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Carl Friedrich Gauss in 1828

Differential geometry of surfacesFrom Wikipedia, the free encyclopedia

In mathematics, the differential geometry of surfaces deals withsmooth surfaces with various additional structures, most often, aRiemannian metric. Surfaces have been extensively studied from variousperspectives: extrinsically, relating to their embedding in Euclideanspace and intrinsically, reflecting their properties determined solely bythe 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 (1825-1827), who showedthat curvature was an intrinsic property of a surface, independent of itsisometric embedding in Euclidean space.

Surfaces naturally arise as graphs of functions of a pair of variables, andsometimes appear in parametric form or as loci associated to spacecurves. An important role in their study has been played by Lie groups (inthe spirit of the Erlangen program), namely the symmetry groups of theEuclidean plane, the sphere and the hyperbolic plane. These Lie groupscan be used to describe surfaces of constant Gaussian curvature; theyalso 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 beenextensively 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 anembedding, one of Lagrange's main applications of the two variable equations was to minimal surfaces, a conceptthat can only be defined in terms of an embedding.

Contents

1 Overview

2 History of surfaces

3 Curvature of surfaces in E3

4 Examples

4.1 Surfaces of revolution

4.2 Quadric surfaces

4.3 Ruled surfaces

4.4 Minimal surfaces

4.5 Surfaces of constant Gaussian curvature5 Local metric structure

5.1 Line and area elements

5.2 Second fundamental form

5.3 Shape operator

6 Geodesic curves on a surface

6.1 Geodesics

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6.2 Geodesic curvature

6.3 Isometric embedding problem

6.4 Orthogonal coordinates

7 Geodesic polar coordinates

7.1 Exponential map

7.2 Computation of normal coordinates7.3 Gauss's lemma

7.4 Theorema Egregium

7.5 Gauss–Jacobi equation

7.6 Laplace–Beltrami operator

8 Gauss–Bonnet theorem

8.1 Geodesic triangles

8.2 Gauss-Bonnet theorem

8.3 Curvature and embeddings

9 Surfaces of constant curvature

9.1 Euclidean geometry

9.2 Spherical geometry9.3 Hyperbolic geometry

9.4 Uniformization10 Surfaces of non-positive curvature

10.1 Alexandrov's comparison inequality10.2 Existence of geodesics10.3 Von Mangoldt-Hadamard theorem

11 Riemannian connection and parallel transport11.1 Covariant derivative

11.2 Parallel transport11.3 Connection 1-form

12 Global differential geometry of surfaces13 Reading guide

14 See also15 Notes

16 References

Overview

See also: Surfaces

Polyhedra in the Euclidean space, such as the boundary of a cube, are among the first surfaces encountered ingeometry. 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 manyresults.

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

Smooth surfaces equipped with Riemannian metrics are of foundational importance in differential geometry. ARiemannian metric endows a surface with notions of geodesic, distance, angle, and area. An important class of suchsurfaces are the developable surfaces: surfaces that can be flattened to a plane An without stretching; examplesinclude 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 complexone-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 asmooth surface, possibly with singularities.

Complex structures on a closed oriented surface correspond to conformal equivalence classes of Riemannianmetrics on the surface. One version of the uniformization theorem (due to Poincaré) states that any Riemannianmetric 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 classificationof Riemann surfaces than the topological one by Euler characteristic alone.

The uniformization theorem states that every smooth Riemannian surface is conformally equivalent to a surfacehaving constant curvature, and the constant may be taken to be 1, 0, or -1. A surface of constant curvature 1 islocally 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 constantcurvature 0 is locally isometric to the Euclidean plane, and a surface of constant curvature -1 is locally isometric tothe hyperbolic plane.

Constant curvature surfaces are the two-dimensional realization of what are known as space forms. These are oftenstudied 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 thisarticle. Neither are algebraic curves or surfaces defined over fields other than the complex numbers.

History of surfaces

Isolated properties of surfaces of revolution were known already to Archimedes. The development of calculus inthe seventeenth century provided a more systematic way of proving them. Curvature of general surfaces was first

studied by Euler. In 1760[1] he proved a formula for the curvature of a plane section of a surface and in 1771[2] heconsidered surfaces represented in a parametric form. Monge laid down the foundations of their theory in his

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The principal curvatures at a point on

a surface

The Gauss map sends a point on the

surface to the outward pointing unit

normal vector, a point on S2

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.[3] This marked anew departure from tradition because for the first time Gauss considered the intrinsic geometry of a surface, theproperties which are determined only by the geodesic distances between points on the surface independently of theparticular way in which the surface is located in the ambient Euclidean space. The crowning result, the TheoremaEgregium of Gauss, established that the Gaussian curvature is an intrinsic invariant, i.e. invariant under localisometries. This point of view was extended to higher-dimensional spaces by Riemann and led to what is knowntoday as Riemannian geometry. The nineteenth century was the golden age for the theory of surfaces, from both thetopological and the differential-geometric point of view, with most leading geometers devoting themselves to their

study.[citation needed] 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 atime Gauss was Cartographer to George III of Great Britain and Hannover; this royal patronage could explain whythese papers contain practical calculations of the curvature of the earth based purely on measurements on thesurface of the planet.

Curvature of surfaces in E3

See also: Gaussian curvature and Mean curvature

Informally Gauss defined the curvature of a surface in terms of thecurvatures of certain plane curves connected with the surface. He laterfound a series of equivalent definitions. One of the first was in terms ofthe area-expanding properties of the Gauss map, a map from the surfaceto a 2-dimensional sphere. However, before obtaining a more intrinsicdefinition in terms of the area and angles of small triangles, Gauss neededto make an in-depth investigation of the properties of geodesics on thesurface, i.e. paths of shortest length between two fixed points on the

surface[4] (see below).

The Gaussian curvature at a point on an embedded smooth surfacegiven locally by the equation

z = F(x,y)

in E3, is defined to be the product of the principal curvatures at the

point;[5] the mean curvature is defined to be their average. Theprincipal curvatures are the maximum and minimum curvatures of theplane curves obtained by intersecting the surface with planes normal tothe 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 setting the coefficient on xy tozero, F will have the Taylor series expansion

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The principal curvatures are k1 and k2 in this case, the Gaussian curvature is given by

and the mean curvature by

Since K and Km are invariant under isometries of E3, in general

and

where the derivatives at the point are given by P = Fx, Q = Fy, R = Fx x, S = Fx y, and T = Fy y.[6]

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

Direct computation shows that: the Gaussian curvature is the Jacobian of the Gauss map.[7]

Examples

Surfaces of revolution

Main article: Surface of revolution

A surface of revolution can be obtained by rotating a curve in the xz plane about the z-axis, assuming the curvedoes not intersect the z-axis. Suppose that the curve is given by

with t lies in (a, b), and is parametrized by arclength, so that

Then the surface of revolution is the point set

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The surface of revolution obtained by

rotating the curve x = 2 + cos z about

the z-axis.

