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| {{dablink|This article discusses surfaces from the point of view of [[topology]]. For uses in other contexts, see [[Differential geometry of surfaces]], [[algebraic surface]], and [[Surface (disambiguation)]]. }}
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| [[Image:Saddle pt.jpg|thumb|225px|right|An [[open surface]] with ''X''-, ''Y''-, and ''Z''-contours shown.]]
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| In [[mathematics]], specifically, in [[topology]], a '''surface''' is a [[two-dimensional]], [[topological manifold]]. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional [[Euclidean space]] '''R'''<sup>3</sup> — for example, the surface of a [[ball]]. On the other hand, there are surfaces, such as the [[Klein bottle]], that cannot be [[embedding|embedded]] in three-dimensional Euclidean space without introducing [[singularity theory|singularities]] or self-intersections.
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| To say that a surface is "two-dimensional" means that, about each point, there is a ''[[coordinate patch]]'' on which a two-dimensional [[coordinate system]] is defined. For example, the surface of the [[Earth]] is (ideally) a two-dimensional [[sphere]], and [[latitude]] and [[longitude]] provide two-dimensional coordinates on it (except at the poles and along the [[180th meridian]]).
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| The concept of surface finds application in [[physics]], [[engineering]], [[computer graphics]], and many other disciplines, primarily in representing the surfaces of physical objects. For example, in analyzing the [[aerodynamics|aerodynamic]] properties of an [[airplane]], the central consideration is the flow of air along its surface.
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| ==Definitions and first examples==
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| A ''(topological) surface'' is a nonempty [[Second-countable space|second countable]] [[Hausdorff space|Hausdorff]] [[topological space]] in which every point has an open [[topological neighbourhood|neighbourhood]] [[homeomorphism|homeomorphic]] to some [[open set|open subset]] of the Euclidean plane '''E'''<sup>2</sup>. 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''.
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| More generally, a ''(topological) surface with boundary'' is a [[Hausdorff space|Hausdorff]] [[topological space]] in which every point has an open [[topological neighbourhood|neighbourhood]] [[homeomorphism|homeomorphic]] to some [[open set|open subset]] of the closure of the [[upper half-plane]] '''H'''<sup>2</sup> 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 set|empty]]. The closed [[disk (mathematics)|disk]] is a simple example of a surface with boundary. The boundary of the disc is a circle.
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| 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]] are examples of closed surfaces.
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| The [[Möbius strip]] is a surface with only one "side". 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 deleting a point or disk from the real projective plane produces the Möbius strip).
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| In [[differential geometry|differential]] and [[algebraic geometry]], extra structure is added upon the topology of the surface. This added structures detects [[Singular point of an algebraic variety|singularities]], such as self-intersections and cusps, that cannot be described solely in terms of the underlying topology.
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| ==Extrinsically defined surfaces and embeddings==
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| [[Image:Sphere wireframe.svg|left|thumb|250px|A sphere can be defined parametrically (by ''x'' = ''r'' sin ''θ'' cos ''φ'',
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| ''y'' = ''r'' sin ''θ'' sin ''φ'', ''z'' = ''r'' cos ''θ'') or implicitly (by {{nowrap|''x''² + ''y''² + ''z''² − ''r''² {{=}} 0}}.)]]
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| Historically, surfaces were initially defined as subspaces of Euclidean spaces. Often, these surfaces were the [[locus (mathematics)|locus]] of [[root of a function|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''.
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| 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''.
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| 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]] asserts every surface can in fact be embedded homeomorphically into Euclidean space, in fact into '''E'''<sup>4</sup>: The extrinsic and intrinsic approaches turn out to be equivalent.
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| In fact, any compact surface that is either orientable or has a boundary can be embedded in '''E'''³; on the other hand, the real projective plane, which is compact, non-orientable and without boundary, cannot be embedded into '''E'''³ (see Gramain). [[Steiner surface]]s, including [[Boy's surface]], the [[Roman surface]] and the [[cross-cap]], are [[embedding|immersions]] of the real projective plane into '''E'''³. These surfaces are singular where the immersions intersect themselves.
