Differential geometry of surfaces

The differential geometry of surfaces is a considerably richer subject than the differential geometry of curves, which is in large part exhausted by the Frenet-Serret formulas. The main distinction occurs in the properties of the curvature, first studied in depth by the German mathematician Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its embedding in Euclidean space.

Surfaces in geometry

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. 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 compact 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. Here are a few important classes of such surfaces.

• 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.
• Developable surfaces are surfaces that can be flattened to a plane without stretching; examples include the cylinder and the cone.
• 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.

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.

Brief history

Isolated properties of surfaces of revolution were known already to Archimedes. The development of calculus in the seventeenth century provided a more systematic way of proving 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 layed 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 the remarkable paper Disquisitiones circa superficies curvas (1827). The break with the tradition occurred in that Gauss considered intrinsic geometry of a surface, the properties which are determined only by the geodesic distances between points on the surface and are independent of a particular way in which the surface is located in the ambient Euclidean space. The signature result, 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 the Riemannian geometry. The nineteenth century was the golden age for the theory of surfaces, both from the topological and from 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 presentation below largely follows Gauss, but with important later contributions from other geometers. For a time Gauss was Cartographer to George III of Great Britain and Hannover; this royal patronage could explain why these papers contain practical calculations of the curvature of the earth based purely on measurements on the surface of the planet.

Gaussian curvature of surfaces in E³

Main article: Gaussian curvature

The Gaussian curvature at a point on an embedded smooth surface given locally by the equation z = F(x,y) in E³, is defined to be the product of the principal curvatures at the point. These are the maximum and minimum curvatures of the plane curves obtained by intersecting the surface with planes normal to the tangent plane at the point. If the point is (0, 0, 0) with tangent plane z = 0, then, after a rotation about the z-axis, F will have the Taylor series expansion

F(x, y) = k₁ x² + k₂ y² + ...

The principal curvatures are k₁ and k₂ and the Gaussian curvature K = k₁k₂. Since K is invariant under isometries of E³, in general

K = (RT − S²)/(1 + P² + Q²)²,

where the derivatives at the point are given by P = F_x, Q = F_y, R = F_{x x}, S = F_{x y}, and T = F_{y y}.

For every oriented embedded surface the Gauss map is the map into the unit sphere sending each point to the (outward pointing) unit vector normal to the oriented tangent plane at the point. In coordinates the map sends (x,y,z) to

N(x, y, z) = (P² + Q² + 1)^−1/2·(P, Q, −1).

Direct computation shows that K is the Jacobian of the Gauss map.

Local metric structure

Taking a local chart, for example by projecting onto the x-y plane (z = 0), the line and area elements can be written in terms of local coordinates as

ds² = E dx² + 2F dx dy + G dy²

and

dA = (EG − F²)^1/2 dx dy.

Similarly line and area elements are associated to an abstract Riemannian 2-manifold. Given a piecewise smooth path c(t) = (x(t), y(t)) in the chart for t in [a, b], its length is defined by

$L(c)=\int _{a}^{b}(E{\dot {x}}^{2}+2F{\dot {x}}{\dot {y}}+G{\dot {y}}^{2})^{1/2}\,dt$

and energy by

$E(c)=\int _{a}^{b}(E{\dot {x}}^{2}+2F{\dot {x}}{\dot {y}}+G{\dot {y}}^{2})\,dt$

The length is independent of the parametrisation of a path. By the Euler-Lagrange equations, if c(t) is a path minimising length, parametrised by arclength, it must satisfy the Euler equations

${\ddot {x}}$ + Γ¹₁₁${\dot {x}}$² + 2Γ¹₁₂ ${\dot {x}}{\dot {y}}$ + Γ¹₂₂ ${\dot {y}}$² =0 and ${\ddot {y}}$ + Γ²₁₁${\dot {x}}$² + 2Γ²₁₂ ${\dot {x}}{\dot {y}}$ + Γ²₂₂ ${\dot {y}}$² =0

where the Christoffel symbols Γ^k_ij are given by

Γ^k_ij =${1 \over 2}\sum _{m}$ g ^km ($\partial$_j g_im +$\partial$_i g_jm – $\partial$_m g_ij)

where g₁₁ = E, g₁₂=F, g₂₂ =G and (g^ij) is the inverse matrix to (g_ij). A path satisfying the Euler equations is called a geodesic. By the Cauchy-Schwarz inequality a path minimising energy is just a geodesic parametrised by arc length; and, for any geodesic, the parameter t is proportional to arclength.

