# Surface

For other senses of this word, see surface (disambiguation).
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An open surface with X-, Y-, and Z-contours shown.

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, roughness.

The two-dimensional character of a surface comes from the fact that, about each point, there is a "coordinate patch" on which a two-dimensional coordinate system is defined; in general, it is not possible to extend this patch to the entire surface, so it will be necessary to define multiple patches which collectively cover the surface.

A surface may have a boundary, where the surface ends. For example, the boundary of a disc or hemisphere would be the circle around the edge.

## Examples

The general concept of a surface, and the richness and variety of surfaces, can be understood by examining a variety of examples. Any formal definition of a surface must be strong enough to encompass this variety.

## 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 two infinite collections:

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 article Poincaré metric.

## Some models

To make some models of various surfaces, attach the sides of these squares (A with A, B with B) so that the directions of the arrows match:

## 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: $\displaystyle A A^{-1}$
• projective plane: $\displaystyle A A$
• Klein bottle: $\displaystyle A B A^{-1} B$
• torus: $\displaystyle A B A^{-1} B^{-1}$

(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, for $\displaystyle g \ge 0$ .
• gP (g-fold projective plane): non-orientable surface of genus g, for $\displaystyle g \ge 1$ .

## 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 real manifold.