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[[File:Bendixen - Carl Friedrich Gauß, 1828.jpg|thumb|right|[[Carl Friedrich Gauss]] in 1828]]
In [[mathematics]], the '''differential geometry of surfaces''' deals with [[smooth manifold|smooth]] [[surface]]s 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]] ([[#CITEREFGauss1825-1827|1825-1827]]), 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 [[Doublepointer|space curves]].  An important role in their study has been played by [[Lie group]]s (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 [[connection (mathematics)|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.


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{{Unreferenced section|date=July 2008}}
{{Off-topic|Surfaces|date=June 2011}}
{{See also|Surfaces}}
 
[[Polyhedron|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 [[diffeomorphism|diffeomorphic]] to some open set in '''E'''<sup>2</sup>, the [[Euclidean plane]]. This elaboration allows [[calculus]] to be applied to surfaces to prove many results.
 
Two smooth surfaces are diffeomorphic if and only if they are [[homeomorphic]]. (The analogous result does not hold for manifolds of dimension greater than three.) It follows that [[closed surface]]s are classified up to diffeomorphism by their [[Euler characteristic]] and [[orientability]].
 
Smooth surfaces equipped with [[Riemannian metric]]s are of foundational importance in differential geometry. A Riemannian metric endows a surface with notions of [[geodesic]], [[distance]], [[angle]], and area. An important class of such surfaces are the [[developable surface]]s: surfaces that can be flattened to a plane An without stretching; examples include the [[cylinder (geometry)|cylinder]] and the [[conical surface|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 surface]]s are surfaces that minimize the surface area for given [[boundary condition]]s; examples include [[soap film]]s stretched across a wire frame, [[catenoid]]s and [[helicoid]]s.
*[[Ruled surface]]s 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 (2''n'')-dimensional real manifold. Thus any complex one-manifold (also called a [[Riemann surface]]) is a smooth oriented surface with an associated [[Linear complex structure|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 [[conformally equivalent|conformal equivalence classes]] of [[Riemannian metric]]s on the surface. One version of the [[uniformization theorem]] (due to [[Henri Poincaré|Poincaré]]) states that any [[Riemannian metric]] on an oriented, closed surface is conformally equivalent to an essentially unique metric of [[constant curvature]]. This provides a starting point for one of the approaches to [[Teichmüller theory]], which provides a finer classification of Riemann surfaces than the topological one by Euler characteristic alone.
 
The [[uniformization theorem]] states that every smooth Riemannian surface <math>S</math> is [[conformal equivalence|conformally equivalent]] to a surface having constant curvature, and the constant may be taken to be 1, 0, or -1. A surface of constant curvature 1 is locally [[isometry|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 '''E'''<sup>3</sup> with its intrinsic Riemannian metric. Likewise, a surface of constant curvature 0 is locally isometric to the Euclidean plane, and a surface of constant curvature -1 is locally isometric to the [[hyperbolic geometry|hyperbolic plane]].
 
Constant curvature surfaces are the two-dimensional realization of what are known as [[space form]]s. These are often studied from the point of view of [[Felix Klein]]'s [[Erlangen programme]], by means of smooth [[transformation group]]s.  Any [[connected space|connected]] surface with a three-dimensional group of isometries is a surface of constant curvature.
 
A ''complex surface'' is a complex two-manifold and thus a real four-manifold; it is not a surface in the sense of this article. Neither are algebraic curves or surfaces defined over [[field (mathematics)|fields]] other than the complex numbers.
 
== History of surfaces ==
Isolated properties of [[surface of revolution|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 [[Leonhard Euler|Euler]]. In 1760<ref>{{harvnb|Euler|1760}}</ref> he proved a formula for the curvature of a plane section of a surface and in 1771<ref>{{harvnb|Euler|1771}}</ref> he considered surfaces represented in a parametric form. [[Gaspard Monge|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 [[Carl Friedrich Gauss|Gauss]] in two remarkable papers written in 1825 and 1827.<ref>{{harvnb|Gauss|1825-1827}}</ref> 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 [[isometry|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.{{Citation needed|date=July 2008}} [[Gaston Darboux|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.
 
== Curvature of surfaces in E<sup>3</sup> ==
{{See also|Gaussian curvature|Mean curvature}}
[[File:Minimal surface curvature planes-en.svg|thumb|right|The principal curvatures at a point on a surface]]
[[File:Surface normal.png|thumb|right|The Gauss map sends a point on the surface to the outward pointing unit normal vector, a point on ''S''<sup>2</sup>]]
 
Informally Gauss defined the curvature of a surface in terms of the curvatures of certain plane curves connected with the surface. He later found a series of equivalent definitions. One of the first was in terms of the area-expanding properties of the Gauss map, a map from the surface to a 2-dimensional sphere. However, before obtaining a more intrinsic definition in terms of the area and angles of small triangles, Gauss needed to make an in-depth investigation of the properties of geodesics on the surface, i.e. paths of shortest length between two fixed points on the surface<ref>This is the final position into which a rubber band stretched between two fixed points on the surface would fall.</ref> (see below).
 
The '''Gaussian curvature''' at a point on an embedded smooth surface given locally by the equation
 
:''z'' = ''F''(''x'',''y'')
 
in '''E'''<sup>3</sup>, is defined to be the product of the [[principal curvature]]s at the point;<ref name="harvnb|Berger|2004">{{harvnb|Berger|2004}}</ref>
the '''mean curvature''' is defined to be their average. The principal curvatures are the maximum and minimum [[curvature]]s of the [[plane curve]]s 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 setting the coefficient on ''xy'' to zero, ''F'' will have the Taylor series expansion
 
:<math>F(x, y) = \frac{1}{2} k_1 x^2 + \frac{1}{2} k_2 y^2 + ...</math>
 
The principal curvatures are ''k''<sub>1</sub> and ''k''<sub>2</sub> in this case, the Gaussian curvature is given by
 
:<math>K=k_1\cdot k_2.</math>
 
and the mean curvature by
 
:<math>K_m=\tfrac{1}{2} (k_1 + k_2).</math>
 
Since ''K'' and ''K''<sub>''m''</sub> are invariant under [[isometry|isometries]] of '''E'''<sup>3</sup>, in general
 
:<math>K = \frac{RT - S^2}{\left( 1 + P^2 + Q^2 \right)^2}</math>
 
and
 
:<math>K_m =\frac{ET + GR - 2FS}{\left(1 + P^2 + Q^2\right)^2}</math>
 
where the derivatives at the point are given by ''P'' = ''F''<sub>''x''</sub>, ''Q'' = ''F''<sub>''y''</sub>, ''R'' = ''F''<sub>''x'' ''x''</sub>, ''S'' = ''F''<sub>''x'' ''y''</sub>, and ''T'' = ''F''<sub>''y'' ''y''</sub>.<ref name="ref_">{{harvnb|Eisenhart|2004|p=123}}</ref>
 
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
 
:<math>N(x, y, z) = \frac{1}{\sqrt{1 + P^2 + Q^2}} (P, Q, -1).</math>
 
Direct computation shows that: ''the Gaussian curvature  is the [[Jacobian variety|Jacobian]] of the Gauss map''.<ref>{{harvnb|Singer|Thorpe|1967|p=223}}</ref>
 
==Examples==
[[File:Surface of revolution illustration.png|thumb|right|The surface of revolution obtained by rotating the curve ''x'' = 2 + cos ''z'' about the ''z''-axis.]]
 
===Surfaces of revolution===
{{main|Surface of revolution}}
A [[surface of revolution]] can be obtained by rotating a curve in the ''xz'' plane about the ''z''-axis, assuming the curve does not intersect the ''z''-axis. Suppose that the curve is given by
 
:<math> x= \varphi(t),\,\, z=\psi(t)</math>
 
with ''t'' lies in (''a'', ''b''), and is parametrized by arclength, so that
 
:<math> \dot{\varphi}^2 + \dot{\psi}^2 = 1.</math>
 
Then the surface of revolution is the point set
 
:<math>M=\{(\varphi(t)\cos \theta, \varphi(t)\sin \theta,\psi(t))\colon t\in (a,b), \theta\in [0,2\pi)\}.</math>
 
The Gaussian curvature and mean curvature are given by<ref>{{harvnb|do Carmo|1976|pp=161–162}}</ref>
 
:<math> K = -{\ddot{\varphi}\over \varphi},\,\, K_m = {-\dot{\psi} +\varphi(\dot{\psi}\ddot{\phi} -\ddot{\psi}\dot{\varphi})\over 2 \varphi}. </math>
 
[[File:Ellipsoid Quadric.png|thumb|right|A quadric [[ellipsoid]]]]
Geodesics on a surface of revolution are governed by [[Clairaut's relation]].
 
