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.