A quadric ellipsoid

The Gaussian curvature and mean curvature are given by[8]

Geodesics on a surface of revolution are governed by Clairaut's relation.

Quadric surfaces

Main article: Quadric surface

Consider the quadric surface defined by[9]

This surface admits a parametrization

The Gaussian curvature and mean curvature are given by

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A single-sheeted quadric hyperboloid

which is a ruled surface in two

different ways.

Ruled surfaces

Main article: Ruled surface

A ruled surface is one which can be generated by the motion of a straight

line in E3.[10] Choosing a directrix on the surface, i.e. a smooth unitspeed 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 velocityvector v=ct and u satisfy

The surface consists of points

as s and t vary.

Then, if

the Gaussian and mean curvature are given by

The Gaussian curvature of the ruled surface vanishes if and only if ut and v are proportional,[11] 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.[12] More generally a surface in E3 has

vanishing Gaussian curvature near a point if and only if it is developable near that point.[6] (An equivalent conditionis given below in terms of the metric.)

Minimal surfaces

Main article: Minimal surface

In 1760 Lagrange extended Euler's results on the calculus of variations involving integrals in one variable to two

variables.[13] He had in mind the following problem:

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Surfaces with (from l. to r.) constant

negative, zero and positive Gaussian

curvature

Given a closed curve in E3, find a surface having the curve as boundary with minimal area.

Such a surface is called a minimal surface. In 1776 Jean Baptiste Meusnier showed that the differential equationderived by Lagrange was equivalent to the vanishing of the mean curvature of the surface:

A surface is minimal if and only if its mean curvature vanishes.

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

in 1936).[14]

Many explicit examples of minimal surface are known explicitly, such as the catenoid, the helicoid, the Scherksurface 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 S2.

Surfaces of constant Gaussian curvature

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

constant curvature.[15]

The unit sphere in E3 has constant Gaussian curvature +1.

The Euclidean plane and the cylinder both have constant Gaussian

curvature 0.

The surfaces of revolution with φtt = φ have constant Gaussian

curvature –1. Particular cases are obtained by taking φ(t)

= C cosh t, C sinh t and C et.[16] The latter case is the classical

pseudosphere generated by rotating a tractrix around a central

axis. In 1868 Beltrami showed that the geometry of the

pseudosphere was directly related to that of the 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.[17] This surface of constant

curvature 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) = x2 + y2 – z2.[18]

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A chart for the upper

hemisphere of the 2-

sphere obtained by

projecting onto the x-y-

plane

Coordinate changes

between different local

charts must be smooth

Each of these surfaces of constant curvature has a transitive Lie group of symmetries. This group theoretic fact hasfar-reaching consequences, all the more remarkable because of the central role these special surfaces play in thegeometry 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 anydevelopable surface.

Local metric structure

Main article: Riemannian manifold

For any surface embedded in Euclidean space of dimension 3 or higher, it is possibleto measure the length of a curve on the surface, the angle between two curves andthe area of a region on the surface. This structure is encoded infinitesimally in aRiemannian metric on the surface through line elements and area elements.Classically in the nineteenth and early twentieth centuries only surfaces embedded in

R3 were considered and the metric was given as a 2×2 positive definite matrixvarying smoothly from point to point in a local parametrization of the surface. Theidea of local parametrization and change of coordinate was later formalized throughthe current abstract notion of a manifold, a topological space where the smoothstructure is given by local charts on the manifold, exactly as the planet Earth ismapped by atlases today. Changes of coordinates between different charts of thesame region are required to be smooth. Just as contour lines on real-life mapsencode changes in elevation, taking into account local distortions of the Earth'ssurface to calculate true distances, so the Riemannian metric describes distances andareas "in the small" in each local chart. In each local chart a Riemannian metric isgiven by smoothly assigning a 2×2 positive definite matrix to each point; when adifferent chart is taken, the matrix is transformed according to the Jacobian matrix ofthe coordinate change. The manifold then has the structure of a 2-dimensionalRiemannian manifold.

Line and area elements

Taking a local chart, for example by projecting onto the x-y plane (z = 0), the lineelement ds and the area element dA 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.

The expression E dx2 + 2F dx dy + G dy2 is called the first fundamental form.[6]

The matrix

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Definition of second fundamental form

is required to be positive-definite and to depend smoothly on x and y.

In a similar way line and area elements can be associated to any abstract Riemannian 2-manifold in a local chart.

Second fundamental form

Main article: Second fundamental form

The extrinsic geometry of surfaces studies the properties of surfaces embedded into a Euclidean space, typically E3.In intrinsic geometry, two surfaces are "the same" if it is possible to unfold one surface onto the other withoutstretching it, i.e. a map of one surface onto the other preserving distance. Thus a cylinder is locally "the same" as theplane. In extrinsic geometry, two surfaces are "the same" if they are congruent in the ambient Euclidean space, i.e.

there is an isometry of E3 carrying one surface onto the other. With this more rigid definition of similitude, thecylinder and the plane are obviously no longer the same.

Although the primary invariant in the study of the intrinsic geometry of surfaces is the metric (the first fundamental

form) and the Gaussian curvature, certain properties of surfaces also depend on an embedding into E3 (or a higherdimensional Euclidean space). The most important example is the second fundamental form, defined classically as

follows.[19]

Take a point (x, y) on thesurface in a local chart.The Euclidean distancefrom a nearby point(x + dx, y + dy) to thetangent plane at (x, y), i.e.the length of theperpendicular droppedfrom the nearby point tothe tangent plane, has theform

e dx2 + 2f dx dy +

g dy2

plus third and higher ordercorrections. The aboveexpression, a symmetricbilinear form at each point,is the second fundamentalform. It is described by a2 × 2 symmetric matrix

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Wilhelm Blaschke

(1885-1962)

which depends smoothly on x and y. The Gaussian curvature can be calculated as the ratio of the determinants ofthe second and first fundamental forms:

Remarkably Gauss proved that it is an intrinsic invariant (see his Theorema Egregium below).

