Surface Guide, Meaning , Facts, Information and Description

In mathematics, a surface is a two-dimensional manifold. Examples arise in three-dimensional space as the boundaries of three-dimensional 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.

Table of contents

1 Topology 2 See also 3 External links

Topology

In what follows, all surfaces are considered to be second-countable two dimensional manifolds.

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 n-tori). These are orientable surfaces with Euler characteristic 2-2n, also called surfaces of genus n.
• Projective planess with n handles attached. These are non-orientable surfaces with Euler characteristic 1-2n.
• Klein bottles with n handles attached. These are non-orientable 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 with boundary are just these with one or more removed disks. A compact surface can be embedded in R³ 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 R⁴.

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)           (sphere with handle)

This notion of a surface is distinct from the notion of an algebraic surface. A non-singular 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

External links

• Math Surfaces Gallery, with 60 ~surfaces and Java Applet for live rotation viewing

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