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| The [[Alexander horned sphere]] is a well-known [[pathological (mathematics)|pathological]] embedding of the two-sphere into the three-sphere.
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| [[Image:KnottedTorus.svg|right|thumb|A knotted torus.]]
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| 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 '''E'''³ in the "standard" manner (which looks like a [[bagel]]) or in a [[knot (mathematics)|knotted]] manner (see figure). The two embedded tori are homeomorphic, but not [[Homotopy#Isotopy|isotopic]]: They are topologically equivalent, but their embeddings are not.
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| The [[image (mathematics)|image]] of a continuous, [[injection (mathematics)|injective]] function from '''R'''<sup>2</sup> to higher-dimensional '''R'''<sup>n</sup> is said to be a [[parametric surface]]. Such an image is so-called because the ''x''- and ''y''- directions of the domain '''R'''<sup>2</sup> 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.
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| If ''f'' is a smooth function from '''R'''³ to '''R''' whose [[gradient]] is nowhere zero, then the [[locus (mathematics)|locus]] of [[root of a function|zeros]] of ''f'' does define a surface, known as an ''[[implicit surface]]''. If the condition of non-vanishing gradient is dropped, then the zero locus may develop singularities.
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| {{clr}}
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| ==Construction from polygons==
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| Each closed surface can be constructed from an oriented polygon with an even number of sides, called a [[fundamental polygon]] 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.
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| <gallery>
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| Image:SphereAsSquare.svg|[[sphere]]
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| Image:ProjectivePlaneAsSquare.svg|[[real projective plane]]
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| Image:TorusAsSquare.svg|[[torus]]
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| Image:KleinBottleAsSquare.svg|[[Klein bottle]]
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| </gallery>
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| 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
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| * sphere: <math>A B B^{-1} A^{-1}</math>
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| * real projective plane: <math>A B A B</math>
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| * torus: <math>A B A^{-1} B^{-1}</math>
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| * Klein bottle: <math>A B A B^{-1}</math>.
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| 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).
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| The expression thus derived from a fundamental polygon of a surface turns out to be the sole relation in a [[presentation of a group|presentation]] of the [[fundamental group]] of the surface with the polygon edge labels as generators. This is a consequence of the [[Seifert–van Kampen theorem]]. | |
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| 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.
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| ==Connected sums==
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| The [[connected sum]] 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]] <math>\chi</math> of {{nowrap|''M'' # ''N''}} is the sum of the Euler characteristics of the summands, minus two:
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| :<math>\chi(M \# N) = \chi(M) + \chi(N) - 2.\,</math>
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| The sphere '''S''' is an [[identity element]] for the connected sum, meaning that {{nowrap|1='''S''' # ''M'' = ''M''}}. This is because deleting a disk from the sphere leaves a disk, which simply replaces the disk deleted from ''M'' upon gluing.
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| 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 {{nowrap|'''T''' # ''M''}}. The connected sum is associative, so the connected sum of a finite collection of surfaces is well-defined.
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| The connected sum of two real projective planes, {{nowrap|'''P''' # '''P'''}}, 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, {{nowrap|1='''P''' # '''K''' = '''P''' # '''T'''}}. 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.
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| == Closed surfaces ==
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| A '''closed surface''' is a surface that is [[compact space|compact]] and without [[Boundary (topology)|boundary]]. Examples are spaces like the [[sphere]], the [[torus]] and the [[Klein bottle]]. Examples of non-closed surfaces are: an [[disk (mathematics)|open disk]], which is a sphere with a puncture; a [[cylinder (geometry)|cylinder]], which is a sphere with two punctures; and the [[Möbius strip]].
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| === Classification of closed surfaces ===
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| [[File:SurfacesWithAndWithoutBoundary.svg|right|thumb|200px|Some examples of orientable closed surfaces (left) and surfaces with boundary (right). Left: Some orientable closed surfaces are the surface of a sphere, the surface of a [[torus]], and the surface of a cube. (The cube and the sphere are topologically equivalent to each other.) Right: Some surfaces with boundary are the [[disk (mathematics)|disk surface]], square surface, and hemisphere surface. The boundaries are shown in red. All three of these are topologically equivalent to each other.]]