===Quadric surfaces===
{{main|Quadric surface}}
Consider the [[quadric surface]] defined by<ref>{{harvnb|Eisenhart|2004|pp=228–229}}</ref>
 
:<math> {x^2\over a} + {y^2\over b} +{z^2\over c}=1.</math>
 
This surface admits a parametrization
 
:<math>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)}.</math>
 
The Gaussian curvature and mean curvature are given by
 
:<math>K={abc\over u^2 v^2} ,\,\,K_m=-(u+v)\sqrt{abc\over u^3v^3}.</math>
 
[[File:Ruled hyperboloid.jpg|thumb|right|A single-sheeted quadric [[hyperboloid]] which is a ruled surface in two different ways.]]
 
===Ruled surfaces===
{{main|Ruled surface}}
A [[ruled surface]] is one which can be generated by the motion of a straight line in '''E'''<sup>3</sup>.<ref>{{harvnb|Eisenhart|2004|pp=241–250}}; {{harvnb|do Carmo|1976|pp=188–197}}.</ref> 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''<sub>t</sub> and ''u'' satisfy
 
:<math>u\cdot v=0, \,\,\|u\|=1,\,\,\|v\|=1.</math>
 
The surface consists of points
 
:<math>c(t) + s\cdot u(t)</math>
 
as ''s'' and ''t'' vary.
 
Then, if
 
:<math>a=\|u_t\|, \,\, b=u_t\cdot v, \,\, \alpha=-b/a^2, \,\, \beta=(a^2-b^2)/a^2,</math>
 
the Gaussian and mean curvature are given by
 
:<math>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]^{3/2}}.</math>
 
The Gaussian curvature of the ruled surface vanishes if and only if ''u''<sub>''t''</sub> and ''v'' are proportional,<ref>{{harvnb|do Carmo|1976|p=194}}.</ref> This condition is equivalent to the surface being the [[Envelope (mathematics)#Envelope of a family of surfaces|envelope]] of the planes along the curve containing the tangent vector ''v'' and the orthogonal vector ''u'', i.e. to the surface being [[developable surface|developable]] along the curve.<ref>{{harvnb|Eisenhart|2004|pp=61–65}}.</ref> More generally a surface in '''E'''<sup>3</sup> has vanishing Gaussian curvature near a point if and only if it is developable near that point.<ref name="ref_">{{harvnb|Eisenhart|2004}}</ref> (An equivalent condition is given below in terms of the metric.)
 
===Minimal surfaces===
{{main|Minimal surface}}
In 1760 [[Lagrange]] extended Euler's results on the [[calculus of variations]] involving integrals in one variable to two variables.<ref>{{harvnb|Eisenhart|2004|pp=250–269}}; {{harvnb|do Carmo|1976|pp=197–213}}.</ref> He had in mind the following problem:
 
{{quotation| Given a closed curve in '''E'''<sup>3</sup>, 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 equation derived by Lagrange was equivalent to the vanishing of the mean curvature of the surface:
 
{{quotation|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 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 one of the first [[Fields medal]]s for this work in 1936).<ref>Douglas' solution is described in {{harvtxt|Courant|1950}}.</ref>
 
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 {{harvtxt|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''<sup>2</sup>.
 
[[File:Gaussian curvature.PNG|thumb|right|Surfaces with (from l. to r.) constant negative, zero and positive Gaussian curvature]]
 
===Surfaces of constant Gaussian curvature===
If a surface has constant Gaussian curvature, it is called a '''surface of constant curvature'''.<ref>{{harvnb|Eisenhart|2004|pp=270–291}}; {{harvnb|O'Neill|pp=249–251}}; {{harvnb|Hilbert|Cohn-Vossen|1952}}.</ref>
 
*The unit [[sphere]] in '''E'''<sup>3</sup> has constant Gaussian curvature +1.
*The Euclidean [[plane (mathematics)|plane]] and the [[cylinder (geometry)|cylinder]] both have constant Gaussian curvature 0.
*The surfaces of revolution with φ<sub>''tt''</sub> =&nbsp;φ have constant Gaussian curvature –1. Particular cases are obtained by taking φ(''t'') =&nbsp;''C''&nbsp;cosh&nbsp;''t'', ''C''&nbsp;sinh&nbsp;''t'' and ''C''&nbsp;''e''<sup>''t''</sup>.<ref>{{harvnb|O'Neill|pp=249–251}}; {{harvnb|do Carmo|pp=168–170}}; {{harvnb|Gray|Abbena|Salamon|2006}}.</ref> The latter case is the classical [[pseudosphere]] generated by rotating a [[tractrix]] around a central axis. In 1868 [[Eugenio Beltrami|Beltrami]] showed that the geometry of the pseudosphere was directly related to that of the [[hyperbolic geometry|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.<ref>{{harvnb|Stillwell|1996|pp=1–5}}.</ref> 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'',&nbsp;''y'',&nbsp;''z'')&nbsp;=&nbsp;&minus;1 in three-dimensional [[Minkowski space]], where ''q''(''x'', ''y'', ''z'') = ''x''<sup>2</sup> + ''y''<sup>2</sup> – ''z''<sup>2</sup>.<ref>{{harvnb|Wilson|2008}}.</ref>
 
Each of these surfaces of constant curvature has a [[Sharply multiply transitive|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 [[Henri Poincaré|Poincaré]]'s [[uniformization theorem]] (see below).
 
Other examples of surfaces with Gaussian curvature 0 include [[cone (geometry)|cones]], [[tangent developable]]s, and more generally any developable surface.
 
== Local metric structure ==
{{main|Riemannian manifold}}
[[File:Sphere with chart.svg|thumb|right|150px|A chart for the upper hemisphere of the 2-sphere obtained by projecting onto the ''x''-''y''-plane]]
[[File:Transition.gif|thumb|right|150px|Coordinate changes between different local charts must be smooth]]
For any surface embedded in Euclidean space of dimension 3 or higher, it is possible to measure the length of a curve on the surface, the angle between two curves and the area of a region on the surface. This structure is encoded infinitesimally in a '''[[Riemannian metric]]''' on the surface through ''line elements'' and ''area elements''. Classically in the nineteenth and early twentieth centuries only surfaces embedded in '''R'''<sup>3</sup> were considered and the metric was given as a 2×2 [[positive definite matrix]] varying smoothly from point to point in a local parametrization of the surface. The idea of local parametrization and change of coordinate was later formalized through the current abstract notion of a [[manifold]], a topological space where the [[smooth structure]] is given by local charts on the manifold, exactly as the planet [[Earth]] is mapped by [[atlas]]es today. Changes of coordinates between different charts of the same region are required to be smooth. Just as contour lines on real-life maps encode changes in elevation, taking into account local distortions of the Earth's surface to calculate true distances, so the Riemannian metric describes distances and areas "in the small" in each local chart. In each local chart a Riemannian metric is given by smoothly assigning a 2×2 positive definite matrix to each point; when a different chart is taken, the matrix is transformed according to the [[Jacobian matrix]] of the coordinate change. The manifold then has the structure of a 2-dimensional [[Riemannian manifold]].
 
===Line and area elements===
Taking a local chart, for example by projecting onto the ''x''-''y'' plane (''z'' = 0), the line element ''ds'' and the area element ''dA'' can be written in terms of local coordinates as
 
:''ds''<sup>2</sup> = ''E'' ''dx''<sup>2</sup> + 2''F'' ''dx'' ''dy'' + ''G'' ''dy''<sup>2</sup>
 
and
 
:''dA'' = (''EG'' − ''F''<sup>2</sup>)<sup>1/2</sup> ''dx'' ''dy''.
 
The expression ''E'' ''dx''<sup>2</sup> + 2''F'' ''dx'' ''dy'' + ''G'' ''dy''<sup>2</sup> is called the '''first fundamental form'''.<ref name="ref_" />
 
The matrix
 
:<math>\begin{pmatrix}
E(x,y) & F(x,y) \\
F(x,y) & G(x,y)\end{pmatrix}</math>
 
is required to be [[positive-definite matrix|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 manifold|Riemannian 2-manifold]] in a local chart.
 
