Definition
Around what follows, completely shells come considered to become second-countable 2-dimensional manifolds.

Supplementary precisely: the topologic surface (by having boundary) occurs as 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. A placed of points which keep close at hand an open front yard homeomorphic to E^north is known as a interior of the manifold; these are universally non-empty. A complement of a interior, is known as the boundary; these are the (Single)-manifold, or even union of closed curves.

The surface by having empty boundary is said to become closed in case these are compact, and open in case these are non compact.

Classification of closed surfaces
There is a complete classification of closed (i personally.e compact without boundary) connected, shells as much as homeomorphism. Any such surface fall under one of deuce infinite collections:

Spheres with g handles tied (known as g-stack tori). Which are actually orientable surfaces with Euler characteristic 2-2g, likewise known as shells of genus g. Spheres with with one thousand projective planes attached. Which are actually non-orientable shells by owning Euler characteristic Ii-k.

So Euler characteristic and orientability describe a compact shells as much as homeomorphism (and in case shells come smooth so as much as diffeomorphism).

Compact surfaces
Compact shells by using boundary come upright these by having a single or even additional flushed open disks whose closures are disjoint.

Embeddings in R³
The compact surface may be embedded around R^Three whenever these are orientable or even in case it has nonempty boundary. These are the symptom of the Whitney embedding theorem that any surface can be embedded within R^Foursome.

Differential geometry
The elementary view of the embedding of the surface around n dimensions, & the computation of the region of such the surface, is provided in the article volume form. Metrical properties of Riemann surfaces are briefly reviewed in the article Poincaré metric.

Some models
To produce a bit of system, seize a sides one (& dislodge a corners to puncture): * * B B five five 5 ^ *>>>>>* *>>>>>* 5 five five ^ 5 5 five v The five 5 The The five ^ The The five v The The v v A five 5 five ^ 5 five 5 v 5 five 5 ^ *<<<<<* *>>>>>* * * B B

sphere real projective plane Klein bottle torus (perforate Köbius band) (sinker)

Fundamental polygon
For each one closed surface may be constructed from either an possibly sided orientated polygonal shape, known as the fundamental polygon by pairwise identification of its edges.

This constructionorth may be represented as a string of length 2n of n distinct symbols in which for each one symbol appears twice by owning exponent either +One or even -1. A exponent -1 signifies that a corresponding edge has a orientation opposing a one of a fundamental polygonal shape.

A above system may be described when follows:

sphere: $The\; A^$ projective plane: $The\; A$ Klein bottle: $The\; B\; A^B$ torus: $The\; B\; A^$

(Look at a independent article fundamental polygon for details.)

Connected sum of surfaces
Given 2 shells M & M', their attached total Thousand # M' is found by removing a disk inside every of the children & gluing the children along the recently formed boundary components.

I personally utilise a as punishment notation. sphere: S torus: T Klein bottle: K Projective plane: P

Information:

S # S = S S # M = M P # P = K P # K = P # T

I apply the tachygraphy floating: nM = K # One thousand # ... # K (north-days) by having 0M = S.

Closed shells come classified when follows:

gT (g-stack torus): orientable surface of genus g, for $g\; \backslash ge\; Zero$. gP (g-stack projective plane): non-orientable surface of genus g, for $g\; \backslash ge\; Unity$.

Algebraic surface
This notion of a surface is distinct from either the notion of an algebraic surface. The non-singular complex projective algebraic curve is a smooth surface. Algebraical shells above a complex number field have dimension Four once considered as a manifold.