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| The ''classification theorem of closed surfaces'' states that any [[connected (topology)|connected]] closed surface is homeomorphic to some member of one of these three families:
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| # the sphere;
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| # the connected sum of ''g'' tori, for <math>g \geq 1</math>;
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| # the connected sum of ''k'' real projective planes, for <math>k \geq 1</math>.
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| The surfaces in the first two families are [[orientability|orientable]]. 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 {{nowrap|2 − 2''g''}}.
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| 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 {{nowrap|2 − ''k''}}.
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| 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.
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| Closed surfaces with multiple [[Connected component (topology)|connected components]] are classified by the class of each of their connected components, and thus one generally assumes that the surface is connected.
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| === Monoid structure ===
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| Relating this classification to connected sums, the closed surfaces up to homeomorphism form a [[monoid]] with respect to the 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 {{nowrap|1='''P''' # '''P''' # '''P''' = '''P''' # '''T'''}}, which may also be written {{nowrap|1='''P''' # '''K''' = '''P''' # '''T'''}}, since {{nowrap|1='''K''' = '''P''' # '''P'''}}. This relation is sometimes known as '''{{visible anchor|Dyck's theorem}}''' after [[Walther von Dyck]], who proved it in {{Harv|Dyck|1888}}, and the triple cross surface {{nowrap|'''P''' # '''P''' # '''P'''}} is accordingly called '''{{visible anchor|Dyck's surface}}'''.<ref name="fw"/>
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| Geometrically, connect-sum with a torus ({{nowrap|# '''T'''}}) adds a handle with both ends attached to the same side of the surface, while connect-sum with a Klein bottle ({{nowrap|# '''K'''}}) adds a handle with the two ends attached to opposite sides of an orientable surface; in the presence of a projective plane ({{nowrap|# '''P'''}}), 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.
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| === Surfaces with boundary ===
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| [[Compact manifold|Compact]] 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.
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| 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 acts [[transitive action|''k''-transitively]] on any connected manifold of dimension at least 2.
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| 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 (topology)|cone]]) yields a closed surface.
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| The unique compact orientable surface of genus ''g'' and with ''k'' boundary components is often denoted <math>\Sigma_{g,k},</math> for example in the study of the [[mapping class group]].
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| === Riemann surfaces ===
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| A closely related example to the classification of compact 2-manifolds is the classification of compact [[Riemann surface]]s, i.e., compact complex 1-manifolds. (Note that the 2-sphere and the torus are both [[complex manifold]]s, in fact [[algebraic variety|algebraic varieties]].) Since every complex manifold is orientable, the connected sums of projective planes are not complex manifolds. Thus, compact Riemann surfaces are characterized topologically simply by their genus. The genus counts the number of holes in the manifold: the sphere has genus 0, the one-holed torus genus 1, etc.
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| === Non-compact surfaces ===
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| 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 the sphere, otherwise known as the [[Cantor tree surface]]. However, not every non-compact surface is a subset of a compact surface; two canonical counterexamples are the [[Jacob's ladder (manifold)|Jacob's ladder]] and the [[Loch Ness monster surface|Loch Ness monster]], which are non-compact surfaces with infinite genus.
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| === Proof ===
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| The classification of closed surfaces has been known since the 1860s,<ref name="fw">{{Harv|Francis|Weeks|1999}}</ref> and today a number of proofs exist.
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| 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 {{Harv|Seifert|Threlfall|1934}},<ref name="fw"/> 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 {{Harv|Francis|Weeks|1999}}.
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| A geometric proof, which yields a stronger geometric result, is the [[uniformization theorem]]. This was originally proven only for Riemann surfaces in the 1880s and 1900s by [[Felix Klein]], [[Paul Koebe]], and [[Henri Poincaré]].
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| ==Surfaces in geometry==
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| {{main|Differential geometry of surfaces}}
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| [[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 [[diffeomorphism|diffeomorphic]] to some open set in '''E'''². This elaboration allows [[calculus]] to be applied to surfaces to prove many results.