It is unknown at present, except in some special cases, whether every metric structure arises from a local embedding in E³. In 1926 Maurice Janet proved that this is always possible locally if E, F and G are analytic; soon afterwards Elie Cartan generalised this to local embeddings of Riemannian n-manifolds in E^m where m = ½(n² +n). To prove Janet's theorem near (0,0), the Cauchy-Kowalevski theorem is used twice to produce analytic geodesics orthogonal to the y-axis and then the x-axis to make an analytic change of coordinate so that E=1 and F=0. An isometric embedding u must satisfy

u_x • u_x =1, u_x • u_y = 0, u_y • u_y = G.

Differentiating gives the three additional equations

u_xx • u_y = 0, u_xx • u_x = 0, u_xx • u_yy = u_xy • u_{x y} - ½ G_xx

with u(0,y) and u_x(0,y) prescribed. These equations can be solved near (0,0) using the Cauchy-Kowalevski theorem and yield a solution of the original embedding equations.

When F=0 in the metric, lines parallel to the x- and y-axes are orthogonal and provide orthogonal coordinates. If in addition E=1 and H=G^½, then the angle $\varphi$ between the geodesic and the line y= constant at their intersection is given by the equation

$\tan \varphi =H\cdot {\dot {y}}/{\dot {x}}$

and satisfies the following equation of Gauss:

${\dot {\varphi }}=-H_{x}\cdot {\dot {y}}.$

Geodesic polar coordinates

The theory of ordinary differential equations shows that if f(t, v) is smooth then the differential equation dv/dt = f(t,v) with initial condition v(0) = v₀ has a unique solution for |t| sufficiently small and the solution depends smoothly on t and v₀. This implies that for sufficiently small tangent vectors v at a given point p = (x₀,y₀), there is a geodeic c_v(t) defined on (-2,2) with c_v(0) = (x₀,y₀) and ${\dot {c}}$_v(0) = v. Moreover if |s| ≤ 1, then c_sv = c_v(st). The exponential map is defined by

exp_p(v) = c_v (1)

and gives a diffeomorphism between a disc ||v|| < δ and a neighbourhood of p; more generally the map sending (p,v) to exp_p(v) gives a local diffeomorphism onto a neighbourhood of (p,p). The exponential map gives geodesic normal coordinates near p. In these coordinates the matrix g(x) satisfies g(0) = I and the lines t $\mapsto$ tv are geodesics through 0. Euler's equations imply the matrix equation

g(v)v = v,

a key result, usually called the Gauss lemma. Geometrically it states that the geodesics through 0 cut the circles centred at 0 orthogonally. Taking polar coordinates (r,θ), it follows that the metric has the form

ds² = dr² + G(r,θ) dθ².

In geodesic coordinates, it is easy to check that the geodesics through zero minimize length. The topology on the Riemannian manifold is then given by a distance function d(p,q), namely the infimum of the lengths of piecewise smooth paths between p and q. This distance is realised locally by geodesics, so that in normal coordinates d(0,v) = ||v||. If the radius δ is taken small enough, a slight sharpening of the Gauss lemma shows that the image U of the disc ||v|| < δ under the exponential map is geodesically convex, i.e. any two points in U are joined by a unique geodesic lying entirely inside U.

There is a standard technique (see for example Berger (2004)) for computing the change of variables to normal coordinates u, v at a point as a formal Taylor series expansion. If the coordinates x, y at (0,0) are locally orthogonal, write

x(u,v) = α u + L(u,v) + λ(u,v) + ···

y(u,v) = β v + M(u,v) + μ(u,v) + ···

where L, M are quadratic and λ, μ cubic homogeneous polynomials in u and v. If u and v are fixed, x(t) = x(tu,tv) and y(t) = y(tu, tv) can be considered as formal power series solutions of the Euler equations: this uniquely determines α, β, L, M, λ and μ.