===Second fundamental form===
{{main|Second fundamental form}}
The extrinsic geometry of surfaces studies the properties of surfaces embedded into a Euclidean space, typically '''E'''<sup>3</sup>.  In intrinsic geometry, two surfaces are "the same" if it is possible to unfold one surface onto the other without stretching it, i.e. a map of one surface onto the other preserving distance. Thus a cylinder is locally "the same" as the plane.  In extrinsic geometry, two surfaces are "the same" if they are [[congruence (geometry)|congruent]] in the ambient Euclidean space, i.e. there is an [[isometry]] of '''E'''<sup>3</sup> carrying one surface onto the other. With this more rigid definition of similitude, the cylinder 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 '''E'''<sup>3</sup> (or a higher dimensional Euclidean space). The most important example is the '''second fundamental form''', defined classically as follows.<ref>{{harvnb|Eisenhart|2004|pp=114–115}}; {{harvnb|Pressley|2001|pp=123–124}}; {{harvnb|Wilson|2008|pp=123–124}}.</ref>
[[File:Secondfundamentalform.jpg|500px|thumb|right|Definition of second fundamental form]]
Take a point (''x'',&nbsp;''y'') on the surface in a local chart. The Euclidean distance from a nearby point (''x''&nbsp;+&nbsp;''dx'',&nbsp;''y''&nbsp;+&nbsp;''dy'') to the tangent plane at (''x'', ''y''), i.e. the length of the perpendicular dropped from the nearby point to the tangent plane, has the form
 
:''e'' ''dx''<sup>2</sup> + 2''f'' ''dx'' ''dy'' + ''g'' ''dy''<sup>2</sup>
 
plus third and higher order corrections. The above expression, a symmetric bilinear form at each point, is the second fundamental form.
It is described by a 2&nbsp;×&nbsp;2 [[symmetric matrix]]
 
:<math>\begin{pmatrix}
e(x,y) & f(x,y) \\
f(x,y) & g(x,y)\end{pmatrix}</math>
 
which depends smoothly on ''x'' and ''y''. The Gaussian curvature can be calculated as the ratio of the determinants of the second and first fundamental forms:
 
:<math>K={eg-f^2\over EG-F^2}</math>
 
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]]''' ''K''<sub>''m''</sub> defined as the ''sum'' of the principal curvatures. It is given by the formula<ref name="ref_" />
 
:<math>K_m={eG + gE - 2 fF\over EG -F^2}</math>
 
The coefficients of the first and second fundamental forms satisfy certain compatibility conditions known as the [[Gauss-Codazzi equations]];
they involve the [[Christoffel symbols]] <math>\Gamma_{ij}^k</math>  associated with the first fundamental form:<ref>{{harvnb|Eisenhart|2004|p=156}}</ref>
 
:<math>e_y-f_x=e\Gamma_{12}^1 + f(\Gamma_{12}^2-\Gamma_{11}^1) - g\Gamma_{11}^2</math>
:<math>f_y-g_x=e\Gamma_{22}^1 + f(\Gamma_{22}^2-\Gamma_{12}^1) - g\Gamma_{12}^2.</math>
 
These equations can also be [[Riemannian connection on a surface#Gauss-Codazzi equations|succinctly expressed and derived]] in the language of [[connection form]]s due to [[Élie Cartan]].<ref>{{harvnb|O'Neill|1997|p=257}}</ref> [[Pierre Ossian Bonnet|Pierre Bonnet]] proved that two quadratic forms satisfying the Gauss-Codazzi equations always uniquely determine an embedded surface locally.<ref>{{harvnb|do Carmo|1976|pp=309–314}}</ref> For this reason the Gauss-Codazzi equations are often called the fundamental equations for embedded surfaces, precisely identifying where the intrinsic and extrinsic curvatures come from. They admit generalizations to surfaces embedded in more general [[Riemannian manifold]]s.
 
===Shape operator===
{{further|Peterson operator}}
[[File:Wilhelm Blaschke.jpg|thumb|right|130px|Wilhelm Blaschke (1885-1962)]]
The [[differential of a function|differential]] ''df'' of the [[Gauss map]] ''f'' can be used to define a type of extrinsic curvature, known as the '''shape operator'''<ref>{{harvnb|O'Neill|1997|pp=195–216}}; {{harvnb|do Carmo|1976|pp=134–153}}; {{harvnb|Singer|Thorpe|1967|pp=216–224}}.</ref> or Weingarten map.  This operator first appeared implicitly in the work of [[Wilhelm Blaschke]] and later explicitly in a treatise by Burali-Forti and Burgati.<ref>{{harvnb|Gray|Abbena|Salamon|2006|p=386}}.</ref>  Since at each point ''x'' of the surface, the tangent space is an [[inner product space]], the shape operator ''S''<sub>''x''</sub> can be defined as a linear operator on this space by the formula
 
:<math> (S_x v, w) =(df(v), w)</math>
 
for tangent vectors ''v'', ''w'' (the inner product makes sense because ''df''(''v'') and ''w'' both lie in '''E'''<sup>3</sup>).<ref>Note that in some more recent texts the symmetric bilinear form on the right hand side is referred to as the second fundamental form; however, it does not in general correspond to the classically defined second fundamental form.</ref>  The right hand side is symmetric in ''v'' and ''w'', so the shape operator is [[symmetric matrix|self-adjoint]] on the tangent space.  The eigenvalues of ''S''<sub>''x''</sub> are just the principal curvatures ''k''<sub>1</sub> and ''k''<sub>2</sub> at ''x''.  In particular the [[determinant]] of the shape operator at a point is the Gaussian curvature, but it also contains other information, since the [[mean curvature]] is half the [[trace (linear algebra)|trace]] of the shape operator.  The mean curvature is an extrinsic invariant.  In intrinsic geometry, a cylinder is developable, meaning that every piece of it is intrinsically indistinguishable from a piece of a plane since its Gauss curvature vanishes identically.  Its mean curvature is not zero, though; hence extrinsically it is different from a plane.
 
In general, the [[Eigenvalue, eigenvector and eigenspace|eigenvectors and eigenvalues]] of the shape operator at each point determine the directions in which the surface bends at each point.  The eigenvalues correspond to the [[principal curvatures]] of the surface and the eigenvectors are the corresponding principal directions.  The principal directions specify the directions that a curve embedded in the surface must travel to have maximum and minimum curvature, these being given by the principal curvatures.
 
The shape operator is given in terms of the components of the first and second fundamental forms by the [[Weingarten equations]]:<ref>{{harvnb|Gray|Abbena|Salamon|2006|p=394}}.</ref>
 
:<math>S= (EG-F^2)^{-1}\begin{pmatrix}
eG-fF& fG-gF \\
fE-eF & gE- fF\end{pmatrix}.</math>
 
==Geodesic curves on a surface==
Curves on a surface which minimize length between the endpoints are called [[geodesic]]s; they are the shape that an [[elastic band]] stretched between the two points would take. Mathematically they are described using [[partial differential equation]]s from the [[calculus of variations]]. The differential geometry of surfaces revolves around the study of geodesics. It is still an open question whether every Riemannian metric on a 2-dimensional local chart arises from an embedding in 3-dimensional Euclidean space: the theory of geodesics has been used to show this is true in the important case when the components of the metric are [[analytic function|analytic]].
 
===Geodesics===
[[File:Spherical triangle.svg|thumb|right|150px|A geodesic triangle on the sphere.
The geodesics are [[great circle]] arcs.]]
 
Given a piecewise smooth path ''c''(''t'') = (''x''(''t''), ''y''(''t''))  in the chart for ''t'' in [''a'', ''b''], its ''length'' is defined by
 
:<math> L(c) = \int_a^b (E\dot{x}^2 + 2F \dot{x}\dot{y} + G \dot{y}^2)^{1/2}\, dt </math>
 
and ''energy'' by
 
:<math> E(c) = \int_a^b (E\dot{x}^2 + 2F \dot{x}\dot{y} + G \dot{y}^2)\, dt. </math>
 
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]]
:<math>\ddot{x}</math> + Γ¹<sub>11</sub><math>\dot{x}</math>² + 2Γ¹<sub>12</sub> <math> \dot{x}\dot{y}</math> + Γ¹<sub>22</sub> <math>\dot{y}</math>² =0 and <math>\ddot{y}</math> + Γ²<sub>11</sub><math>\dot{x}</math>² + 2Γ²<sub>12</sub> <math> \dot{x}\dot{y}</math> + Γ²<sub>22</sub> <math>\dot{y}</math>² =0
where the [[Christoffel symbols]] Γ<sup>''k''</sup><sub>''ij''</sub> are given by
:Γ<sup>''k''</sup><sub>''ij''</sub> =<math>{1\over 2}\sum_m</math> ''g''<sup> ''km''</sup> (<math>\partial</math><sub>''j''</sub> ''g''<sub>''im''</sub> +<math>\partial</math><sub>''i''</sub> ''g''<sub>''jm''</sub> – <math>\partial</math><sub>''m''</sub> ''g''<sub>''ij''</sub>)
where ''g''<sub>11</sub> = ''E'', ''g''<sub>12</sub>=''F'', ''g''<sub>22</sub> =''G'' and (''g''<sup>''ij''</sup>) is the inverse matrix to (''g''<sub>''ij''</sub>). 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.<ref>{{harvnb|Berger|2004}}; {{harvnb|Wilson|2008}}; {{harvnb|Milnor|1963}}.</ref>
 
===Geodesic curvature===
{{See also|Geodesic curvature|Darboux frame}}
The '''geodesic curvature''' <math>k_g</math> at a point of a curve ''c''(''t''), parametrised by arc length, on an oriented surface is defined to be<ref>{{harvnb|Eisenhart|2002|p=131}}; {{harvnb|Berger|2004|p=39}}; {{harvnb|do Carmo|1976|p=248}}; {{harvnb|O'Neill|1997|p=237}}</ref>
 