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| 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]]s are classified up to diffeomorphism by their Euler characteristic and orientability.
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| Smooth surfaces equipped with [[Riemannian metric]]s are of fundational importance in [[differential geometry]]. A Riemannian metric endows a surface with notions of [[geodesic]], [[distance]], [[angle]], 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:
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| :<math>\int_S K \; dA = 2 \pi \chi(S).</math>
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| This result exemplifies the deep relationship between the geometry and topology of surfaces (and, to a lesser extent, higher-dimensional manifolds).
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| 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]] viewed as a complex manifold is a Riemann surface.
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| Every closed orientable surface admits a complex structure. Complex structures on a closed oriented surface correspond to [[conformally equivalent|conformal equivalence classes]] of Riemannian metrics 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.
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| 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 (mathematics)|field]]s other than the complex numbers,
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| nor are algebraic surfaces defined over [[field (mathematics)|field]]s other than the real numbers.
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| ==See also==
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| *[[Volume form]], for volumes of surfaces in '''E'''''<sup>n</sup>''
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| *[[Poincaré metric]], for metric properties of Riemann surfaces
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| *[[Area element]], the area of a differential element of a surface
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| *[[Roman surface]]
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| *[[Boy's surface]]
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| *[[Tetrahemihexahedron]]
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| ==Notes==
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| {{reflist|group = note}}
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| *{{Citation | first = Walther | last = Dyck | authorlink = Walther von Dyck | title = Beiträge zur Analysis situs I | journal = Math. Ann. | volume = 32 | year = 1888 | pages = 459–512 }}
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| *{{cite book| author= Gramain, André|title=Topology of Surfaces| publisher=BCS Associates| year=1984|isbn = 0-914351-01-X}} [http://www.math.u-psud.fr/~biblio/numerisation/docs/G_GRAMAIN-55/pdf/G_GRAMAIN-55.pdf (Original 1969-70 Orsay course notes in French for "Topologie des Surfaces")]
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| *{{cite book| author=Bredon, Glen E.|authorlink = Glen Bredon| title=Topology and Geometry| publisher=Springer-Verlag| year=1993| isbn= 0-387-97926-3}}
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| *{{cite book| author=Massey, William S.| title=A Basic Course in Algebraic Topology| publisher=Springer-Verlag| year=1991| isbn= 0-387-97430-X}}
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| *{{citation | title = Conway's ZIP Proof | first1 = George K. | last1 = Francis | first2 = Jeffrey R. | last2 = Weeks | journal = [[American Mathematical Monthly]] | volume = 106 | number = 5 |date=May 1999 | url = http://new.math.uiuc.edu/zipproof/zipproof.pdf | postscript =, page discussing the paper: [http://new.math.uiuc.edu/zipproof/ On Conway's ZIP Proof] }}
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| {{refend}}
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| ==External links==
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| {{wiktionary|surface}}
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| *[http://www.maths.ed.ac.uk/~aar/jordan/ The Classification of Surfaces and the Jordan Curve Theorem] in Home page of Andrew Ranicki
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| *[http://xahlee.org/surface/gallery.html Math Surfaces Gallery, with 60 ~surfaces and Java Applet for live rotation viewing]
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| *[http://wokos.nethium.pl/surfaces_en.net Math Surfaces Animation, with JavaScript (Canvas HTML) for tens surfaces rotation viewing]
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| *[http://www.math.ohio-state.edu/~fiedorow/math655/classification.html The Classification of Surfaces] Lecture Notes by Z.Fiedorowicz
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| *[http://maxwelldemon.com/2009/03/21/surfaces-1-the-ooze-of-the-past/ History and Art of Surfaces and their Mathematical Models]
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| *[http://www.map.mpim-bonn.mpg.de/2-manifolds 2-manifolds] at the Manifold Atlas
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| [[Category:Surfaces]]
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| [[Category:Geometric topology]]
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| [[Category:Differential geometry of surfaces]]
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| [[Category:Analytic geometry]]
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