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In the part of mathematics referred to as topology, a surface is a two-dimensional
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
. Some surfaces arise as the
boundaries Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space. Topological surfaces are sometimes equipped with additional information, such as a Riemannian metric or a complex structure, that connects them to other disciplines within mathematics, such as differential geometry and complex analysis. The various mathematical notions of surface can be used to model surfaces in the physical world.


In general

In mathematics, a surface is a geometrical shape that resembles a deformed
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
. The most familiar examples arise as boundaries of solid objects in ordinary three-dimensional Euclidean space R3, such as spheres. The exact definition of a surface may depend on the context. Typically, in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
, a surface may cross itself (and may have other singularities), while, in topology and differential geometry, it may not. A surface is a two-dimensional space; this means that a moving point on a surface may move in two directions (it has two degrees of freedom). In other words, around almost every point, there is a ''
coordinate patch In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an ...
'' on which a two-dimensional
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
is defined. For example, the surface of the Earth resembles (ideally) a two-dimensional sphere, and
latitude In geography, latitude is a coordinate that specifies the north–south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pole ...
and longitude provide two-dimensional coordinates on it (except at the poles and along the
180th meridian The 180th meridian or antimeridian is the meridian 180° both east and west of the prime meridian in a geographical coordinate system. The longitude at this line can be given as either east or west. On Earth, these two meridians form a grea ...
). The concept of surface is widely used in physics, engineering, computer graphics, and many other disciplines, primarily in representing the surfaces of physical objects. For example, in analyzing the
aerodynamic Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dyn ...
properties of an
airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft that is propelled forward by thrust from a jet engine, propeller, or rocket engine. Airplanes come in a variety of sizes, shapes, and wing configurations. The broad spectr ...
, the central consideration is the flow of air along its surface.


Definitions and first examples

A ''(topological) surface'' is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
in which every point has an open neighbourhood homeomorphic to some open subset of the Euclidean plane E2. Such a neighborhood, together with the corresponding homeomorphism, is known as a ''(coordinate) chart''. It is through this chart that the neighborhood inherits the standard coordinates on the Euclidean plane. These coordinates are known as ''local coordinates'' and these homeomorphisms lead us to describe surfaces as being ''locally Euclidean''. In most writings on the subject, it is often assumed, explicitly or implicitly, that as a topological space a surface is also nonempty, second-countable, and Hausdorff. It is also often assumed that the surfaces under consideration are connected. The rest of this article will assume, unless specified otherwise, that a surface is nonempty, Hausdorff, second-countable, and connected. More generally, a ''(topological) surface with boundary'' is a Hausdorff
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
in which every point has an open neighbourhood homeomorphic to some open subset of the closure of the
upper half-plane In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0. Complex plane Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds t ...
H2 in C. These homeomorphisms are also known as ''(coordinate) charts''. The boundary of the upper half-plane is the ''x''-axis. A point on the surface mapped via a chart to the ''x''-axis is termed a ''boundary point''. The collection of such points is known as the ''boundary'' of the surface which is necessarily a one-manifold, that is, the union of closed curves. On the other hand, a point mapped to above the ''x''-axis is an ''interior point''. The collection of interior points is the ''interior'' of the surface which is always non- empty. The closed disk is a simple example of a surface with boundary. The boundary of the disc is a circle. The term ''surface'' used without qualification refers to surfaces without boundary. In particular, a surface with empty boundary is a surface in the usual sense. A surface with empty boundary which is compact is known as a 'closed' surface. The two-dimensional sphere, the two-dimensional torus, and the
real projective plane In mathematics, the real projective plane is an example of a compact non- orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
are examples of closed surfaces. The
Möbius strip In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Aug ...
is a surface on which the distinction between clockwise and counterclockwise can be defined locally, but not globally. In general, a surface is said to be ''orientable'' if it does not contain a homeomorphic copy of the Möbius strip; intuitively, it has two distinct "sides". For example, the sphere and torus are orientable, while the real projective plane is not (because the real projective plane with one point removed is homeomorphic to the open Möbius strip). In differential and
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
, extra structure is added upon the topology of the surface. This added structure can be a smoothness structure (making it possible to define differentiable maps to and from the surface), a Riemannian metric (making it possible to define length and angles on the surface), a complex structure (making it possible to define holomorphic maps to and from the surface—in which case the surface is called a Riemann surface), or an algebraic structure (making it possible to detect singularities, such as self-intersections and cusps, that cannot be described solely in terms of the underlying topology).