:<math>k_g= \ddot{c}(t)\cdot \mathbf{n}(t).</math>
 
where '''n'''(''t'') is the "principal" unit normal to the curve in the surface, constructed by rotating the unit tangent vector <math>\dot{c}(t)</math> 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 the curve.
*A unit speed curve ''c''(''t'') in an embedded surface is a geodesic  if and only if its acceleration vector <math>\ddot{c}(t)</math> 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 {{harvtxt|Jacobowitz|1972}} and {{harvtxt|Poznjak|1973}} shows that every metric structure on a surface arises from a local embedding in '''E'''<sup>4</sup>.
Apart from some special cases, whether this is possible in '''E'''<sup>3</sup> remains an open question, the so-called "Weyl problem".<ref>{{harvnb|Han|Hong|2006}}</ref> In 1926 [[Maurice Janet]] proved that it is always possible locally if ''E'', ''F'' and ''G'' are [[Analytic function|analytic]]; soon afterwards [[Élie Cartan]] generalised this to local embeddings of [[Riemannian manifold|Riemannian ''n''-manifolds]] in '''E'''<sup>''m''</sup> where ''m'' = ½(''n''² +''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''<sub>''x''</sub> • ''u''<sub>''x''</sub> =1, ''u''<sub>''x''</sub> • ''u''<sub>''y''</sub> = 0, ''u''<sub>''y''</sub> • ''u''<sub>''y''</sub> = G.
Differentiating gives the three additional equations
:''u''<sub>''xx''</sub> • ''u''<sub>''y''</sub> = 0, ''u''<sub>''xx''</sub> • ''u''<sub>''x''</sub> = 0, ''u''<sub>''xx''</sub> • ''u''<sub>''yy''</sub> = ''u''<sub>''xy''</sub> • ''u''<sub>''x y''</sub> - ½ G<sub>''xx''</sub>
with ''u''(0,''y'') and ''u''<sub>''x''</sub>(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.
[[File:Tangency Example 2.svg|thumb|right|In orthogonal coordinates φ is the angle the tangent ''L'' to the geodesic ''C'' makes with the ''x''-axis]]
 
===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'')<sup>½</sup>, then the Gaussian curvature is given by<ref>{{harvnb|Eisenhart|2004}}; {{harvnb|Taylor|1996a}}, Appendix C.</ref>
 
:<math> K=-{1\over 2H} \left[\partial_x(G_x/H) +\partial_y(E_y/H)\right].</math>
 
If in addition ''E''=1, so that ''H''=''G''<sup>½</sup>, then the angle <math>\varphi</math> at the intersection between geodesic (''x''(''t''),''y''(''t'')) and the line ''y'' = constant is given by the equation
:<math>\tan \varphi = H\cdot \dot{y}/\dot{x}.</math>
The derivative of <math>\varphi</math> is given by a classical derivative formula of Gauss:<ref>{{harvnb|Eisenhart|2004}}; {{harvnb|Berger|2004}}.</ref>
 
:<math> \dot{\varphi} = -H_x \cdot \dot{y}.</math>
 
== Geodesic polar coordinates ==
[[File:Carl Jacobi.jpg|thumb|right|[[Carl Gustav Jakob Jacobi|Carl Jacobi]] (1804&ndash;1851)]]
[[File:GaussJacobi.jpg|thumb|right|Contour lines tracking the motion of points on a fixed curve moving along geodesics towards a basepoint]]
Once a metric is given on a surface and a base point is fixed, there is a unique geodesic connecting the base point to each sufficiently nearby point. The direction of the geodesic at the base point and the distance uniquely determine the other endpoint. These two bits of data, a direction and a magnitude, thus determine a tangent vector at the base point. The map from tangent vectors to endpoints smoothly sweeps out a neighbourhood 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 Euclidean space, namely any two points in it are joined by a unique geodesic. This property is called "geodesic convexity" and the coordinates are called "normal coordinates". The explicit calculation of normal coordinates can be accomplished by considering the differential equation satisfied by geodesics. The convexity properties are consequences of [[Gauss's lemma (Riemannian geometry)|Gauss's lemma]] and its generalisations. Roughly speaking this lemma states that geodesics starting at the base point must cut the spheres of fixed radius centred on the base point at right angles. Geodesic polar coordinates are obtained by combining the exponential map with polar coordinates on tangent vectors at the base point.
The Gaussian curvature of the surface is then given by the second order deviation of the metric at the point from the Euclidean metric. 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 by Gauss and later generalized by Jacobi, arising from the change of normal coordinates about two different points. The Gauss&ndash;Jacobi equation provides another way of computing the Gaussian curvature. Geometrically it explains what 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.<ref>
{{harvnb|doCarmo|1976|p=357}}</ref> One and a quarter centuries after Gauss and Jacobi, [[Marston Morse]] gave a more conceptual interpretation of the Jacobi field in terms of second derivatives of the energy function on the infinite-dimensional [[Hilbert manifold]] of paths.<ref>{{harvnb|Milnor|1963}}</ref>
 
===Exponential map===
{{main|Normal coordinates}}
The theory of [[ordinary differential equation]]s shows that if ''f''(''t'', ''v'')  is smooth then the differential equation
''dv''/''dt'' = ''f''(''t'',''v'') with initial condition ''v''(0) = v<sub>0</sub> has a unique solution for ''|t|'' sufficiently small and the solution depends smoothly
on ''t'' and ''v''<sub>0</sub>. This implies that for sufficiently small [[tangent vector]]s ''v'' at a given point ''p'' = (''x''<sub>0</sub>,''y''<sub>0</sub>), there is a geodesic ''c''<sub>''v''</sub>(''t'') defined on (&minus;2,2) with ''c''<sub>''v''</sub>(0) =  (''x''<sub>0</sub>,''y''<sub>0</sub>) and <math>\dot{c}</math><sub>''v''</sub>(0) = ''v''. Moreover if |''s''| ≤ 1, then ''c''<sub>''sv''</sub> = ''c''<sub>''v''</sub>(''st''). The '''[[exponential map#Riemannian geometry|exponential map]]''' is defined by
:exp<sub>''p''</sub>(''v'') = ''c''<sub>''v''</sub> (1)
and gives a diffeomorphism between a disc ||''v''|| < δ and a neighbourhood of ''p''; more generally the map sending (''p'',''v'') to exp<sub>''p''</sub>(''v'') gives a local diffeomorphism onto a neighbourhood of (''p'',''p''). The exponential map gives '''[[geodesic normal coordinates]]''' near ''p''.<ref>{{harvnb|Wilson|2008}}</ref>
 
===Computation of normal coordinates===
There is a standard technique (see for example {{harvtxt|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.
 
===Gauss's lemma===
{{main|Gauss's lemma (Riemannian geometry)}}
[[File:Polar coordinates.PNG|thumb|right|200px|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)<sup>½</sup> times the angle they subtend.]]
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's lemma (Riemannian geometry)|Gauss lemma]]. Geometrically it states that
 
:{| border="1" cellspacing="0" cellpadding="5"
|'''''the geodesics through 0 cut the circles centred at 0 [[orthogonal]]ly.'''''
|}
 
Taking [[polar coordinates]] (''r'',θ), it follows that the metric has the form
:''ds''<sup>2</sup> = ''dr''<sup>2</sup> + ''G''(''r'',θ) ''d''θ<sup>2</sup>.
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''.<ref name="harvnb|Berger|2004"/>
 
===Theorema Egregium===
{{main|Theorema Egregium}}
Taking ''x'' and ''y'' coordinates of a surface in '''E'''<sup>3</sup> corresponding to ''F''(''x'',''y'') =
''k''<sub>1</sub> ''x''<sup>2</sup>  + ''k''<sub>2</sub> ''y''<sup>2</sup> + ···, the power series expansion of the metric is given in normal coordinates (''u'', ''v'') as
 
:''ds''<sup>2</sup> = ''du''<sup>2</sup> + ''dv''<sup>2</sup> +  ''K''(''u dv'' –  ''v du'')<sup>2</sup> + ···
 
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 '''E'''³ and unchanged under coordinate transformations. In particular isometries of surfaces preserve Gaussian curvature.<ref name="harvnb|Berger|2004" />
 
===Gauss&ndash;Jacobi equation===
{{main|Jacobi field}}
 
Taking a coordinate change from normal coordinates at ''p'' to normal coordinates at a nearby point ''q'', yields the [[Sturm&ndash;Liouville equation]] satisfied by ''H''(''r'',θ) = ''G''(''r'',θ)<sup>½</sup>, discovered by Gauss and [[Jacobi field|later generalised]] by [[Carl Gustav Jakob Jacobi|Jacobi]],
 
