, z.

The sphere, the plane and the hyperbolic plane have transitive Lie group of symmetries. This group theoretic fact has far-reaching consequences, all the more remarkable because of the central role these special surfaces play in the geometry of surfaces, due to Poincaré's uniformization theorem (see below).

Other examples of surfaces with Gaussian curvature 0 include cones, tangent developables, and more generally any developable surface.