# 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.

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## 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 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: [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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