Taking x and y coordinates of a surface in E³ corresponding to F(x,y) = k₁ x² + k₂ y² + ···, the power series expansion of the metric is given in normal coordinates as

ds² = du² + dv² – K(u dv – v du)²/12 + ···

This extraordinary result — Gauss' Theorema Egregium — shows that the Gaussian curvature of a surface can be computed solely in terms of the metric and is thus an intrinsic invariant of the surface, independent of any embedding in E³ and unchanged under coordinate transformations. In particular isometries of surfaces preserve Gaussian curvature.

Taking a coordinate change from normal coordinates at p to normal coordinates at a nearby point q, yields the Sturm-Liouville equation satisfied by H(r,θ) = G(r,θ)^½, discovered by Gauss and later generalised by Jacobi,

H_rr = – K H.

The Jacobian of this coordinate change at q is equal to H_r.

Gauss–Bonnet theorem

Main article: Gauss–Bonnet theorem

Gauss proved that, if Δ is a geodesic triangle on a surface with angles α, β and γ at vertices A, B and C, then

$\int$_Δ K dA = α + β + γ - π.

In fact taking geodesic polar coordinates with origin A and AB, AC the radii at polar angles 0 and α

$\int$_Δ K dA =$\int$_Δ KH dr dθ = – $\int _{0}^{\alpha }\int _{0}^{r_{\theta }}$ H_rr dr dθ = $\int _{0}^{\alpha }$ 1- H_r(r_θ,θ) dθ = $\int _{0}^{\alpha }$ dθ + $\int _{\pi -\beta }^{\gamma }$ dφ = α + β + γ - π.

Gauss' formula shows that the curvature at a point can be calculated as the limit of angle excess α + β + γ - π over area for successively smaller geodesic triangles near the point. Qualitatively a surface is positively or negatively curved according to the sign of the angle excess for arbitrarily small geodesic triangles.

Since every compact oriented 2-manifold M can be triangulated by small geodesic triangles, it follows that

$\int$_M K dA = 2π·χ(M).

In fact if there are t triangles, e edges and v vertices, then 3t = 2e and the left hand side equals 2π·v – π·t = 2π·(v – e + t) = 2π·χ(M).

This is the celebrated Gauss-Bonnet theorem: it shows that the integral of the Gaussian curvature is a topological invariant of the manifold, namely the Euler characteristic. This theorem can be interpreted in many ways; perhaps one of the most far-reaching has been as the index theorem for an elliptic differential operator on M, one of the simplest cases of the Atiyah-Singer index theorem. Another related result, which can be proved using the Gauss-Bonnet theorem, is the Poincaré-Hopf index theorem for vector fields on M which vanish at only a finite number of points: the sum of the indices at these points equals the Euler characteristic. (On a small circle round each isolated zero, the vector field defines a map into the unit circle; the index is just the winding number of this map.)

If the Gaussian curvature of a surface M is everywhere positive, then the Euler characteristic is positive so M is homeomorphic (and therefore diffeomorphic) to S². If in addition the surface is isometrically embedded in E³, the Gauss map provides an explicit diffeomorphism. As Hadamard observed, in this case the surface is convex; this criterion for convexity can be viewed as a 2-dimensional generalisation of the well-known second derivative criterion for convexity of plane curves. Hilbert proved that every isometrically embedded closed surface must have a point of positive curvature. Thus a closed Riemannian 2-manifold of non-positive curvature can never be embedded isometrically in E³; however, as Adriano Garsia showed using the Beltrami equation for quasiconformal mappings, this is always possible for some conformally equivalent metric.

Surfaces of non-positive curvature

In a region where the curvature of the surface satisfies K≤0, geodesic triangles satisfy the CAT(0) inequalities of comparison geometry, studied by Cartan, Alexandrov and Toponogov and considered later from a different point of view by Bruhat and Tits; thanks to the vision of Gromov, this characterisation of non-positive curvature in terms of the underlying metric space has had a profound impact on modern geometry and in particular geometric group theory.

