In mathematics (topology), a surface is a two-dimensional manifold. Examples arise in three-dimensional space as the boundaries of three-dimensional solid objects.

• For the nature of real surfaces see surface tension, surface chemistry, surface energy, roughness.

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.

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

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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.

• Parametric surfaces are surfaces defined by parametric equations, generally the images of maps from a sub set of R² to R³.
• Developable surfaces are surfaces that can be flattened to a plane without stretching; examples include the cylinder, the cone, and the torus.
• Ruled surfaces are surfaces that have at least one straight line running through every point. Examples include the cylinder and the hyperboloid of one sheet.
• Surfaces of revolution are surfaces with cylindrical symmetry.
• Minimal surfaces are surfaces that minimize the surface area. Examples include soap films stretched across a wire frame, catenoids and helicoids.
• Algebraic surfaces are surfaces defined by the locus of zeros of a set of algebraic equations. Examples include the quadrics, the cubic surfaces, and the Veronese surface.
• Implicit surfaces are surfaces defined as the locus of zeros of a general equation.
• The Klein bottle and the Möbius strip are examples of non-orientable manifolds.
• Riemann surfaces are surfaces that admit a complex analytic structure; in particular, complex analytic functions may be defined between them. Examples include the sphere and the torus.
• Projective surfaces are defined in projective spaces. Examples include Boy's surface, the Roman surface and the cross-cap, all of which are Steiner surfaces.
• The Alexander horned sphere is an example of a surface with a topology that combines an ordinary smooth surface with a Cantor set.

Definition

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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

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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:

• Spheres with g handles attached (called g-fold tori). These are orientable surfaces with Euler characteristic 2-2g, also called surfaces of genus g.
• Spheres with k projective planes attached. These are non-orientable surfaces with Euler characteristic 2-k.

Therefore Euler characteristic and orientability describe a compact surfaces up to homeomorphism (and if surfaces are smooth then up to diffeomorphism).

Compact surfaces

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

Embeddings in R³

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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

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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

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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:

Image:SphereAsSquare.svg|sphere Image:ProjectivePlaneAsSquare.svg|real projective plane Image:KleinBottleAsSquare.svg|Klein bottle Image:TorusAsSquare.svg|torus

Fundamental polygon

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

(See the main article fundamental polygon for details.)

Connected sum of surfaces

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

Algebraic surface

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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. Algebraic surfaces over the real numbers will give normal surfaces, however these may contain singular points, where the algebraic surface forms a degenerate lines or points.

External links

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• Math Surfaces Gallery, with 60 ~surfaces and Java Applet for live rotation viewing

Surfaces | Geometric topology

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