One of the other extrinsic numerical invariants of a surface is the mean curvature Km defined as the sum of the

principal curvatures. It is given by the formula[6]

The coefficients of the first and second fundamental forms satisfy certain compatibility conditions known as the

Gauss-Codazzi equations; they involve the Christoffel symbols associated with the first fundamental form:[20]

These equations can also be succinctly expressed and derived in the language of connection forms due to Élie

Cartan.[21] Pierre Bonnet proved that two quadratic forms satisfying the Gauss-Codazzi equations always uniquely

determine an embedded surface locally.[22] For this reason the Gauss-Codazzi equations are often called thefundamental equations for embedded surfaces, precisely identifying where the intrinsic and extrinsic curvaturescome from. They admit generalizations to surfaces embedded in more general Riemannian manifolds.

Shape operator

Further information: Peterson operator

The differential df of the Gauss map f can be used to define a type of extrinsic curvature,

known as the shape operator[23] or Weingarten map. This operator first appearedimplicitly in the work of Wilhelm Blaschke and later explicitly in a treatise by Burali-Forti

and Burgati.[24] Since at each point x of the surface, the tangent space is an innerproduct space, the shape operator Sx can be defined as a linear operator on this space

by the formula

for tangent vectors v, w (the inner product makes sense because df(v) and w both lie in

E3).[25] The right hand side is symmetric in v and w, so the shape operator is self-adjoint on the tangent space. The eigenvalues of Sx are just the principal curvatures k1

and k2 at x. In particular the determinant of the shape operator at a point is the

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A geodesic triangle on the

sphere. The geodesics

are great circle arcs.

Gaussian curvature, but it also contains other information, since the mean curvature is half the trace of the shapeoperator. The mean curvature is an extrinsic invariant. In intrinsic geometry, a cylinder is developable, meaning thatevery piece of it is intrinsically indistinguishable from a piece of a plane since its Gauss curvature vanishesidentically. Its mean curvature is not zero, though; hence extrinsically it is different from a plane.

In general, the eigenvectors and eigenvalues of the shape operator at each point determine the directions in whichthe surface bends at each point. The eigenvalues correspond to the principal curvatures of the surface and theeigenvectors are the corresponding principal directions. The principal directions specify the directions that a curveembedded in the surface must travel to have maximum and minimum curvature, these being given by the principalcurvatures.

The shape operator is given in terms of the components of the first and second fundamental forms by the

Weingarten equations:[26]

Geodesic curves on a surface

Curves on a surface which minimize length between the endpoints are called geodesics; they are the shape that anelastic band stretched between the two points would take. Mathematically they are described using partialdifferential equations from the calculus of variations. The differential geometry of surfaces revolves around the studyof geodesics. It is still an open question whether every Riemannian metric on a 2-dimensional local chart arises froman embedding in 3-dimensional Euclidean space: the theory of geodesics has been used to show this is true in theimportant case when the components of the metric are analytic.

Geodesics

Given a piecewise smooth path c(t) = (x(t), y(t)) in the chart for t in [a, b], its lengthis defined by

and energy by

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

+ Γ¹11 ² + 2Γ¹12 + Γ¹22 ² =0 and + Γ²11 ² + 2Γ²12 + Γ²22 ² =0

where the Christoffel symbols Γkij are given by

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Γkij = g km ( j gim + i gjm – m gij)

where g11 = E, g12=F, g22 =G and (gij) is the inverse matrix to (gij). 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.[27]

Geodesic curvature

See also: Geodesic curvature and Darboux frame

The geodesic curvature at a point of a curve c(t), parametrised by arc length, on an oriented surface is defined

to be[28]

where n(t) is the "principal" unit normal to the curve in the surface, constructed by rotating the unit tangent vector through an angle of + 90°.

The geodesic curvature at a point is an intrinsic invariant depending only on the metric near the point.

A unit speed curve on a surface is a geodesic if and only if its geodesic curvature vanishes at all points on thecurve.

A unit speed curve c(t) in an embedded surface is a geodesic if and only if its acceleration vector is

normal to the surface.

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

Isometric embedding problem

A result of Jacobowitz (1972) and Poznjak (1973) shows that every metric structure on a surface arises from a

local embedding in E4. Apart from some special cases, whether this is possible in E3 remains an open question, the

so-called "Weyl problem".[29] In 1926 Maurice Janet proved that it is always possible locally if E, F and G are

analytic; soon afterwards Élie Cartan generalised this to local embeddings of Riemannian n-manifolds in Em wherem = ½(n² +n). To prove Janet's theorem near (0,0), the Cauchy-Kowalevski theorem is used twice to produceanalytic geodesics orthogonal to the y-axis and then the x-axis to make an analytic change of coordinate so thatE=1 and F=0. An isometric embedding u must satisfy

ux • ux =1, ux • uy = 0, uy • uy = G.

Differentiating gives the three additional equations

uxx • uy = 0, uxx • ux = 0, uxx • uyy = uxy • ux y - ½ Gxx

with u(0,y) and ux(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.

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In orthogonal coordinates φ is the

angle the tangent L to the geodesic C

makes with the x-axis

Carl Jacobi (1804–1851)

Orthogonal coordinates

When F=0 in the metric, lines parallel to the x- and y-axes are

orthogonal and provide orthogonal coordinates. If H=(EG)½, then the

Gaussian curvature is given by[30]

If in addition E=1, so that H=G½, then the angle at the intersectionbetween geodesic (x(t),y(t)) and the line y = constant is given by theequation

The derivative of is given by a classical derivative formula of Gauss:[31]

Geodesic polar coordinates

Once a metric is given on a surface and a base point is fixed, there is aunique geodesic connecting the base point to each sufficiently nearbypoint. The direction of the geodesic at the base point and the distanceuniquely determine the other endpoint. These two bits of data, a directionand a magnitude, thus determine a tangent vector at the base point. Themap from tangent vectors to endpoints smoothly sweeps out aneighbourhood of the base point and defines what is called the"exponential map", defining a local coordinate chart at that base point.The neighbourhood swept out has similar properties to balls in Euclideanspace, namely any two points in it are joined by a unique geodesic. Thisproperty is called "geodesic convexity" and the coordinates are called"normal coordinates". The explicit calculation of normal coordinates canbe accomplished by considering the differential equation satisfied bygeodesics. The convexity properties are consequences of Gauss's lemmaand its generalisations. Roughly speaking this lemma states that geodesicsstarting at the base point must cut the spheres of fixed radius centred onthe base point at right angles. Geodesic polar coordinates are obtainedby combining the exponential map with polar coordinates on tangent vectors at the base point. The Gaussiancurvature of the surface is then given by the second order deviation of the metric at the point from the Euclideanmetric. In particular the Gaussian curvature is an invariant of the metric, Gauss's celebrated Theorema Egregium.A convenient way to understand the curvature comes from an ordinary differential equation, first considered byGauss and later generalized by Jacobi, arising from the change of normal coordinates about two different points.The Gauss–Jacobi equation provides another way of computing the Gaussian curvature. Geometrically it explainswhat happens to geodesics from a fixed base point as the endpoint varies along a small curve segment through data

recorded in the Jacobi field, a vector field along the geodesic.[32] One and a quarter centuries after Gauss andJacobi, Marston Morse gave a more conceptual interpretation of the Jacobi field in terms of second derivatives of