The simplest form of the comparison inequality, first proved for surfaces by Alexandrov around 1940, states that the distance between a vertex of a geodesic triangle and the midpoint of the opposite side is always less than the corresponding distance in the triangle in the plane with the same side-lengths. The inequality follows from the fact that if c(t) describes a geodesic parametrised by arclength and a is a fixed point, then

f(t) = d(a,c(t))² - t²

is a convex function, i.e.

${\ddot {f}}(t)\geq 0.$

Taking geodesic polar coordinates with origin at a so that ||c(t)|| = r(t), convexity is equivalent to

$r{\ddot {r}}+{\dot {r}}^{2}\geq 1.$

Changing to normal coordinates u, v at c(t), this inequality becomes

u² +H ^{- 1} H_r v² ≥ 1,

where (u,v) corresponds to the unit vector ${\dot {c}}(t)$. This follows from the inequality H_r ≥ H, a consequence of the non-negativity of the derivative of the Wronskian of H and r from Sturm-Liouville theory.

For closed surfaces of non-positive curvature, Hans von Mangoldt (1881) and Jacques Hadamard (1898) proved that the exponential map at a point is a covering map, so that the universal covering space of the manifold is R². This result was generalised to higher dimensions by Elie Cartan and is usually referred to in this form as the Cartan–Hadamard Theorem. For surfaces, this result follows from three important facts:

• The exponential map has non-zero Jacobian everywhere for non-positively curved surfaces, a consequence of the non-vanishing of H_r.

• Every geodesic is infinitely extendible, a result known as the Hopf-Rinow theorem for n-dimensional manifolds. In two dimensions, if a geodesic tended at infinity towards a point x, a closed disc D centred on a nearby point y with x removed would be contractible to y along geodesics, a topological impossibility.

• Every two points are connected by a unique geodesic. This can be deduced from the curve shortening process of George Birkhoff, published in 1917, that eventually won him the prestigious Bôcher memorial prize and was to have a profound influence on Marston Morse's development of Morse theory in infinite dimensions and also on the theory of dynamical systems.

Birkhoff's curve shortening process replaces a loop or a path, with a given subdivision into segments, by geodesics between the points of subdivision, and then repeats this process for the midpoints of the subdivision. The process diminishes the energy. Since the resulting curves are equicontinuous and therefore by the generalised Arzela-Ascoli theorem form a compact subset in the space of continuous loops or paths, iteration of the process produces a geodesic in each homotopy class of loop or path. If the surface is negatively curved, this geodesic is unique. In fact on a negatively curved suface, the distance between corresponding points on two geodesics is a convex function of arclength; thus, if two geodesics have the same endpoints, they must coincide everywhere. The method of Birkhoff allows the study of the energy flow on the infinite-dimensional loop space to be reduced to a discrete dynamical system on a finite-dimensional space.

References

• Berger, Marcel (2004). A Panoramic View of Riemannian Geometry. Springer-Verlag. ISBN 3-540-65317-1.
• Singer, Isadore M. and Thorpe, John A. (1967). Lecture Notes on Elementary Topology and Geometry. Springer-Verlag. ISBN 0-387-90202-3.{{cite book}}: CS1 maint: multiple names: authors list (link)
• O'Neill, Barrett (1997). Elementary Differential Geometry. Academic Press. ISBN 0-12-526745-2.
• do Carmo, Manfredo P. (1976). Differential Geometry of Curves and Surfaces. Prentice-Hall. ISBN 0132125897.
• Eisenhart, Luther P. (2004). A Treatise on the Differential Geometry of Curves and Surfaces. Dover. ISBN 0486438201.
• Han, Qing and Hong, Jia-Xing (2006). Isometric Embedding of Riemannian Manifolds in Euclidean Spaces. American Mathematical Society. ISBN 0821840711.{{cite book}}: CS1 maint: multiple names: authors list (link)