Extrinsically defined surfaces and embeddings

Historically, surfaces were initially defined as subspaces of Euclidean spaces. Often, these surfaces were the
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of zeros of certain functions, usually polynomial functions. Such a definition considered the surface as part of a larger (Euclidean) space, and as such was termed ''extrinsic''. In the previous section, a surface is defined as a topological space with certain properties, namely Hausdorff and locally Euclidean. This topological space is not considered a subspace of another space. In this sense, the definition given above, which is the definition that mathematicians use at present, is ''intrinsic''. A surface defined as intrinsic is not required to satisfy the added constraint of being a subspace of Euclidean space. It may seem possible for some surfaces defined intrinsically to not be surfaces in the extrinsic sense. However, the
Whitney embedding theorem In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney: *The strong Whitney embedding theorem states that any smooth real -dimensional manifold (required also to be Hausdorff ...
asserts every surface can in fact be embedded homeomorphically into Euclidean space, in fact into E4: The extrinsic and intrinsic approaches turn out to be equivalent. In fact, any compact surface that is either orientable or has a boundary can be embedded in E3; on the other hand, the real projective plane, which is compact, non-orientable and without boundary, cannot be embedded into E3 (see Gramain). Steiner surfaces, including
Boy's surface In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901. He discovered it on assignment from David Hilbert to prove that the projective plane ''could not'' be immersed in 3-space ...
, the Roman surface and the cross-cap, are models of the real projective plane in E3, but only the Boy surface is an immersed surface. All these models are singular at points where they intersect themselves. The
Alexander horned sphere The Alexander horned sphere is a pathological object in topology discovered by . Construction The Alexander horned sphere is the particular embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting ...
is a well-known
pathological Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in th ...
embedding of the two-sphere into the three-sphere. The chosen embedding (if any) of a surface into another space is regarded as extrinsic information; it is not essential to the surface itself. For example, a torus can be embedded into E3 in the "standard" manner (which looks like a
bagel A bagel ( yi, בײגל, translit=beygl; pl, bajgiel; also spelled beigel) is a bread roll originating in the Jewish communities of Poland. It is traditionally shaped by hand into a roughly hand-sized ring from yeasted wheat dough that is fi ...
) or in a knotted manner (see figure). The two embedded tori are homeomorphic, but not isotopic: They are topologically equivalent, but their embeddings are not. The
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of a continuous, injective function from R2 to higher-dimensional Rn is said to be a
parametric surface A parametric surface is a surface in the Euclidean space \R^3 which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occ ...
. Such an image is so-called because the ''x''- and ''y''- directions of the domain R2 are 2 variables that parametrize the image. A parametric surface need not be a topological surface. A surface of revolution can be viewed as a special kind of parametric surface. If ''f'' is a smooth function from R3 to R whose gradient is nowhere zero, then the
locus Locus (plural loci) is Latin for "place". It may refer to: Entertainment * Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front * ''Locus'' (magazine), science fiction and fantasy magazine ** ''Locus Award' ...
of zeros of ''f'' does define a surface, known as an ''
implicit surface In mathematics, an implicit surface is a surface in Euclidean space defined by an equation : F(x,y,z)=0. An ''implicit surface'' is the set of zeros of a function of three variables. '' Implicit'' means that the equation is not solved for ...
''. If the condition of non-vanishing gradient is dropped, then the zero locus may develop singularities.


Construction from polygons

Each closed surface can be constructed from an oriented polygon with an even number of sides, called a
fundamental polygon In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal equ ...
of the surface, by pairwise identification of its edges. For example, in each polygon below, attaching the sides with matching labels (''A'' with ''A'', ''B'' with ''B''), so that the arrows point in the same direction, yields the indicated surface. Image:SphereAsSquare.svg, sphere Image:ProjectivePlaneAsSquare.svg,
real projective plane In mathematics, the real projective plane is an example of a compact non- orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
Image:TorusAsSquare.svg, torus Image:KleinBottleAsSquare.svg, Klein bottle
Any fundamental polygon can be written symbolically as follows. Begin at any vertex, and proceed around the perimeter of the polygon in either direction until returning to the starting vertex. During this traversal, record the label on each edge in order, with an exponent of -1 if the edge points opposite to the direction of traversal. The four models above, when traversed clockwise starting at the upper left, yield * sphere: A B B^ A^ * real projective plane: A B A B * torus: A B A^ B^ * Klein bottle: A B A B^. Note that the sphere and the projective plane can both be realized as quotients of the 2-gon, while the torus and Klein bottle require a 4-gon (square). The expression thus derived from a fundamental polygon of a surface turns out to be the sole relation in a
presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
of the fundamental group of the surface with the polygon edge labels as generators. This is a consequence of the Seifert–van Kampen theorem. Gluing edges of polygons is a special kind of quotient space process. The quotient concept can be applied in greater generality to produce new or alternative constructions of surfaces. For example, the real projective plane can be obtained as the quotient of the sphere by identifying all pairs of opposite points on the sphere. Another example of a quotient is the connected sum.