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Contour lines tracking the motion of

points on a fixed curve moving along

geodesics towards a basepoint

the energy function on the infinite-dimensional Hilbert manifold of

paths.[33]

Exponential map

Main article: Normal coordinates

The theory of ordinary differential equations shows that if f(t, v) issmooth then the differential equation dv/dt = f(t,v) with initial conditionv(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 geodesic cv(t)

defined on (−2,2) with cv(0) = (x0,y0) and v(0) = v. Moreover if |s| ≤

1, then csv = cv(st). The exponential map is defined by

expp(v) = cv (1)

and gives a diffeomorphism between a disc ||v|| < δ and a neighbourhood of p; more generally the map sending(p,v) to expp(v) gives a local diffeomorphism onto a neighbourhood of (p,p). The exponential map gives geodesic

normal coordinates near p.[34]

Computation of normal coordinates

There is a standard technique (see for example Berger (2004)) for computing the change of variables to normalcoordinates u, v at a point as a formal Taylor series expansion. If the coordinates x, y at (0,0) are locallyorthogonal, 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 uniquelydetermines α, β, L, M, λ and μ.

Gauss's lemma

Main article: Gauss's lemma (Riemannian geometry)

In these coordinates the matrix g(x) satisfies g(0) = I and the lines t ↦ tv are geodesics through 0. Euler'sequations 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.

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In geodesic polar coordinates the

geodesics radiating from the origin

cut the circles of constant radius

orthogonally. The distances along

radii are true distances but on the

concentric circles small arcs have

length H(r,θ) = G(r,θ)½ times the

angle they subtend.

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 theRiemannian manifold is then given by a distance function d(p,q), namely the infimum of the lengths of piecewisesmooth 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 thedisc ||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.[5]

Theorema Egregium

Main article: Theorema Egregium

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 (u, v) as

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

This extraordinary result — Gauss' Theorema Egregium — shows that theGaussian curvature of a surface can be computed solely in terms of themetric and is thus an intrinsic invariant of the surface, independent of anyembedding in E³ and unchanged under coordinate transformations. In

particular isometries of surfaces preserve Gaussian curvature.[5]

Gauss–Jacobi equation

Main article: Jacobi field

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,

Hrr = – K H

The Jacobian of this coordinate change at q is equal to Hr. This gives another way of establishing the intrinsic nature

of Gaussian curvature. Because H(r,θ) can be interpreted as the length of the line element in the θ direction, theGauss–Jacobi equation shows that the Gaussian curvature measures the spreading of geodesics on a geometric

surface as they move away from a point.[35]

Laplace–Beltrami operator

On a surface with local metric

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A triangulation of the torus

and Laplace–Beltrami operator

where H2 = EG – F 2, the Gaussian curvature at a point is given by the formula[36]

where r is the denotes the geodesic distance from the point.

Since Δ is manifestly an intrinsic invariant, this gives yet another proof that the Gaussian curvature is an intrinsicinvariant.

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

In this case the Laplace–Beltrami operator is given by

and φ satisfies Liouville's equation[37]

Isothermal coordinates are known to exist in a neighbourhood of any point on the surface, although all proofs to

date rely on non-trivial results on partial differential equations.[38] There is an elementary proof for minimal

surfaces.[39]

Gauss–Bonnet theorem

On a sphere or a hyperboloid, the area of a geodesic triangle, i.e. a triangle all thesides of which are geodesics, is proportional to the difference of the sum of theinterior angles and π. The constant of proportionality is just the Gaussian curvature,a constant for these surfaces. For the torus, the difference is zero, reflecting the factthat its Gaussian curvature is zero. These are standard results in spherical,hyperbolic and high school trigonometry (see below). Gauss generalised theseresults to an arbitrary surface by showing that the integral of the Gaussian curvatureover the interior of a geodesic triangle is also equal to this angle difference orexcess. His formula showed that the Gaussian curvature could be calculated near apoint as the limit of area over angle excess for geodesic triangles shrinking to thepoint. Since any closed surface can be decomposed up into geodesic triangles, theformula could also be used to compute the integral of the curvature over the wholesurface. As a special case of what is now called the Gauss-Bonnet theorem, Gauss proved that this integral wasremarkably always 2π times an integer, a topological invariant of the surface called the Euler characteristic. This

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The area of a spherical triangle on the

unit sphere is α + β + γ - π.

The Euler characteristic of a sphere,

triangulated like an icosahedron, is V -

E + F = 12 - 30 + 20 =2.

invariant is easy to compute combinatorially in terms of the number of vertices, edges, and faces of the triangles inthe decomposition, also called a triangulation. This interaction between analysis and topology was the forerunner ofmany later results in geometry, culminating in the Atiyah-Singer indextheorem. In particular properties of the curvature impose restrictions onthe topology of the surface.

Geodesic triangles

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

Δ K dA = α + β + γ − π.

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

Δ K dA = Δ KH dr dθ = – Hrr dr dθ = 1 −

Hr(rθ,θ) dθ = dθ + dφ = α + β + γ − π,

where the second equality follows from the Gauss–Jacobi equation and the fourth from Gauss' derivative formula inthe orthogonal coordinates (r,θ).

Gauss' formula shows that the curvature at a point can be calculated as the limit of angle excess α + β + γ − π overarea 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.[6]

Gauss-Bonnet theorem

Main article: Gauss-Bonnet theorem

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

M K dA = 2π·χ(M)

where χ(M) denotes the Euler characteristic of the surface.

In fact if there are F faces, E edges and V vertices, then 3F = 2E and theleft hand side equals 2π·V – π·F = 2π·(V – E + F) = 2π·χ(M).

