Surface

 For other uses, see Surface (disambiguation).
In mathematics (topology), a surface is a twodimensional manifold. Examples arise in threedimensional space as the boundaries of threedimensional solid objects. The surface of a fluid object, such as a rain drop or soap bubble, is an idealisation. To speak of the surface of a snowflake, which has a great deal of fine structure, is to go beyond the simple mathematical definition. For the nature of real surfaces see surface tension, surface chemistry, surface energy.
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Definition
In what follows, all surfaces are considered to be secondcountable 2dimensional manifolds.
More precisely: a topological surface (with boundary) is a Hausdorff space in which every point has an open neighbourhood homeomorphic to either an open subset of E^{2} (Euclidean 2space) or an open subset of the closed half of E^{2}. The set of points which have an open neighbourhood homeomorphic to E^{n} is called the interior of the manifold; it is always nonempty. The complement of the interior, is called the boundary; it is a (1)manifold, or union of closed curves.
A surface with empty boundary is said to be closed if it is compact, and open if it is not compact.
Classification of closed surfaces
There is a complete classification of closed (i.e compact without boundary) connected, surfaces up to homeomorphism. Any such surface falls into one of three infinite collections:
 Spheres with n handles attached (called ntori). These are orientable surfaces with Euler characteristic 22n, also called surfaces of genus n.
 Projective planes with n handles attached. These are nonorientable surfaces with Euler characteristic 12n.
 Klein bottles with n handles attached. These are nonorientable surfaces with Euler characteristic 2n.
Therefore Euler characteristic and orientability describe a compact surfaces up to homeomorphism (and if surfaces are smooth then up to diffeomorphism).
Compact surfaces
Compact surfaces with boundary are just these with one or more removed open disks whose closures are disjoint.
Embeddings in R^{3}
A compact surface can be embedded in R^{3} if it is orientable or if it has nonempty boundary. It is a consequence of the Whitney embedding theorem that any surface can be embedded in R^{4}.
Differential geometry
A simple review of the embedding of a surface in n dimensions, and a computation of the area of such a surface, is provided in the article volume form. Metric properties of Riemann surfaces are briefly reviewed in the the article Poincaré metric.
Some models
To make some models, attach the sides of these (and remove the corners to puncture):
* * B B v v v ^ *>>>>>* *>>>>>* v v v ^ v v v v A v v A A v ^ A A v v A A v v A v v v ^ v v v v v v v ^ *<<<<<* *>>>>>* * * B B
sphere real projective plane Klein bottle torus (punctured Möbius band) (donut)
Fundamental polygon
Each closed surface can be constructed from an even sided oriented polygon, called a fundamental polygon by pairwise identification of its edges.
This construction can be represented as a string of length 2n of n distinct symbols where each symbol appears twice with exponent either +1 or 1. The exponent 1 signifies that the corresponding edge has the orientation opposing the one of the fundamental polygon.
The above models can be described as follows:
 sphere: <math>A A^{1}<math>
 projective plane: <math>A A<math>
 Klein bottle: <math>A B A^{1} B<math>
 torus: <math>A B A^{1} B^{1}<math>
(See the main article fundamental polygon for details.)
Connected sum of surfaces
Given two surfaces M and M', their connected sum M # M' is obtained by removing a disk in each of them and gluing them along the newly formed boundary components.
We use the following notation.
 sphere: S
 torus: T
 Klein bottle: K
 Projective plane: P
Facts:
 S # S = S
 S # M = M
 P # P = K
 P # K = P # T
We use a shorthand natation: nM = M # M # ... # M (ntimes) with 0M = S.
Closed surfaces are classified as follows:
 gT (gfold torus): orientable surface of genus g.
 gP (gfold projective plane): nonorientable surface of genus g.
Algebraic surface
This notion of a surface is distinct from the notion of an algebraic surface. A nonsingular complex projective algebraic curve is a smooth surface. Algebraic surfaces over the complex number field have dimension 4 when considered as a manifold.
See also
 minimal surface
 Riemann surface
 algebraic surface
 Klein bottle
 Torus
 Sphere
 Cylinder
 Projective plane
External links
 Math Surfaces Gallery, with 60 ~surfaces and Java Applet for live rotation viewing (http://xahlee.org/surface/gallery.html)ast:Superficie
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