:{| border="1" cellspacing="0" cellpadding="5"
|''H''<sub>''rr''</sub>  =  – ''K H''
|}
 
The [[Jacobian matrix|Jacobian]] of this coordinate change at ''q'' is equal to ''H''<sub>''r''</sub>. 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, the Gauss&ndash;Jacobi equation shows that the Gaussian curvature measures the spreading of geodesics on a geometric surface as they move away from a point.<ref>{{harvnb|O'Neill|1997|p=395}}</ref>
 
===Laplace&ndash;Beltrami operator===
On a surface with local metric
 
:<math> ds^2 = E \, dx^2 + 2F \, dx \, dy + G \, dy^2 </math>
 
and [[Laplace&ndash;Beltrami operator]]
 
:<math>\Delta f = {1\over H} (\partial_x {G\over H} \partial_x f - \partial_x {F\over H}\partial_y f -\partial_y {F\over H}\partial_x f + \partial_y {E\over H}\partial_yf),</math>
 
where ''H''<sup>2</sup> =&nbsp;''EG''&nbsp;–&nbsp;''F ''<sup>2</sup>, the Gaussian curvature at a point is given by the formula<ref>{{harvnb|Helgason|1978|p=92}}</ref>
 
:<math> K=- 3 \lim_{r\rightarrow 0} \Delta (\log r),</math>
 
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 intrinsic invariant.
 
In [[isothermal coordinates]], first considered by Gauss, the metric is required to be of the special form
 
:<math>ds^2 = e^\varphi (dx^2+dy^2). \, </math>
 
In this case the Laplace&ndash;Beltrami operator is given by
 
:<math>\Delta = e^{-\varphi} \left(\frac{\partial^2 }{\partial x^2} + \frac{\partial^2 }{\partial y^2}\right)</math>
 
and φ satisfies [[Liouville's equation]]<ref>{{harvnb|O'Niell|1997|p=286}}</ref>
 
:<math>\Delta \varphi=-2K. \, </math>
 
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 equation]]s.<ref>{{harvnb|do Carmo|1976|p=227}}</ref> There is an elementary proof for minimal surfaces.<ref>{{harvnb|Osserman|2002|pp=31&ndash;32}}</ref>
 
== Gauss–Bonnet theorem ==
[[File:Toroidal polyhedron.gif|thumb|right|A triangulation of the [[torus]]]]
[[File:Spherical triangle 3d opti.png|thumb|right|The area of a [[spherical triangle]] on the unit sphere is α + β + γ - π.]]
On a [[sphere]] or a [[hyperboloid]], the area of a geodesic triangle, i.e. a triangle all the sides of which are geodesics, is proportional to the difference of the sum of the interior 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 fact that its Gaussian curvature is zero. These are standard results in spherical, hyperbolic and high school trigonometry (see below). Gauss generalised these results to an arbitrary surface by showing that the integral of the Gaussian curvature over the interior of a geodesic triangle is also equal to this angle difference or excess. His formula showed that the Gaussian curvature could be calculated near a point as the limit of area over angle excess for geodesic triangles shrinking to the point. Since any closed surface can be decomposed up into geodesic triangles, the formula could also be used to compute the integral of the curvature over the whole surface. As a special case of what is now called the [[Gauss-Bonnet theorem]], Gauss proved that this integral was remarkably always 2π times an integer, a topological invariant of the surface called the [[Euler characteristic]]. This invariant is easy to compute combinatorially in terms of the number of vertices, edges, and faces of the triangles in the decomposition, also called a [[triangulation (topology)|triangulation]]. This interaction between analysis and topology was the forerunner of many later results in geometry, culminating in the [[Atiyah-Singer index theorem]]. In particular properties of the curvature impose restrictions on the 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
 
:{| border="1" cellspacing="0" cellpadding="5"
|<math>\int</math><sub>Δ</sub> ''K'' ''dA'' =  α + β + γ &minus; π.
|}
 
In fact taking geodesic polar coordinates with origin ''A'' and ''AB'', ''AC'' the radii at polar angles 0 and α
 
:<math>\int</math><sub>Δ</sub> ''K''&nbsp;''dA'' =<math>\int</math><sub>Δ</sub> ''KH'' ''dr'' ''d''θ  =  – <math>\int_0^\alpha \int_0^{r_\theta}</math> ''H''<sub>''rr''</sub> ''dr'' ''d''θ = <math>\int_0^\alpha</math> 1 &minus; ''H''<sub>''r''</sub>(''r''<sub>θ</sub>,θ) ''d''θ =  <math>\int_0^\alpha</math> ''d''θ + <math>\int_{\pi-\beta}^\gamma</math> ''d''φ = α + β + γ &minus; π,
 
where the second equality follows from the Gauss&ndash;Jacobi equation and the fourth from Gauss' derivative formula in the 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 &minus; 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.<ref name="ref_" />
 
===Gauss-Bonnet theorem===
[[File:Icosahedron.svg|thumb|right|The Euler characteristic of a sphere, triangulated like an [[icosahedron]], is ''V'' - ''E'' + ''F'' = 12 - 30 + 20 =2.]]
{{Main|Gauss-Bonnet theorem}}
Since every compact oriented 2-manifold ''M'' can be [[triangulation (topology)|triangulated]] by small geodesic triangles, it follows that
 
:{| border="1" cellspacing="0" cellpadding="5"
|<math> \int</math><sub>''M''</sub> 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 3''F'' = 2''E'' and the left hand side equals  2π·''V'' – π·''F'' = 2π·(''V'' – ''E'' + ''F'') =  2π·χ(''M''). </sub>
 
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 [[index (mathematics)|indices]]{{Disambiguation needed|date=February 2012}} 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.)<ref name="ref_" />
 
===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 '''S'''<sup>2</sup>. If in addition the surface is isometrically embedded in '''E'''<sup>3</sup>, the Gauss map provides an explicit diffeomorphism. As [[Jacques Hadamard|Hadamard]] observed, in this case the surface is [[Convex set|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. [[David Hilbert|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 '''E'''<sup>3</sup>; however, as [[Adriano Garsia]] showed using the [[Isothermal coordinates#Beltrami equation|Beltrami equation]] for [[quasiconformal mapping]]s, this is always possible for some [[conformal equivalence|conformally equivalent]] metric.<ref name="Garsia1961">{{harvnb|Singer|Thorpe|1967}}; {{citation|last=Garsia|first=Adriano M.|title=An imbedding of closed Riemann surfaces in Euclidean space|journal=Comment. Math. Helv.|volume= 35|year= 1961|pages= 93–110|doi=10.1007/BF02567009}}</ref>
 
==Surfaces of constant curvature==
The [[simply connected]] surfaces of constant curvature  0, +1 and –1 are the Euclidean plane, the unit sphere in '''E'''<sup>3</sup>, and the [[hyperbolic geometry|hyperbolic plane]]. Each of these has a transitive three-dimensional [[Lie group]] of orientation preserving [[isometry|isometries]] ''G'', which can be used to study their geometry.  Each of the two non-compact surfaces can be identified with the quotient ''G'' / ''K'' where ''K'' is a [[maximal compact subgroup]] of ''G''.  Here ''K'' is isomorphic to [[circle group|SO(2)]]. Any other closed Riemannian 2-manifold ''M'' of constant Gaussian curvature, after scaling the metric by a constant factor if necessary, will have one of these three surfaces as its [[universal covering space]].  In the orientable case, the [[fundamental group]] Γ of ''M'' can be identified with a [[torsion (algebra)|torsion-free]] [[lattice (discrete subgroup)|uniform subgroup]] of ''G'' and ''M'' can then be identified with the [[double coset|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 '''R'''<sup>2</sup> by discrete rank 2 subgroups.  For closed surfaces of [[genus]] <math>g\geq 2</math>, the [[moduli space]] of Riemann surfaces obtained as Γ varies over all such subgroups, has real dimension 6''g'' - 6 .<ref>{{harvnb|Imayoshi|Taniguchi|1992|pp=47–49}}</ref> By Poincaré's [[uniformization theorem]], any orientable closed 2-manifold is [[conformal equivalence|conformally equivalent]] to a surface of constant curvature 0, +1 or –1. In other words, by multiplying the metric by a positive scaling factor, the Gaussian curvature can be made to take exactly one of these values (the sign of the [[Euler characteristic]] of ''M'').<ref>{{harvnb|Berger|1977}}; {{harvnb|Taylor|1996}}.</ref>
 
===Euclidean geometry===
[[File:Triangle with notations 2.svg|thumb|right|A triangle in the plane]]
In the case of the Euclidean plane, the symmetry group is the [[Euclidean group|Euclidean motion group]], the [[semidirect product]] of
the two dimensional group of translations by the group of rotations.<ref>{{harvnb|Wilson|2008|pp=1–23}}, Chapter I, ''Euclidean geometry''.</ref> Geodesics are straight lines and the geometry is encoded in the elementary formulas of [[trigonometry]], such as the [[cosine rule]] for a triangle with sides ''a'', ''b'', ''c'' and angles α, β, γ:
 