This is the celebrated Gauss-Bonnet theorem: it shows that the integralof the Gaussian curvature is a topological invariant of the manifold,namely the Euler characteristic. This theorem can be interpreted in many

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A triangle in the plane

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 theGauss-Bonnet theorem, is the Poincaré-Hopf index theorem for vector fields on M which vanish at only a finitenumber 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.)[6]

Curvature and embeddings

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, theGauss map provides an explicit diffeomorphism. As Hadamard observed, in this case the surface is convex; thiscriterion for convexity can be viewed as a 2-dimensional generalisation of the well-known second derivativecriterion for convexity of plane curves. Hilbert proved that every isometrically embedded closed surface must havea 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.[40]

Surfaces of constant curvature

The simply connected surfaces of constant curvature 0, +1 and –1 are the Euclidean plane, the unit sphere in E3,and the hyperbolic plane. Each of these has a transitive three-dimensional Lie group of orientation preservingisometries G, which can be used to study their geometry. Each of the two non-compact surfaces can be identifiedwith the quotient G / K where K is a maximal compact subgroup of G. Here K is isomorphic to SO(2). Any otherclosed Riemannian 2-manifold M of constant Gaussian curvature, after scaling the metric by a constant factor ifnecessary, will have one of these three surfaces as its universal covering space. In the orientable case, thefundamental group Γ of M can be identified with a torsion-free uniform subgroup of G and M can then be identifiedwith the double coset space Γ \ G / K. In the case of the sphere and the Euclidean plane, the only possible

examples are the sphere itself and tori obtained as quotients of R2 by discrete rank 2 subgroups. For closedsurfaces of genus , the moduli space of Riemann surfaces obtained as Γ varies over all such subgroups, has

real dimension 6g - 6 .[41] By Poincaré's uniformization theorem, any orientable closed 2-manifold is conformallyequivalent to a surface of constant curvature 0, +1 or –1. In other words, by multiplying the metric by a positivescaling factor, the Gaussian curvature can be made to take exactly one of these values (the sign of the Euler

characteristic of M).[42]

Euclidean geometry

In the case of the Euclidean plane, the symmetry group is the Euclideanmotion group, the semidirect product of the two dimensional group of

translations by the group of rotations.[43] Geodesics are straight lines andthe geometry is encoded in the elementary formulas of trigonometry, suchas the cosine rule for a triangle with sides a, b, c and angles α, β, γ:

Flat tori can be obtained by taking the quotient of R2 by a lattice, i.e. a free Abelian subgroup of rank 2. These

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A spherical triangle

Flat tori can be obtained by taking the quotient of R2 by a lattice, i.e. a free Abelian subgroup of rank 2. These

closed surfaces have no isometric embeddings in E3. They do nevertheless admit isometric embeddings in E4; in theeasiest case this follows from the fact that the torus is a product of two circles and each circle can be isometrically

embedded in E2.[44]

Spherical geometry

See also: spherical trigonometry and spherical triangle

The isometry group of the unit sphere S2 in E3 isthe orthogonal group O(3), with the rotation groupSO(3) as the subgroup of isometries preservingorientation. It is the direct product of SO(3) with

the antipodal map, sending x to –x.[45] The group

SO(3) acts transitively on S2. The stabilizersubgroup of the unit vector (0,0,1) can be

identified with SO(2), so that S2 = SO(3)/SO(2).

The geodesics between two points on the sphereare the great circle arcs with these givenendpoints. If the points are not antipodal, there is aunique shortest geodesic between the points. Thegeodesics can also be described grouptheoretically: each geodesic through the Northpole (0,0,1) is the orbit of the subgroup ofrotations about an axis through antipodal points onthe equator.

A spherical triangle is a geodesic triangle on thesphere. It is defined by points A, B, C on thesphere with sides BC, CA, AB formed from greatcircle arcs of length less than π. If the lengths of the sides are a, b, c and the angles between the sides α, β, γ, thenthe spherical cosine law states that

The area of the triangle is given by

Area = α + β + γ - π.

Using stereographic projection from the North pole, the sphere can be identified with the extended complex planeC {∞}. The explicit map is given by

Under this correspondence every rotation of S2 corresponds to a Möbius transformation in SU(2), unique up to

sign.[46] With respect to the coordinates (u, v) in the complex plane, the spherical metric becomes[47]

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Eugenio Beltrami

(1835-1899)

Felix Klein (1849-

1925)

The unit sphere is the unique closed orientable surface with constant curvature +1. The quotient SO(3)/O(2) can be

identified with the real projective plane. It is non-orientable and can be described as the quotient of S2 by theantipodal map (multiplication by –1). The sphere is simply connected, while the real projective plane hasfundamental group Z2. The finite subgroups of SO(3), corresponding to the finite subgroups of O(2) and the

symmetry groups of the platonic solids, do not act freely on S2, so the corresponding quotients are not 2-manifolds,just orbifolds.

Hyperbolic geometry

See also: hyperbolic triangle and hyperbolic geometry

Non-Euclidean geometry[48] was first discussed in letters of Gauss, who made extensivecomputations at the turn of the nineteenth century which, although privately circulated, hedecided not to put into print. In 1830 Lobachevsky and independently in 1832 Bolyai, theson of one Gauss' correspondents, published synthetic versions of this new geometry, forwhich they were severely criticized. However it was not until 1868 that Beltrami, followedby Klein in 1871 and Poincaré in 1882, gave concrete analytic models for what Kleindubbed hyperbolic geometry. The four models of 2-dimensional hyperbolic geometrythat emerged were:

the Beltrami-Klein model;

the Poincaré disk;the Poincaré upper half-plane;

the hyperboloid model of Wilhelm Killing in 3-dimensional Minkowski space.

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

Let

be the Poincaré disk in the complex plane with Poincaré metric

In polar coordinates (r, θ) the metric is given by

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Henri Poincaré

(1854-1912)

A hyperbolic triangle in the Poincaré

disk model

The length of a curve γ:[a,b] D is given by the formula

The group G = SU(1,1) given by

acts transitively by Möbius transformations on D and the stabilizersubgroup of 0 is the rotation group

The quotient group SU(1,1)/±I is the group of orientation-preservingisometries of D. Any two points z, w in D are joined by a uniquegeodesic, given by the portion of the circle or straight line passing throughz and w and orthogonal to the boundary circle. The distance between zand w is given by

In particular d(0,r) = 2 tanh−1 r and c(t) = tanh t/2 is the geodesicthrough 0 along the real axis, parametrized by arclength.

The topology defined by this metric is equivalent to the usual Euclidean topology, although as a metric space (D,d)is complete.

A hyperbolic triangle is a geodesic triangle for this metric: any three points in D are vertices of a hyperbolic triangle.If the sides have length a, b, c with corresponding angles α, β, γ, then the hyperbolic cosine rule states that

The area of the hyperbolic triangle is given by[49]

Area = π – α – β – γ.

