Surface 27865 223344813 2008-07-03T17:49:17Z 12.30.114.67 {{wiktionarypar|surface}} {{dablink|This article discusses surfaces from the point of view of [[topology]]. For other uses, see [[Differential geometry of surfaces]], [[algebraic surface]], and [[Surface (disambiguation)]]}} [[Image:Saddle pt.jpg|thumb|225px|right|An open surface with ''X''-, ''Y''-, and ''Z''-contours shown.]] In [[mathematics]], specifically in [[topology]], a '''surface''' is a [[two-dimensional]] [[manifold]]. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional [[Euclidean space]], '''E'''³. On the other hand, there are also more exotic surfaces, that are so "contorted" that they cannot be [[embedding| embedded]] in three-dimensional space at all. 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 coordinates on it &mdash; except at the [[International Date Line]] and the poles, where longitude is undefined. This example illustrates that in general it is not possible to extend any one coordinate patch to the entire surface; surfaces, like manifolds of all dimensions, are usually constructed by patching together multiple coordinate systems. Surfaces find application in [[physics]], [[engineering]], [[computer graphics]], and many other disciplines, primarily when they represent 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. ==Definitions and first examples== 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 closed half space of '''E'''² (Euclidean 2-space). The neighborhood, along with the homeomorphism to Euclidean space, is called a ''(coordinate) chart''. The set of points that have an open neighbourhood homeomorphic to '''E'''² is called the ''interior'' of the surface; it is always non-[[empty set|empty]]. The [[complement (set theory)| complement]] of the interior is called the ''boundary''; it is a one-manifold, or union of closed curves. The simplest example of a surface with boundary is the closed [[disk (mathematics)| disk]] in '''E'''²; its boundary is a circle. A surface with an empty boundary is called ''boundaryless''. (Sometimes the word surface, used alone, refers only to boundaryless surfaces.) A ''closed'' surface is one that is boundaryless and [[compact space|compact]]. The two-dimensional sphere, the two-dimensional [[torus]], and the [[real projective plane]] are examples of closed surfaces. 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). More generally, it is common in [[differential geometry| differential]] and [[algebraic geometry]] to study surfaces with [[Singular point of an algebraic variety|singularities]], such as self-intersections, cusps, etc. ==Extrinsically defined surfaces and embeddings== [[Image:Sphere-wireframe.png|left|thumb|250px|A sphere can be defined parametrically (by ''x'' = ''r'' sin ''θ'' cos ''φ'', ''y'' = ''r'' sin ''θ'' sin ''φ'', ''z'' = ''r'' cos ''θ'') or implicitly (by ''x''²&nbsp;+&nbsp;''y''²&nbsp;+&nbsp;''z''²&nbsp;&minus;&nbsp;''r''²&nbsp;=&nbsp;0.)]] Historically, surfaces were originally defined and constructed not using the abstract, ''intrinsic'' definition given above, but ''extrinsically'', as subsets of Euclidean spaces such as '''E'''³. Let ''f'' be a continuous, [[injection (mathematics)| injective]] function from '''R'''² to '''R'''³. Then the [[image (mathematics)| image]] of ''f'' is said to be a [[parametric surface]]. A [[surface of revolution]] can be viewed as a special kind of parametric surface. On the other hand, suppose that ''f'' is a smooth function from '''R'''³ to '''R''' whose [[gradient]] is nowhere zero. Then the [[locus (mathematics)|locus]] of [[root (mathematics)|zeros]] of ''f'' is said to be an ''[[implicit surface]]''. If the condition of non-vanishing gradient is dropped then the zero locus may develop singularities. One can also define parametric and implicit surfaces in higher-dimensional Euclidean spaces '''E'''''<sup>n</sup>''. It is natural to ask whether all surfaces (defined abstractly, as in the preceding section) arise as subsets of some '''E'''''<sup>n</sup>''. The answer is yes; the [[Whitney embedding theorem]], in the case of surfaces, states that any surface can be embedded homeomorphically into '''E'''<sup>4</sup>. Therefore 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 '''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. The [[Alexander horned sphere]] is a well-known [[pathological (mathematics)| pathological]] embedding of the two-sphere into the three-sphere. [[Image:KnottedTorus.png|right|thumb|A knotted torus.]] 