:<math> c^2 = a^2 +b^2 -2ab \,\cos \gamma.</math>
 
Flat tori can be obtained by taking the quotient of '''R'''<sup>2</sup> by a [[lattice (mathematics)|lattice]]{{Disambiguation needed|date=April 2012}}, i.e. a free Abelian subgroup of rank 2. These closed surfaces have no isometric embeddings in '''E'''<sup>3</sup>. They do nevertheless  admit isometric embeddings in '''E'''<sup>4</sup>; in the easiest case this follows from the fact that the torus is a product of two circles and each circle can be isometrically embedded in '''E'''<sup>2</sup>.<ref>{{harvnb|do Carmo|1976}}.</ref>
 
===Spherical geometry===
{{See also|spherical trigonometry|spherical triangle}}
[[File:RechtwKugeldreieck.svg|frame|right|A spherical triangle]]
The isometry group of the unit sphere ''S''<sup>2</sup> in '''E'''<sup>3</sup> is the orthogonal group [[O(3)]], with the [[rotation group SO(3)]] as the subgroup of isometries preserving orientation. It is the direct product of SO(3) with the [[antipodal map]], sending ''x'' to –''x''.<ref>{{harvnb|Wilson|2008|pp=25–49}}, Chapter II, ''Spherical geometry''.</ref> The group SO(3) acts transitively on ''S''<sup>2</sup>. The [[stabilizer subgroup]] of the unit vector (0,0,1) can be identified with SO(2), so that ''S''<sup>2</sup> = SO(3)/SO(2).
 
The geodesics between two points on the sphere are the [[great circle]] arcs with these given endpoints. If the points are not antipodal, there is a unique shortest geodesic between the points. The geodesics can also be described group theoretically: each geodesic through the North pole (0,0,1) is the orbit of the subgroup of rotations about an axis through antipodal points on the equator.
 
A '''[[spherical triangle]]''' is a geodesic triangle on the sphere. It is defined by points ''A'', ''B'', ''C'' on the sphere with sides ''BC'', ''CA'', ''AB'' formed from great circle 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,
then the [[spherical triangle|spherical cosine law]] states that
 
:<math>\cos c  = \cos a \, \cos b + \sin a\, \sin b \,\cos \gamma.</math>
 
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 plane]] '''C''' <math>\cup</math> {∞}. The explicit map is given by
 
:<math>\pi(x,y,z)={x+iy\over 1-z}\equiv u + iv.</math>
 
Under this correspondence every rotation of ''S''<sup>2</sup> corresponds to a [[Möbius transformation]] in [[SU(2)]], unique up to sign.<ref>{{harvnb|Wilson| 2008}}, Chapter 2.</ref> With respect to the coordinates (''u'', ''v'') in the complex plane, the spherical metric becomes<ref>{{harvnb|Eisenhart|2004|p=110}}.</ref>
 
:<math> ds^2 = {4(du^2 + dv^2)\over (1+u^2+v^2)^2}.</math>
 
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 ''S''<sup>2</sup> by the antipodal map (multiplication by –1). The sphere is simply connected, while the real projective plane has fundamental group '''Z'''<sub>2</sub>.  The [[point groups in three dimensions|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 ''S''<sup>2</sup>, so the corresponding quotients are not 2-manifolds, just [[orbifold]]s.
 
[[File:Beltrami.jpg|thumb|right|120px|[[Eugenio Beltrami]] (1835-1899)]]
 
===Hyperbolic geometry===
{{See also|hyperbolic triangle|hyperbolic geometry}}
[[File:Felix Klein.jpeg|thumb|right|120px|[[Felix Klein]] (1849-1925)]]
[[File:Poincare.jpg|thumb|right|120px|[[Henri Poincaré]] (1854-1912)]]
[[File:Triangolo iperbolico.svg|thumb|right|A hyperbolic triangle in the Poincaré disk model]]
[[Non-Euclidean geometry]]<ref>{{harvnb|Stillwell|1990}}; {{harvnb|Bonola|Carslaw|Enriques|1955}}.</ref> was first discussed in letters of Gauss, who made extensive computations at the turn of the nineteenth century which, although privately circulated, he decided not to put into print. In 1830 [[Lobachevsky]] and independently in 1832 [[Bolyai]], the son of one Gauss' correspondents, published synthetic versions of this new geometry, for which they were severely criticized. However it was not until 1868 that Beltrami, followed by [[Felix Klein|Klein]] in 1871 and Poincaré in 1882, gave concrete analytic models for what Klein dubbed '''[[hyperbolic geometry]]'''. The four models of 2-dimensional hyperbolic geometry that emerged were:
 
*the [[Beltrami-Klein model]];
*the [[Poincaré disk]];
*the [[Upper half-plane|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 line segments (that is, intersections of Euclidean lines with the open unit disk).The last model has the advantage that it gives a construction which is completely parallel to that of the unit sphere in 3-dimensional Euclidean space.  Because of their application in complex analysis and geometry, however, the models of Poincaré are the most widely used: they are interchangeable thanks to the Möbius transformations between the disk and the upper half-plane.
 
Let
 
:<math>D=\{z\,\colon |z|<1\}</math>
 
be the [[Poincaré disk]] in the complex plane with [[Poincaré metric]]
 
:<math>ds^2= {4(dx^2 +dy^2)\over (1-x^2-y^2)^2}.</math>
 
In polar coordinates (''r'', θ) the metric is given by
 
:<math> ds^2= {4(dr^2 + r^2\, d\theta^2)\over (1-r^2)^2}.</math>
 
The length of a curve γ:[''a'',''b''] <math>\rightarrow</math> ''D'' is given by the formula
 
:<math>\ell(\gamma)=\int_a^b {2|\gamma^\prime(t)|\, dt\over 1 -|\gamma(t)|^2}.</math>
 
The group ''G'' = [[SL(2,R)|SU(1,1)]] given by
 
:<math>G=\left\{ \begin{pmatrix}
\alpha & \beta \\
\overline{\beta} & \overline{\alpha}
\end{pmatrix} : \alpha,\beta\in\mathbf{C},\,|\alpha|^2 -|\beta|^2=1 \right\}</math>
 
acts transitively by [[Möbius transformation]]s on ''D'' and the [[stabilizer subgroup]] of 0 is the rotation group
 
:<math> K=\left\{ \begin{pmatrix}
\zeta & 0 \\
0 & \overline{\zeta}
\end{pmatrix} : \zeta\in\mathbf{C},\,|\zeta| =1 \right\}.</math>
 
The quotient group SU(1,1)/±''I'' is the group of orientation-preserving isometries of ''D''. Any two points ''z'', ''w''  in ''D'' are joined by a unique geodesic, given by the portion of the circle or straight line passing through ''z'' and ''w'' and orthogonal to the boundary circle. The distance between ''z'' and ''w'' is given by
 
:<math>d(z,w)=2 \tanh^{-1} |z-w|/|1-\overline{w}z|.</math>
 
In particular ''d''(0,''r'') = 2 tanh<sup>−1</sup> ''r'' and ''c''(''t'') = tanh ''t''/2 is the geodesic through 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
 
:<math>\cosh c = \cosh a\, \cosh b - \sinh a \,\sinh b \,\cos \gamma.</math>
 
The area of the hyperbolic triangle is given by<ref>{{harvnb|Wilson|2008}}, Chapter 5.</ref>
 
:Area = π – α – β – γ.
 
The unit disk and the upper half-plane
 
:<math>H=\{w=x+iy \,\colon\, y >0\}</math>
 
are conformally equivalent by the Möbius transformations
 
:<math> w=i {1+z\over 1-z},\,\, z={w-i\over w+i}.</math>
 
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
 
:<math> ds^2 = {dx^2 + dy^2\over y^2}.</math>
 
Since lines or circles are preserved under Möbius transformations, geodesics are again described by lines or circles orthogonal to the real axis.
 