Connected sums

The
connected sum In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifica ...
of two surfaces ''M'' and ''N'', denoted ''M'' # ''N'', is obtained by removing a disk from each of them and gluing them along the boundary components that result. The boundary of a disk is a circle, so these boundary components are circles. The Euler characteristic \chi of is the sum of the Euler characteristics of the summands, minus two: :\chi(M \mathbin N) = \chi(M) + \chi(N) - 2.\, The sphere S is an
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
for the connected sum, meaning that . This is because deleting a disk from the sphere leaves a disk, which simply replaces the disk deleted from ''M'' upon gluing. Connected summation with the torus T is also described as attaching a "handle" to the other summand ''M''. If ''M'' is orientable, then so is . The connected sum is associative, so the connected sum of a finite collection of surfaces is well-defined. The connected sum of two real projective planes, , is the Klein bottle K. The connected sum of the real projective plane and the Klein bottle is homeomorphic to the connected sum of the real projective plane with the torus; in a formula, . Thus, the connected sum of three real projective planes is homeomorphic to the connected sum of the real projective plane with the torus. Any connected sum involving a real projective plane is nonorientable.


Closed surfaces

A closed surface is a surface that is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
and without
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film * Boundary (cricket), the edge of the pl ...
. Examples of closed surfaces include the sphere, the torus and the Klein bottle. Examples of non-closed surfaces include an
open disk In geometry, a disk (also spelled disc). is the region in a plane bounded by a circle. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not. For a radius, r, an open disk is usu ...
(which is a sphere with a puncture), a cylinder (which is a sphere with two punctures), and the
Möbius strip In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Aug ...
. A surface embedded in three-dimensional space is closed if and only if it is the boundary of a solid. As with any
closed manifold In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example ...
, a surface embedded in Euclidean space that is closed with respect to the inherited Euclidean topology is ''not'' necessarily a closed surface; for example, a disk embedded in \mathbb^3 that contains its boundary is a surface that is topologically closed but not a closed surface.


Classification of closed surfaces

The ''classification theorem of closed surfaces'' states that any
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closed surface is homeomorphic to some member of one of these three families: # the sphere, # the
connected sum In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifica ...
of ''g'' tori for ''g'' ≥ 1, # the
connected sum In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifica ...
of ''k'' real projective planes for ''k'' ≥ 1. The surfaces in the first two families are
orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
. It is convenient to combine the two families by regarding the sphere as the connected sum of 0 tori. The number ''g'' of tori involved is called the ''genus'' of the surface. The sphere and the torus have Euler characteristics 2 and 0, respectively, and in general the Euler characteristic of the connected sum of ''g'' tori is . The surfaces in the third family are nonorientable. The Euler characteristic of the real projective plane is 1, and in general the Euler characteristic of the connected sum of ''k'' of them is . It follows that a closed surface is determined, up to homeomorphism, by two pieces of information: its Euler characteristic, and whether it is orientable or not. In other words, Euler characteristic and orientability completely classify closed surfaces up to homeomorphism. Closed surfaces with multiple connected components are classified by the class of each of their connected components, and thus one generally assumes that the surface is connected.


Monoid structure

Relating this classification to connected sums, the closed surfaces up to homeomorphism form a commutative
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids a ...
under the operation of connected sum, as indeed do manifolds of any fixed dimension. The identity is the sphere, while the real projective plane and the torus generate this monoid, with a single relation , which may also be written , since . This relation is sometimes known as after Walther von Dyck, who proved it in , and the triple cross surface is accordingly called . Geometrically, connect-sum with a torus () adds a handle with both ends attached to the same side of the surface, while connect-sum with a Klein bottle () adds a handle with the two ends attached to opposite sides of an orientable surface; in the presence of a projective plane (), the surface is not orientable (there is no notion of side), so there is no difference between attaching a torus and attaching a Klein bottle, which explains the relation.