The unit disk and the upper half-plane

are conformally equivalent by the Möbius transformations

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Under this correspondence the action of SL(2,R) by Möbius transformations on H corresponds to that of SU(1,1)on D. The metric on H becomes

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

The unit disk with the Poincaré metric is the unique simply connected oriented 2-dimensional Riemannian manifoldwith constant curvature -1. Any oriented closed surface M with this property has D as its universal covering space.Its fundamental group can be identified with a torsion-free concompact subgroup Γ of SU(1,1), in such a way that

In this case Γ is a finitely presented group. The generators and relations are encoded in a geodesically convexfundamental geodesic polygon in D (or H) corresponding geometrically to closed geodesics on M.

Examples.

the Bolza surface of genus 2;

the Klein quartic of genus 3;the Macbeath surface of genus 7;the First Hurwitz triplet of genus 14.

Uniformization

Given an oriented closed surface M with Gaussian curvature K, the metric on M can be changed conformally by

scaling it by a factor e2u. The new Gaussian curvature K' is then given by

where Δ is the Laplacian for the original metric. Thus to show that a given surface is conformally equivalent to ametric with constant curvature K' it suffices to solve the following variant of Liouville's equation:

When M has Euler characteristic 0, so is diffeomorphic to a torus, K' = 0, so this amounts to solving

By standard elliptic theory, this is possible because the integral of K over M is zero, by the Gauss-Bonnet

theorem.[50]

When M has negative Euler characteristic, K' = -1, so the equation to be solved is:

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Using the continuity of the exponential map on Sobolev space due to Neil Trudinger, this non-linear equation can

always be solved.[51]

Finally in the case of the 2-sphere, K' = 1 and the equation becomes:

So far this non-linear equation has not been analysed directly, although classical results such as the Riemann-Rochtheorem imply that it always has a solution. The method of Ricci flow, developed by Richard Hamilton, gives

another proof of existence based on non-linear partial differential equations to prove existence.[52] In fact the Ricci

flow on conformal metrics on S2 is defined on functions u(x, t) by

After finite time, Chow showed that K' becomes positive; previous results of Hamilton could then be used to show

that K' converges to +1.[53]

A simple proof using only elliptic operators discovered in 1988 can be found in Ding (2001). Let G be the Green's

function on S2 satisfying ΔG = 1 + 4πδP, where δP is the point measure at a fixed point P of S2. The equation Δv =

2K – 2, has a smooth solution v, because the right hand side has integral 0 by the Gauss-Bonnet theorem. Thus φ =

2G + v satisfies Δφ = 2K away from P. It follows that g1 = eφg is a complete metric of constant curvature 0 on the

complement of P, which is therefore isometric to the plane. Composing with stereographic projection, it follows that

there is a smooth function u such that e2ug has Gaussian curvature +1 on the complement of P. The function u

automatically extends to a smooth function on the whole of S2.[54]

Surfaces of non-positive curvature

In a region where the curvature of the surface satisfies K≤0, geodesic triangles satisfy the CAT(0) inequalities ofcomparison geometry, studied by Cartan, Alexandrov and Toponogov, and considered later from a differentpoint of view by Bruhat and Tits; thanks to the vision of Gromov, this characterisation of non-positive curvature interms of the underlying metric space has had a profound impact on modern geometry and in particular geometricgroup theory. Many results known for smooth surfaces and their geodesics, such as Birkhoff's method ofconstructing geodesics by his curve-shortening process or van Mangoldt and Hadamard's theorem that a simplyconnected surface of non-positive curvature is homeomorphic to the plane, are equally valid in this more generalsetting.

Alexandrov's comparison inequality

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 isalways less than the corresponding distance in the comparison triangle in the plane with the sameside-lengths.

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The median in the comparison triangle

is always longer than the actual

median.

George Birkhoff (1884-1944)

The inequality follows from the fact that if c(t) describes a geodesic parametrised by arclength and a is a fixedpoint, then

f(t) = d(a,c(t))2 − t2

is a convex function, i.e.

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

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

u2 + H − 1 Hr v2 ≥ 1,

where (u,v) corresponds to the unit vector . This follows from the

inequality Hr ≥ H, a consequence of the non-negativity of the derivative

of the Wronskian of H and r from Sturm–Liouville theory.[55]

Existence of geodesics

On a complete curved surface any two points can be joined by ageodesic. This is a special case of the Hopf-Rinow theorem, which alsoapplies in higher dimensions. The completeness assumption isautomatically fulfilled for a surface which is embedded as a closed subsetof Euclidean space. However, it is no longer fulfilled if, for example, weremove an isolated point from a surface. For example, the complement ofthe origin in the Euclidean plane is an example of a non-complete surface;in this example two points which are diametrically opposite across theorigin cannot be joined by a geodesic without leaving the puncturedplan).

Von Mangoldt-Hadamard theorem

For closed surfaces of non-positive curvature, von Mangoldt (1881) andHadamard (1898) proved that the exponential map at a point is acovering map, so that the universal covering space of the manifold is E². This result was generalised to higherdimensions by Cartan and is usually referred to in this form as the Cartan–Hadamard theorem. For surfaces, this

result follows from three important facts:[56]

The exponential map has non-zero Jacobian everywhere for non-positively curved surfaces, a consequence

of the non-vanishing of Hr.

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Tullio Levi-Civita

(1873-1941)

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

Every two points in a homotopy class are connected by a unique geodesic (see above).

Riemannian connection and parallel transport

Main article: Riemannian connection on a surface

The classical approach of Gauss to the differential geometry of surfaces was the

standard elementary approach[57] which predated the emergence of the concepts ofRiemannian manifold initiated by Bernhard Riemann in the mid-nineteenth century and ofconnection developed by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the earlytwentieth century. The notion of connection, covariant derivative and parallel transportgave a more conceptual and uniform way of understanding curvature, which not onlyallowed generalisations to higher dimensional manifolds but also provided an important

tool for defining new geometric invariants, called characteristic classes.[58] The approachusing covariant derivatives and connections is nowadays the one adopted in more

advanced textbooks.[59]

Covariant derivative

Connections on a surface can be defined from various equivalent but equally important points of view. The

Riemannian connection or Levi-Civita connection[6] is perhaps most easily understood in terms of lifting vectorfields, considered as first order differential operators acting on functions on the manifold, to differential operators onthe tangent bundle or frame bundle. In the case of an embedded surface, the lift to an operator on vector fields,called the covariant derivative, is very simply described in terms of orthogonal projection. Indeed a vector field

on a surface embedded in can be regarded as a function from the surface into R3. Another vector field act as a

differential operator component-wise. The resulting vector field will not be tangent to the surface, but this can becorrected taking its orthogonal projection onto the tangent space at each point of the surface. As Ricci and Levi-Civita realised at the turn of the twentieth century, this process depends only on the metric and can be locallyexpressed in terms of the Christoffel symbols.