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 (that looks like a [[bagel]]) or in a [[knot (mathematics)|knotted]] manner (see figure). The two embedded tori are homeomorphic but not [[isotopy|isotopic]]; they are topologically equivalent, but their embeddings are not. {{clr}} ==Construction from polygons== 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. <gallery> Image:SphereAsSquare.svg|[[sphere]] Image:ProjectivePlaneAsSquare.svg|[[real projective plane]] Image:TorusAsSquare.svg|[[torus]] Image:KleinBottleAsSquare.svg|[[Klein bottle]] </gallery> 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: <math>A B B^{-1} A^{-1}</math> * real projective plane: <math>A B A B</math> * torus: <math>A B A^{-1} B^{-1}</math> * Klein bottle: <math>A B A B^{-1}</math>. 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]]. ==Quotients and connected summation== 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. 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 [[Euler characteristic]] <math>\chi</math> of ''M'' # ''N'' is the sum of the Euler characteristics of the summands, minus two: :<math>\chi(M \# N) = \chi(M) + \chi(N) - 2.\,</math> The sphere '''S''' is an [[identity element]] for the connected sum, meaning that '''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. Connected summation with the torus '''T''' has the effect of attaching a "handle" to the other summand ''M''. If ''M'' is orientable, then so is '''T''' # ''M''. The connected sum can be iterated to attach any number ''g'' of handles to ''M''. The connected sum of two real projective planes is the Klein bottle. 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. Any connected sum involving a real projective plane is nonorientable. ==Classification of closed surfaces== The ''classification of closed surfaces'' states that any closed surface is homeomorphic to some member of one of these three families: # the sphere; # the connected sum of ''g'' tori, for <math>g \geq 1</math>; # the connected sum of ''k'' real projective planes, for <math>k \geq 1</math>. The surfaces in the first two families are 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. Since the sphere and the torus have Euler characteristics 2 and 0, respectively, it follows that the Euler characteristic of the connected sum of ''g'' tori is 2&nbsp;&minus;&nbsp;2''g''. The surfaces in the third family are nonorientable. Since the Euler characteristic of the real projective plane is 1, the Euler characteristic of the connected sum of ''k'' of them is 2&nbsp;&minus;&nbsp;''k''. 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. ==Surfaces in geometry== {{main|Differential geometry of surfaces}} [[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. 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. 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. 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: :<math>\int_S K \; dA = 2 \pi \chi(S).</math> 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]] viewed as a real manifold is a Riemann surface. Every closed surface admits complex structures. 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. 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)| field]]s other than the complex numbers. ==See also== *[[Volume form]], for volumes of surfaces in '''E'''''<sup>n</sup>'' *[[Poincaré metric]], for metric properties of Riemann surfaces *[[Area element]], the area of a differential element of a surface ==References== *{{cite book | author= Gramain, André |title=Topology of Surfaces | publisher=BCS Associates | year=1984 |id = ISBN 091435101X}} [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")] *{{cite book | author=Bredon, Glen E. | title=Topology and Geometry | publisher=Springer-Verlag | year=1993 | id= ISBN 0-387-97926-3 }} *{{cite book | author=Massey, William S. | title=A Basic Course in Algebraic Topology | publisher=Springer-Verlag | year=1991 | id= ISBN 0-387-97430-X }} ==External links== *[http://xahlee.org/surface/gallery.html Math Surfaces Gallery, with 60 ~surfaces and Java Applet for live rotation viewing] [[Category:Surfaces]] [[Category:Geometric topology]] [[an:Superfizie]] [[ast:Superficie]] [[ca:Superfície]] [[cs:Povrch]] [[de:Fläche (Topologie)]] [[es:Superficie]] [[fr:Surface]] [[fur:Superficie]] [[gl:Superficie]] [[io:Surfaco]] [[ia:Superficie]] [[it:Superficie (matematica)]] [[lv:Virsma]] [[ja:表面]] [[pl:Powierzchnia]] [[pt:Superfície]] [[ro:Suprafaţă]] [[ru:Поверхность]] [[simple:Surface]] [[sk:Povrch]] [[sl:Ploskev]] [[sr:Површ]] [[vi:Mặt]] [[uk:Поверхня]] [[vec:Superficie]] [[zh:曲面]]