Proof

The classification of closed surfaces has been known since the 1860s, and today a number of proofs exist. Topological and combinatorial proofs in general rely on the difficult result that every compact 2-manifold is homeomorphic to a simplicial complex, which is of interest in its own right. The most common proof of the classification is , which brings every triangulated surface to a standard form. A simplified proof, which avoids a standard form, was discovered by John H. Conway circa 1992, which he called the "Zero Irrelevancy Proof" or "ZIP proof" and is presented in . A geometric proof, which yields a stronger geometric result, is the
uniformization theorem In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization ...
. This was originally proven only for Riemann surfaces in the 1880s and 1900s by
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
, Paul Koebe, and Henri Poincaré.


Surfaces with boundary

Compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
surfaces, possibly with boundary, are simply closed surfaces with a finite number of holes (open discs that have been removed). Thus, a connected compact surface is classified by the number of boundary components and the genus of the corresponding closed surface – equivalently, by the number of boundary components, the orientability, and Euler characteristic. The genus of a compact surface is defined as the genus of the corresponding closed surface. This classification follows almost immediately from the classification of closed surfaces: removing an open disc from a closed surface yields a compact surface with a circle for boundary component, and removing ''k'' open discs yields a compact surface with ''k'' disjoint circles for boundary components. The precise locations of the holes are irrelevant, because the
homeomorphism group In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. Homeomorphism groups are very important ...
acts ''k''-transitively on any connected manifold of dimension at least 2. Conversely, the boundary of a compact surface is a closed 1-manifold, and is therefore the disjoint union of a finite number of circles; filling these circles with disks (formally, taking the
cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines conn ...
) yields a closed surface. The unique compact orientable surface of genus ''g'' and with ''k'' boundary components is often denoted \Sigma_, for example in the study of the
mapping class group In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. Mo ...
.


Non-compact surfaces

Non-compact surfaces are more difficult to classify. As a simple example, a non-compact surface can be obtained by puncturing (removing a finite set of points from) a closed manifold. On the other hand, any open subset of a compact surface is itself a non-compact surface; consider, for example, the complement of a
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. T ...
in the sphere, otherwise known as the
Cantor tree surface In dynamical systems, the Cantor tree is an infinite-genus surface homeomorphic to a sphere with a Cantor set removed. The blooming Cantor tree is a Cantor tree with an infinite number of handles added in such a way that every end is a limit of ha ...
. However, not every non-compact surface is a subset of a compact surface; two canonical counterexamples are the Jacob's ladder and the
Loch Ness monster The Loch Ness Monster ( gd, Uilebheist Loch Nis), affectionately known as Nessie, is a creature in Scottish folklore that is said to inhabit Loch Ness in the Scottish Highlands. It is often described as large, long-necked, and with one or ...
, which are non-compact surfaces with infinite genus. A non-compact surface ''M'' has a non-empty space of ends ''E''(''M''), which informally speaking describes the ways that the surface "goes off to infinity". The space ''E''(''M'') is always topologically equivalent to a closed subspace of the
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. T ...
. ''M'' may have a finite or countably infinite number Nh of handles, as well as a finite or countably infinite number ''N''''p'' of projective planes. If both ''N''''h'' and ''N''''p'' are finite, then these two numbers, and the topological type of space of ends, classify the surface ''M'' up to topological equivalence. If either or both of ''N''''h'' and ''N''''p'' is infinite, then the topological type of M depends not only on these two numbers but also on how the infinite one(s) approach the space of ends. In general the topological type of M is determined by the four subspaces of ''E''(''M'') that are limit points of infinitely many handles and infinitely many projective planes, limit points of only handles, limit points of only projective planes, and limit points of neither.


Assumption of second-countability

If one removes the assumption of second-countability from the definition of a surface, there exist (necessarily non-compact) topological surfaces having no countable base for their topology. Perhaps the simplest example is the Cartesian product of the
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with the space of real numbers. Another surface having no countable base for its topology, but ''not'' requiring the Axiom of Choice to prove its existence, is the Prüfer manifold, which can be described by simple equations that show it to be a real-analytic surface. The Prüfer manifold may be thought of as the upper half plane together with one additional "tongue" ''T''''x'' hanging down from it directly below the point (''x'',0), for each real ''x''. In 1925, Tibor Radó proved that all Riemann surfaces (i.e., one-dimensional complex manifolds) are necessarily second-countable ( Radó's theorem). By contrast, if one replaces the real numbers in the construction of the Prüfer surface by the complex numbers, one obtains a two-dimensional complex manifold (which is necessarily a 4-dimensional real manifold) with no countable base.