Parallel transport

Parallel transport of tangent vectors along a curve in the surface was the next major advance in the subject, due

to Levi-Civita.[6] It is related to the earlier notion of covariant derivative, because it is the monodromy of theordinary differential equation on the curve defined by the covariant derivative with respect to the velocity vector ofthe curve. Parallel transport along geodesics, the "straight lines" of the surface, can also easily be described directly.A vector in the tangent plane is transported along a geodesic as the unique vector field with constant length andmaking a constant angle with the velocity vector of the geodesic. For a general curve, this process has to be

modified using the geodesic curvature, which measures how far the curve departs from being a geodesic.[5]

A vector field v(t) along a unit speed curve c(t), with geodesic curvature kg(t), is said to be parallel along the curve

if

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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.

Élie Cartan in 1904

it has constant length

the angle θ(t) that it makes with the velocity vector satisfies

This recaptures the rule for parallel transport along a geodesic or piecewisegeodesic curve, because in that case kg = 0, so that the angle θ(t) should

remain constant on any geodesic segment. The existence of parallel transportfollows because θ(t) can be computed as the integral of the geodesic

curvature. Since it therefore depends continuously on the L2 norm of kg, it

follows that parallel transport for an arbitrary curve can be obtained as thelimit of the parallel transport on approximating piecewise geodesic

curves.[60]

The connection can thus be described in terms of lifting paths in the manifoldto paths in the tangent or orthonormal frame bundle, thus formalising the

classical theory of the "moving frame", favoured by French authors.[61] Liftsof loops about a point give rise to the holonomy group at that point. TheGaussian curvature at a point can be recovered from parallel transportaround increasingly small loops at the point. Equivalently curvature can becalculated directly at an infinitesimal level in terms of Lie brackets of lifted vector fields.

Connection 1-form

The approach of Cartan and Weyl, using connection 1-forms on theframe bundle of M, gives a third way to understand the Riemannianconnection. They noticed that parallel transport dictates that a path in thesurface be lifted to a path in the frame bundle so that its tangent vectorslie in a special subspace of codimension one in the three-dimensionaltangent space of the frame bundle. The projection onto this subspace isdefined by a differential 1-form on the orthonormal frame bundle, theconnection form. This enabled the curvature properties of the surface tobe encoded in differential forms on the frame bundle and formulasinvolving their exterior derivatives.

This approach is particularly simple for an embedded surface. Thanks toa result of Kobayashi (1956), the connection 1-form on a surface

embedded in Euclidean space E3 is just the pullback under the Gauss

map of the connection 1-form on S2.[62] Using the identification of S2

with the homogeneous space SO(3)/SO(2), the connection 1-form is just

a component of the Maurer-Cartan 1-form on SO(3).[63]

Global differential geometry of surfaces

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Shortest loop on a torus

Although the characterisation of curvature involves only the local geometry of a surface, there are important globalaspects such as the Gauss-Bonnet theorem, the uniformization theorem, the von Mangoldt-Hadamard theorem, and

the embeddability theorem. There are other important aspects of the global geometry of surfaces.[64] These include:

Injectivity radius, defined as the largest r such that two points at a distance less than r are joined by aunique geodesic. Wilhelm Klingenberg proved in 1959 that the injectivity radius of a closed surface is

bounded below by the minimum of and the length of its smallest closed geodesic. This

improved a theorem of Bonnet who showed in 1855 that the diameter of a closed surface of positiveGaussian curvature is always bounded above by δ; in other words a geodesic realising the metric distancebetween two points cannot have length greater than δ.

Rigidity. In 1927 Cohn-Vossen proved that two ovaloids – closed surfaces with positive Gaussian

curvature – that are isometric are necessarily congruent by an isometry of E3. Moreover a closed embeddedsurface with positive Gaussian curvature and constant mean curvature is necessarily a sphere; likewise aclosed embedded surface of constant Gaussian curvature must be a sphere (Liebmann 1899). Heinz Hopf

showed in 1950 that a closed embedded surface with constant mean curvature and genus 0, i.e.homeomorphic to a sphere, is necessarily a sphere; five years later Alexandrov removed the topologicalassumption. In the 1980s, Wente constructed immersed tori of constant mean curvature in Euclidean 3-space.

Carathéodory conjecture: This conjecture states that a closed convex three times differentiable surface

admits at least two umbilic points. The first work on this conjecture was in 1924 by Hans Hamburger, whonoted that it follows from the following stronger claim : the half-integer valued index of the principal curvaturefoliation of an isolated umbilic is at most one. The contribution of Hamburger and those of subsequentauthors to proving this local conjecture are inconclusive.

Zero Gaussian curvature: a complete surface in E3 with zero Gaussian curvature must be a cylinder or aplane.

Hilbert's theorem (1901): no complete surface with constant negative curvature can be immersed

isometrically in E3.

The Willmore conjecture. This conjecture states that the integral of

the square of the mean curvature of a torus immersed in E3 should be

bounded below by 2 π2. The conjecture has been proved for largeclasses of torus immersions. It is also known that the integral is a

conformal invariant.

Isoperimetric inequalities. In 1939 Schmidt proved that theclassical isoperimetric inequality for curves in the Euclidean plane isalso valid on the sphere or in the hyperbolic plane: namely he showedthat among all closed curves bounding a domain of fixed area, the

perimeter is minimized by when the curve is a circle for the metric. In

one dimension higher, it is known that among all closed surfaces in E3

arising as the boundary of a bounded domain of unit volume, the surface area is minimized for a Euclideanball.

