Surface

For other uses, see Surface (disambiguation).

In mathematics (topology), 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.

 Contents

Definition

In what follows, all surfaces are considered to be second-countable 2-dimensional 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 E2 (Euclidean 2-space) or an open subset of the closed half of E2. The set of points which have an open neighbourhood homeomorphic to En is called the interior of the manifold; it is always non-empty. 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 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

Compact surfaces with boundary are just these with one or more removed open disks whose closures are disjoint.

Embeddings in R3

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.

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 Poincar� 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: [itex]A A^{-1}[itex]
• projective plane: [itex]A A[itex]
• Klein bottle: [itex]A B A^{-1} B[itex]
• torus: [itex]A B A^{-1} B^{-1}[itex]

(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 (n-times) with 0M = S.

Closed surfaces are classified as follows:

• gT (g-fold torus): orientable surface of genus g.
• gP (g-fold projective plane): non-orientable surface of genus g.

Algebraic surface

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.

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