Surfaces in geometry

Polyhedra, such as the boundary of a cube, are among the first surfaces encountered in geometry. It is also possible to define ''smooth surfaces'', in which each point has a neighborhood diffeomorphic to some open set in E2. This elaboration allows
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
to be applied to surfaces to prove many results. Two smooth surfaces are diffeomorphic if and only if they are homeomorphic. (The analogous result does not hold for higher-dimensional manifolds.) Thus
closed surface In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as ...
s are classified up to diffeomorphism by their Euler characteristic and orientability. Smooth surfaces equipped with Riemannian metrics are of foundational importance in differential geometry. A Riemannian metric endows a surface with notions of geodesic,
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
,
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
, and area. It also gives rise to Gaussian curvature, which describes how curved or bent the surface is at each point. Curvature is a rigid, geometric property, in that it is not preserved by general diffeomorphisms of the surface. However, the famous Gauss–Bonnet theorem for closed surfaces states that the integral of the Gaussian curvature ''K'' over the entire surface ''S'' is determined by the Euler characteristic: :\int_S K \; dA = 2 \pi \chi(S). This result exemplifies the deep relationship between the geometry and topology of surfaces (and, to a lesser extent, higher-dimensional manifolds). Another way in which surfaces arise in geometry is by passing into the complex domain. A complex one-manifold is a smooth oriented surface, also called a Riemann surface. Any complex nonsingular
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
viewed as a complex manifold is a Riemann surface. In fact, every compact orientable surface is realizable as a Riemann surface. Thus compact Riemann surfaces are characterized topologically by their genus: 0, 1, 2, .... On the other hand, the genus does not characterize the complex structure. For example, there are uncountably many non-isomorphic compact Riemann surfaces of genus 1 (the
elliptic curves In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If t ...
). Complex structures on a closed oriented surface correspond to conformal equivalence classes of Riemannian metrics on the surface. One version of the
uniformization theorem In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization ...
(due to Poincaré) states that any Riemannian metric on an oriented, closed surface is conformally equivalent to an essentially unique metric of
constant curvature In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry. The sectional curvature ...
. This provides a starting point for one of the approaches to
Teichmüller theory Teichmüller is a German surname ( German for ''pond miller'') and may refer to: * Anna Teichmüller (1861–1940), German composer * :de:Frank Teichmüller (19?? – now), former German IG Metall district manager "coast" * Gustav Teichmüller (1 ...
, which provides a finer classification of Riemann surfaces than the topological one by Euler characteristic alone. A ''complex surface'' is a complex two-manifold and thus a real four-manifold; it is not a surface in the sense of this article. Neither are algebraic curves defined over
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
s other than the complex numbers, nor are algebraic surfaces defined over
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
s other than the real numbers.


See also

*
Boundary (topology) In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the closure of not belonging to the interior of . An element of the boundary of is called a boundary point of . The term bound ...
* Volume form, for volumes of surfaces in E''n'' *
Poincaré metric In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometry ...
, for metric properties of Riemann surfaces * Roman surface *
Boy's surface In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901. He discovered it on assignment from David Hilbert to prove that the projective plane ''could not'' be immersed in 3-space ...
*
Tetrahemihexahedron In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices. Its vertex figure is a crossed quadrilateral. Its Coxeter–Dynkin dia ...
* Crumpled surface, a non-differentiable surface obtained by deforming (crumpling) a differentiable surface


Notes


References

*


Simplicial proofs of classification up to homeomorphism

*, English translation of 1934 classic German textbook *, Chapter I *, Cambridge undergraduate course * * *, for closed oriented Riemannian manifolds


Morse theoretic proofs of classification up to diffeomorphism

* * *, careful proof aimed at undergraduates
(Original 1969-70 Orsay course notes in French for "Topologie des Surfaces")
*


Other proofs

*, similar to Morse theoretic proof using sliding of attached handles * *, short elementary proof using spanning graphs *, contains short account of Thomassen's proof


External links



i
Mathifold ProjectThe Classification of Surfaces and the Jordan Curve Theorem
in Home page of Andrew Ranicki

* ttp://wokos.nethium.pl/surfaces_en.net Math Surfaces Animation, with JavaScript (Canvas HTML) for tens surfaces rotation viewingbr>The Classification of Surfaces
Lecture Notes by Z.Fiedorowicz
History and Art of Surfaces and their Mathematical Models2-manifolds
at the Manifold Atlas {{Authority control Geometric topology Differential geometry of surfaces Analytic geometry