The unit disk with the Poincaré metric is the unique simply connected oriented 2-dimensional Riemannian manifold with 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
 
:<math> M= \Gamma\backslash G /K.</math>
 
In this case Γ is a [[finitely presented group]]. The generators and relations are encoded in a geodesically convex [[fundamental polygon#Fundamental polygon of a compact Riemann surface|fundamental 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 ''e''<sup>2''u''</sup>. The new Gaussian curvature ''K''' is then given by
 
:<math>K^\prime(x)= e^{-2u} (K(x) - \Delta u),</math>
 
where Δ is the Laplacian for the original metric. Thus to show that a given surface is conformally equivalent to a metric with constant curvature ''K'''
it suffices to solve the following variant of [[Liouville's equation]]:
 
:<math>\Delta u = K^\prime e^{2u} + K(x).</math>
 
When ''M'' has Euler characteristic 0, so is diffeomorphic to a [[torus]], ''K''' = 0, so this amounts to solving
 
:<math> \Delta u = K(x).</math>
 
By standard elliptic theory, this is possible because  the integral of ''K'' over ''M'' is zero, by the Gauss-Bonnet theorem.<ref>{{harvnb|Taylor|1996b|p=107}}; {{harvnb|Berger|1977|pp=341–343}}.</ref>
 
When ''M'' has negative Euler characteristic, ''K''' = -1, so the equation to be solved is:
 
:<math>\Delta u = -e^{2u} + K(x).</math>
 
Using the continuity of the exponential map on [[Sobolev space]] due to [[Neil Trudinger]], this non-linear equation can always be solved.<ref>{{harvnb|Berger|1977|pp=222–225}}; {{harvnb|Taylor|1996b|pp=101–108}}.</ref>
 
Finally in the case of the 2-sphere, ''K''' = 1 and the equation becomes:
 
:<math>\Delta u = e^{2u} + K(x).</math>
 
So far this non-linear equation has not been analysed directly, although classical results such as the [[Riemann-Roch theorem]] imply that it always has a solution. The method of [[Ricci flow]], developed by [[Richard Hamilton (professor)|Richard Hamilton]], gives another proof of existence based on non-linear [[partial differential equations]] to prove existence.<ref>{{harvnb|Chow|1991}}; {{harvnb|Taylor|1996b}}.</ref> In fact the Ricci flow on conformal metrics on ''S''<sup>2</sup> is defined on functions ''u''(''x'', ''t'') by
 
:<math> u_t = 4\pi -  K'(x,t) = 4\pi -e^{-2u} (K(x) - \Delta u). </math>
 
After finite time, Chow showed that ''K' '' becomes positive; previous results of Hamilton could then be used to show that ''K' '' converges to +1.<ref>
{{harvtxt|Chen|Lu| Tian|2006}} pointed out a missing step in the approach of Hamilton and Chow.</ref>
 
A simple proof using only elliptic operators discovered in 1988 can be found in {{harvtxt|Ding|2001}}. Let ''G'' be the [[Green's function]] on
''S''<sup>2</sup> satisfying  Δ''G'' = 1 + 4πδ<sub>''P''</sub>, where δ<sub>''P''</sub> is the point measure at  a fixed point ''P'' of ''S''<sup>2</sup>. The equation Δ''v'' = 2''K'' – 2, has a smooth solution ''v'', because the right hand side has integral 0 by the Gauss-Bonnet theorem. Thus φ = 2''G'' + ''v'' satisfies Δφ = 2''K'' away from ''P''. It follows that ''g''<sub>1</sub> = ''e''<sup>φ</sup>''g'' 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 ''e''<sup>2''u''</sup>''g'' has Gaussian curvature +1 on the complement of ''P''. The function ''u'' automatically extends to a smooth function on the whole of ''S''<sup>2</sup>.<ref>This follows by an argument involving a theorem of {{harvtxt|Sacks|Uhlenbeck|1981}} on removable singularities of [[harmonic map]]s of finite energy.</ref>
 
==Surfaces of non-positive curvature==
In a region where the curvature of the surface satisfies ''K''≤0, geodesic triangles satisfy the [[CAT(k) space|CAT(0)]] inequalities of '''comparison geometry''', studied by [[Élie Cartan|Cartan]], [[Aleksandr Danilovich Aleksandrov|Alexandrov]] and [[Victor Andreevich Toponogov|Toponogov]], and considered later from [[Bruhat-Tits building|a different point of view]] by [[François Bruhat|Bruhat]] and [[Jacques Tits|Tits]]; thanks to the vision of [[Mikhail Gromov (mathematician)|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]]. Many results known for smooth surfaces and their geodesics, such as Birkhoff's method of constructing geodesics by his curve-shortening process or van Mangoldt and Hadamard's theorem that a [[simply connected]] surface of non-positive curvature is homeomorphic to the plane, are equally valid in this more general setting.
 
===Alexandrov's comparison inequality===
[[File:Median.jpg|thumb|right|The [[median]] in the comparison triangle is always longer than the actual median.]]
The simplest form of the comparison inequality, first proved for surfaces by Alexandrov around 1940, states that
 
{{quotation|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 comparison 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''))<sup>2</sup> &minus; ''t''<sup>2</sup>
 
is a [[convex function]], i.e.
 
:<math>\ddot{f}(t) \ge 0.</math>
 
Taking geodesic polar coordinates with origin at ''a'' so that ||''c''(''t'')||  = ''r''(''t''), convexity is equivalent to
 
:<math> r\ddot{r} + \dot{r}^2 \ge 1.</math>
 
Changing to normal coordinates ''u'', ''v'' at ''c''(''t''), this inequality becomes
 
:''u''<sup>2</sup> + ''H''<sup> &minus; 1</sup> ''H''<sub>r</sub> ''v''<sup>2</sup> ≥ 1,
 
where (''u'',''v'') corresponds to the unit vector <math>\dot{c}(t)</math>.
This follows from the inequality ''H''<sub>''r''</sub> ≥ ''H'', a consequence of the non-negativity of the derivative of the [[Wronskian]] of ''H'' and ''r'' from [[Sturm&ndash;Liouville theory]].<ref name="Jost1997">{{harvnb|Berger|2004}}; {{citation|last=Jost|first=Jürgen|title= Nonpositive curvature: geometric and analytic aspects  |publisher=Birkhäuser|year=1997|isbn=0-8176-5736-3|series=Lectures in Mathematics, ETH Zurich}}</ref>
 
[[File:George David Birkhoff 1.jpg|thumb|right|[[George Birkhoff]] (1884-1944)]]
 
===Existence of geodesics===
On a complete curved surface any two points can be joined by a  geodesic. This is a special case of the [[Hopf-Rinow theorem]], which also
applies in higher dimensions. The completeness assumption is automatically fulfilled for a surface which is embedded as a closed subset of Euclidean space.
However,  it is no longer fulfilled if, for example, we remove an isolated point from a surface. For example, the complement of the origin in the Euclidean plane
is an example of a non-complete surface; in this example two points which are diametrically opposite across the origin cannot be
joined by a geodesic without leaving the punctured plan).
 
===Von Mangoldt-Hadamard theorem===
For closed surfaces of non-positive curvature, [[Hans Carl Friedrich von Mangoldt|von Mangoldt]] (1881) and [[Jacques Hadamard|Hadamard]] (1898) proved that the exponential map at a point is a covering map, so that the [[universal covering space]] of the manifold is '''E'''². This result was generalised to higher dimensions by [[Élie Cartan|Cartan]] and is usually referred to in this form as the '''[[Cartan–Hadamard theorem]]'''. For surfaces, this result follows from three important facts:<ref>{{harvnb|do Carmo|1976}}; {{harvnb|Berger|2004}}.</ref>
 
* The exponential map has ''non-zero Jacobian'' everywhere for non-positively curved surfaces, a consequence of the non-vanishing of ''H''<sub>''r''</sub>.
 
* 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 in a homotopy class are ''connected by a unique geodesic'' (see above).
 
==Riemannian connection and parallel transport==
{{main|Riemannian connection on a surface}}
[[File:Levi-Civita 1930.jpeg|thumb|right|130px|[[Tullio Levi-Civita]] (1873-1941)]]
 
The classical approach of Gauss to the differential geometry of surfaces was the standard elementary approach<ref>{{harvnb|Eisenhart|2004}}; {{harvnb|Kreyszig|1991}}; {{harvnb|Berger|2004}}; {{harvnb|Wilson|2008}}.</ref> which predated the emergence of the concepts of [[Riemannian manifold]] initiated by [[Bernhard Riemann]] in the mid-nineteenth century and of [[connection (mathematics)|connection]] developed by [[Tullio Levi-Civita]], [[Élie Cartan]] and [[Hermann Weyl]] in the early twentieth century. The notion of connection, [[covariant derivative]] and [[parallel transport]] gave a more conceptual and uniform way of understanding curvature, which not only allowed generalisations to higher dimensional manifolds but also provided an important tool for defining new geometric invariants, called [[characteristic classes]].<ref>{{harvnb|Kobayashi|Nomizu|1969}}, Chapter XII.</ref> The approach using covariant derivatives and connections is nowadays the one adopted in more advanced textbooks.<ref>{{harvnb|do Carmo|1976}}; {{harvnb|O'Neill|1997}}; {{harvnb|Singer|Thorpe|1967}}.</ref>
 
===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]]'''<ref name="ref_">{{harvnb|Levi-Civita|1917}}.</ref> is perhaps most easily understood in terms of lifting [[vector field]]s, considered as first order [[differential operator]]s acting on functions on the manifold, to differential operators on the [[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 <math>\mathbb{R}^3</math> can be regarded as a function from the surface into '''R'''<sup>3</sup>. Another vector field act as a differential operator component-wise. The resulting vector field will not be tangent to the surface, but this can be corrected 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 locally expressed in terms of the Christoffel symbols.
[[File:Parallel transport.png|200px|thumb|right|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.]]
 