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Systolic inequalities for curves on surfaces. Given a closed surface, its systole is defined to be thesmallest length of any non-contractible closed curve on the surface. In 1949 Loewner proved a torusinequality for metrics on the torus, namely that the area of the torus over the square of its systole is boundedbelow by , with equality in the flat (constant curvature) case. A similar result is given by Pu's inequality

for the real projective plane from 1952, with a lower bound of 2/π also attained in the constant curvaturecase. For the Klein bottle, Blatter and Bavard later obtained a lower bound of . For a closed surface

of genus g, Hebda and Burago showed that the ratio is bounded below by 1/2. Three years later Mikhail

Gromov found a lower bound given by a constant times g1/2, although this is not optimal. Asymptotically

sharp upper and lower bounds given by constants times g/(log g)2 are due to Gromov and Buser-Sarnak,and can be found in Katz (2007). There is also a version for metrics on the sphere, taking for the systole the

length of the smallest closed geodesic. Gromov conjectured a lower bound of in 1980: the best

result so far is the lower bound of 1/8 obtained by Regina Rotman in 2006.[65]

Reading guide

One of the most comprehensive introductory surveys of the subject, charting the historical development from beforeGauss to modern times, is by Berger (2004). Accounts of the classical theory are given in Eisenhart (2004),Kreyszig (1991) and Struik (1988); the more modern copiously illustrated undergraduate textbooks by Gray,Abbena & Salamon (2006), Pressley (2001) and Wilson (2008) might be found more accessible. An accessibleaccount of the classical theory can be found in Hilbert & Cohn-Vossen (1952). More sophisticated graduate-leveltreatments using the Riemannian connection on a surface can be found in Singer & Thorpe (1967), do Carmo(1976) and O'Neill (1997).

See also

Zoll surface

Notes

1. ^ Euler 1760

2. ^ Euler 1771

3. ^ Gauss 1825-1827

4. ^ This is the final position into which a rubber band stretched between two fixed points on the surface would fall.

5. ̂a b c d Berger 2004

6. ̂a b c d e f g h Eisenhart 2004, p. 123

7. ^ Singer & Thorpe 1967, p. 223

8. ^ do Carmo 1976, pp. 161–162

9. ^ Eisenhart 2004, pp. 228–229

10. ^ Eisenhart 2004, pp. 241–250; do Carmo 1976, pp. 188–197.

11. ^ do Carmo 1976, p. 194.

12. ^ Eisenhart 2004, pp. 61–65.

13. ^ Eisenhart 2004, pp. 250–269; do Carmo 1976, pp. 197–213.

14. ^ Douglas' solution is described in Courant (1950).

15. ^ Eisenhart 2004, pp. 270–291; O'Neill, pp. 249–251; Hilbert & Cohn-Vossen 1952.

16. ^ O'Neill, pp. 249–251; do Carmo, pp. 168–170; Gray, Abbena & Salamon 2006.

^ Stillwell 1996, pp. 1–5.

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17. ^ Stillwell 1996, pp. 1–5.

18. ^ Wilson 2008.

19. ^ Eisenhart 2004, pp. 114–115; Pressley 2001, pp. 123–124; Wilson 2008, pp. 123–124.

20. ^ Eisenhart 2004, p. 156

21. ^ O'Neill 1997, p. 257

22. ^ do Carmo 1976, pp. 309–314

23. ^ O'Neill 1997, pp. 195–216; do Carmo 1976, pp. 134–153; Singer & Thorpe 1967, pp. 216–224.

24. ^ Gray, Abbena & Salamon 2006, p. 386.

25. ^ Note that in some more recent texts the symmetric bilinear form on the right hand side is referred to as thesecond fundamental form; however, it does not in general correspond to the classically defined secondfundamental form.

26. ^ Gray, Abbena & Salamon 2006, p. 394.

27. ^ Berger 2004; Wilson 2008; Milnor 1963.

28. ^ Eisenhart 2002, p. 131; Berger 2004, p. 39; do Carmo 1976, p. 248; O'Neill 1997, p. 237

29. ^ Han & Hong 2006

30. ^ Eisenhart 2004; Taylor 1996a, Appendix C.

31. ^ Eisenhart 2004; Berger 2004.

32. ^ doCarmo 1976, p. 357

33. ^ Milnor 1963

34. ^ Wilson 2008

35. ^ O'Neill 1997, p. 395

36. ^ Helgason 1978, p. 92

37. ^ O'Niell 1997, p. 286

38. ^ do Carmo 1976, p. 227

39. ^ Osserman 2002, pp. 31–32

40. ^ Singer & Thorpe 1967; Garsia, Adriano M. (1961), "An imbedding of closed Riemann surfaces in Euclidean

space", Comment. Math. Helv. 35: 93–110, doi:10.1007/BF02567009 (http://dx.doi.org/10.1007%2FBF02567009)

41. ^ Imayoshi & Taniguchi 1992, pp. 47–49

42. ^ Berger 1977; Taylor 1996.

43. ^ Wilson 2008, pp. 1–23, Chapter I, Euclidean geometry.

44. ^ do Carmo 1976.

45. ^ Wilson 2008, pp. 25–49, Chapter II, Spherical geometry.

46. ^ Wilson 2008, Chapter 2.

47. ^ Eisenhart 2004, p. 110.

48. ^ Stillwell 1990; Bonola, Carslaw & Enriques 1955.

49. ^ Wilson 2008, Chapter 5.

50. ^ Taylor 1996b, p. 107; Berger 1977, pp. 341–343.

51. ^ Berger 1977, pp. 222–225; Taylor 1996b, pp. 101–108.

52. ^ Chow 1991; Taylor 1996b.

53. ^ Chen, Lu & Tian (2006) pointed out a missing step in the approach of Hamilton and Chow.

54. ^ This follows by an argument involving a theorem of Sacks & Uhlenbeck (1981) on removable singularities ofharmonic maps of finite energy.

55. ^ Berger 2004; Jost, Jürgen (1997), Nonpositive curvature: geometric and analytic aspects, Lectures inMathematics, ETH Zurich, Birkhäuser, ISBN 0-8176-5736-3

56. ^ do Carmo 1976; Berger 2004.

57. ^ Eisenhart 2004; Kreyszig 1991; Berger 2004; Wilson 2008.

58. ^ Kobayashi & Nomizu 1969, Chapter XII.

59. ^ do Carmo 1976; O'Neill 1997; Singer & Thorpe 1967.

60. ^ Arnold 1989, pp. 301–306, Appendix I.; Berger 2004, pp. 263–264.

61. ^ Darboux 1887,1889,1896

62. ^ Kobayashi & Nomizu 1969

63. ^ Ivey & Landsberg 2003.

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63. ^ Ivey & Landsberg 2003.

64. ^ Berger 2004, pp. 145–161; do Carmo 1976; Chern 1967; Hopf 1989.

65. ^ Rotman, R. (2006) "The length of a shortest closed geodesic and the area of a 2-dimensional sphere," Proc.

Amer. Math. Soc. 134: 3041-3047. Previous lower bounds had been obtained by Croke, Rotman-Nabutovsky andSabourau.

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