Surface - Definition and Overview

For other uses, see Surface (disambiguation).

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.

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 planes 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 (http://xahlee.org/surface/gallery.html)