===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]].<ref name="ref_" /> It is related to the earlier notion of covariant derivative, because it is the [[monodromy]] of the [[ordinary differential equation]] on the curve defined by the covariant derivative with respect to the velocity vector of the 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 and making 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.<ref name="harvnb|Berger|2004"/>
 
A vector field ''v''(''t'') along a unit speed curve ''c''(''t''), with geodesic curvature ''k''<sub>g</sub>(''t''), is said to be parallel along the curve if
 
* it has constant length
* the angle θ(''t'') that it makes with the velocity vector <math>\dot{c}(t)</math> satisfies
 
:<math>\dot{\theta}(t) = - k_g(t)</math>
 
This recaptures the rule for parallel transport along a geodesic or piecewise geodesic curve, because in that case ''k''<sub>g</sub> = 0, so  that the angle θ(''t'') should remain constant on any geodesic segment. The existence of parallel transport follows because θ(''t'') can be computed as the [[integral]] of the geodesic curvature. Since it therefore depends continuously on the L<sup>2</sup> norm of ''k''<sub>g</sub>, it follows that parallel transport for an arbitrary curve can be obtained as the limit of the parallel transport on approximating piecewise geodesic curves.<ref>{{harvnb|Arnold|1989|pp=301–306}}, Appendix I.; {{harvnb|Berger|2004|pp=263–264}}.</ref>
 
The connection can thus be described in terms of lifting paths in the manifold to paths in the tangent or orthonormal frame bundle, thus formalising the classical theory of the "[[moving frame]]", favoured by French authors.<ref>{{harvnb|Darboux|1887,1889,1896}}</ref> Lifts of loops about a point give rise to the [[holonomy group]] at that point. The Gaussian curvature at a point can be recovered from parallel transport around increasingly small loops at the point. Equivalently curvature can be calculated directly at an infinitesimal level in terms of [[Lie bracket of vector fields|Lie brackets]] of lifted vector fields.
[[File:Elie-Cartan-1904.png|thumb|[[Élie Cartan]] in 1904]]
 
===Connection 1-form===
The approach of Cartan and Weyl, using connection 1-forms on the [[frame bundle]] of ''M'', gives a third way to understand the Riemannian connection.
They noticed that parallel transport dictates that a path in the surface be lifted to a path in the frame bundle so that its tangent vectors lie in a
special subspace of codimension one in the three-dimensional tangent space of the frame bundle. The projection onto this subspace is defined by a differential 1-form on the orthonormal frame bundle, the '''[[connection form]]'''. This enabled the curvature properties of the surface to be encoded in [[differential forms]] on the frame bundle and formulas involving their [[exterior derivative]]s.
 
This approach is particularly simple for an embedded surface.  Thanks to a result of {{harvtxt|Kobayashi|1956}}, the connection 1-form on a surface embedded in Euclidean space ''E''<sup>3</sup> is just the pullback under the Gauss map of the connection 1-form on ''S''<sup>2</sup>.<ref>{{harvnb|Kobayashi|Nomizu|1969}}</ref>  Using the identification of ''S''<sup>2</sup> with the [[homogeneous space]] SO(3)/SO(2), the connection 1-form is just a component of the [[Maurer-Cartan form|Maurer-Cartan 1-form]] on SO(3).<ref>{{harvnb|Ivey|Landsberg|2003}}.</ref>
 
==Global differential geometry of surfaces==
Although the characterisation of curvature involves only the local geometry of a surface, there are important global aspects 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.<ref>{{harvnb|Berger|2004|pp=145–161}}; {{harvnb|do Carmo|1976}}; {{harvnb|Chern|1967}}; {{harvnb|Hopf|1989}}.</ref> These include:
 
*'''[[Injectivity radius]]''', defined as the largest ''r'' such that two points at a distance less than ''r'' are joined by a unique geodesic. Wilhelm Klingenberg proved in 1959 that the injectivity radius of a closed surface is bounded below by the minimum of <math>\delta=\pi/\sqrt{\sup K}</math> 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 positive Gaussian curvature is always bounded above by δ; in other words a geodesic realising the metric distance between two points cannot have length greater than δ.
 
*'''Rigidity'''. In 1927 [[Cohn-Vossen]] proved that two [[ovaloid]]s – closed surfaces with positive Gaussian curvature – that are isometric are necessarily congruent by an isometry of '''E'''<sup>3</sup>. Moreover a closed embedded surface with positive Gaussian curvature and constant mean curvature is necessarily a sphere; likewise a closed 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 topological assumption.  In the 1980s, Wente constructed [[immersion (mathematics)|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 point]]s. The first  work on this conjecture was in 1924 by [[Hans Hamburger]], who noted that it follows from the following stronger claim : the half-integer valued index of the principal curvature foliation of an isolated umbilic is at most one. The contribution of Hamburger and those of subsequent authors to proving this local conjecture are inconclusive.
 
*'''Zero Gaussian curvature''': a complete surface in '''E'''<sup>3</sup> with zero Gaussian curvature must be a cylinder or a plane.
 
*'''Hilbert's theorem''' (1901): no complete surface with constant negative curvature can be [[immersion (mathematics)|immersed]] isometrically in '''E'''<sup>3</sup>.
[[File:TorusSystoleLoop.png|right|thumb|200px|Shortest loop on a torus]]
*'''The [[Willmore conjecture]]'''. This conjecture states that the integral of the square of the mean curvature of a torus immersed in '''E'''<sup>3</sup> should be bounded below by 2 π<sup>2</sup>. The conjecture has been proved for large classes of torus immersions. It is also known that the integral is a conformal invariant.
 
*'''[[isoperimetric inequality|Isoperimetric inequalities]]'''. In 1939 Schmidt proved that the classical isoperimetric inequality for curves in the Euclidean plane is also valid on the sphere or in the hyperbolic plane: namely he showed that 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 '''E'''<sup>3</sup> arising as the boundary of a bounded domain of unit volume, the surface area is minimized for a Euclidean ball.
 
*'''[[Systoles of surfaces|Systolic inequalities for curves on surfaces]]'''. Given a closed surface, its [[systolic geometry|systole]] is defined to be the smallest length of any non-contractible closed curve on the surface.  In 1949 [[Loewner]] proved a [[Loewner's torus inequality|torus inequality]] for metrics on the torus, namely that the area of the torus over the square of its systole is bounded below by <math>\sqrt{3}/2</math>, with equality in the flat (constant curvature) case. A similar result is given by [[Pu's inequality|Pu's inequality for the real projective plane]] from 1952, with a lower bound of 2/π also attained in the constant curvature case. For the [[Klein bottle]], Blatter and Bavard later obtained a lower bound of <math> \sqrt{8}/\pi</math>. 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 (mathematician)|Mikhail Gromov]] found a lower bound given by a constant times ''g''<sup>1/2</sup>, although this is not optimal.  Asymptotically sharp upper and lower bounds given by constants times ''g''/(log ''g'')<sup>2</sup> are due to Gromov and Buser-Sarnak, and can be found in {{harvtxt|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 <math>1/2\sqrt{3}</math> in 1980: the best result so far is the lower bound of 1/8 obtained by Regina Rotman in 2006.<ref>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 and Sabourau.</ref>
 
==Reading guide==
One of the most comprehensive introductory surveys of the subject, charting the historical development from before Gauss to modern times, is by {{harvtxt|Berger|2004}}. Accounts of the classical theory are given in {{harvtxt|Eisenhart|2004}}, {{harvtxt|Kreyszig|1991}} and {{harvtxt|Struik|1988}}; the more modern copiously illustrated undergraduate textbooks by {{harvtxt|Gray|Abbena|Salamon|2006}}, {{harvtxt|Pressley|2001}} and {{harvtxt|Wilson|2008}} might be found more accessible.  An accessible account of the classical theory can be found in {{harvtxt|Hilbert|Cohn-Vossen|1952}}. More sophisticated graduate-level treatments using the [[Riemannian connection on a surface]] can be found in {{harvtxt|Singer|Thorpe|1967}}, {{harvtxt|do Carmo|1976}} and {{harvtxt|O'Neill|1997}}.
 
==See also==
*[[Zoll surface]]
 
== Notes ==
{{reflist|colwidth=30em}}
 
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{{curvature}}
 
{{DEFAULTSORT:Differential Geometry Of Surfaces}}
[[Category:Differential geometry of surfaces| ]]
 
{{link GA|fr}}
[[nl:Oppervlak (meetkunde)]]

Latest revision as of 22:22, 15 April 2014


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