Surface (topology)

surfaceclosed surfacesurfaces2-manifoldclassification of closed surfacestopological surfaceClassification of two-dimensional closed manifoldsclosed surfaces2-dimensionalBirkhoff curve shortening argument
In mathematics, a surface is a two-dimensional manifold.wikipedia
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Klein bottle

Klein bottlesbottleKlein
For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.
In topology, a branch of mathematics, the Klein bottle is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined.

Plane (geometry)

planeplanarplanes
In mathematics, a surface is a geometrical shape that resembles a deformed plane.
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.

Sphere

sphericalhemisphereglobose
The most familiar examples arise as boundaries of solid objects in ordinary three-dimensional Euclidean space R 3, such as spheres.
While outside mathematics the terms "sphere" and "ball" are sometimes used interchangeably, in mathematics the above distinction is made between a sphere, which is a two-dimensional closed surface embedded in a three-dimensional Euclidean space, and a ball, which is a three-dimensional shape that includes the sphere and everything inside the sphere (a closed ball), or, more often, just the points inside, but not on the sphere (an open ball).

Differential geometry

differentialdifferential geometerdifferential geometry and topology
Topological surfaces are sometimes equipped with additional information, such as a Riemannian metric or a complex structure, that connects them to other disciplines within mathematics, such as differential geometry and complex analysis. Typically, in algebraic geometry, a surface may cross itself (and may have other singularities), while, in topology and differential geometry, it may not.
In dimension 2, a symplectic manifold is just a surface endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism.

Surface (mathematics)

surfacesurfaces2-dimensional shape
The various mathematical notions of surface can be used to model surfaces in the physical world.
This is formalized by the concept of manifold: in the context of manifolds, typically in topology and differential geometry, a surface is a manifold of dimension two; this means that a surface is a topological space such that every point has a neighborhood which is homeomorphic to an open subset of the Euclidean plane (see Surface (topology) and Surface (differential geometry)).

Möbius strip

Moebius stripMobius stripMöbius band
The Möbius strip is a surface on which the distinction between clockwise and counterclockwise can be defined locally, but not globally.
A Möbius strip, Möbius band, or Möbius loop, also spelled Mobius or Moebius, is a surface with only one side (when embedded in three-dimensional Euclidean space) and only one boundary.

Real projective plane

projective planeFlat projective planeprojective manifolds
The two-dimensional sphere, the two-dimensional torus, and the real projective plane are examples of closed surfaces.
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface.

Riemann surface

Riemann surfacescompact Riemann surfaceconformally invariant
This added structures can be a smoothness structure (making it possible to define differentiable maps to and from the surface), a Riemannian metric (making it possible to define length and angles on the surface), a complex structure (making it possible to define holomorphic maps to and from the surface—in which case the surface is called a Riemann surface), or an algebraic structure (making it possible to detect singularities, such as self-intersections and cusps, that cannot be described solely in terms of the underlying topology).
Every Riemann surface is a two-dimensional real analytic manifold (i.e., a surface), but it contains more structure (specifically a complex structure) which is needed for the unambiguous definition of holomorphic functions.

Topology

topologicaltopologicallytopologist
Typically, in algebraic geometry, a surface may cross itself (and may have other singularities), while, in topology and differential geometry, it may not.
Two-dimensional manifolds are also called surfaces, although not all surfaces are manifolds.

Locus (mathematics)

locuslocilocus of points
Often, these surfaces were the locus of zeros of certain functions, usually polynomial functions.
In geometry, a locus (plural: loci) (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.

Upper half-plane

upper half planelower half-planeupper half of the complex plane
More generally, a (topological) surface with boundary is a Hausdorff topological space in which every point has an open neighbourhood homeomorphic to some open subset of the closure of the upper half-plane H 2 in C.
The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative Gaussian curvature.

Torus

toroidaltoriflat torus
The two-dimensional sphere, the two-dimensional torus, and the real projective plane are examples of closed surfaces.
Topologically, a torus is a closed surface defined as the product of two circles: S 1 × S 1.

Manifold

manifoldsboundarymanifold with boundary
In mathematics, a surface is a two-dimensional manifold.
Two-dimensional manifolds are also called surfaces.

Connected sum

knot sumconnect-sumconnected summing
The connected sum of two surfaces M and N, denoted M # N, is obtained by removing a disk from each of them and gluing them along the boundary components that result.
This construction plays a key role in the classification of closed surfaces.

Fundamental group

fundamental groupoidfundamental groupsfirst homotopy group
The expression thus derived from a fundamental polygon of a surface turns out to be the sole relation in a presentation of the fundamental group of the surface with the polygon edge labels as generators.
It describes the monodromy properties of complex-valued functions, as well as providing a complete topological classification of closed surfaces.

Orientability

orientableorientednon-orientable
The surfaces in the first two families are orientable.
In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point.

Two-dimensional space

Euclidean planetwo-dimensional2D
A surface is a two-dimensional space; this means that a moving point on a surface may move in two directions (it has two degrees of freedom).
In topology, the plane is characterized as being the unique contractible 2-manifold.

Cross-cap

Cross capcrosscap
Steiner surfaces, including Boy's surface, the Roman surface and the cross-cap, are models of the real projective plane in E 3, but only the Boy surface is an immersed surface.
An important theorem of topology, the classification theorem for surfaces, states that each two-dimensional compact manifold without boundary is homeomorphic to a sphere with some number (possibly 0) of "handles" and 0, 1, or 2 cross-caps.

Gaussian curvature

Gauss curvaturecurvatureLiebmann's theorem
It also gives rise to Gaussian curvature, which describes how curved or bent the surface is at each point.
In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures,

Gauss–Bonnet theorem

Gauss-Bonnet theoremGauss–Bonnet formulaChern–Gauss–Bonnet formula
However, the famous Gauss–Bonnet theorem for closed surfaces states that the integral of the Gaussian curvature K over the entire surface S is determined by the Euler characteristic:
The Gauss–Bonnet theorem, or Gauss–Bonnet formula, is an important statement about surfaces in differential geometry, connecting their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic).

Teichmüller space

Teichmüller theoryBers compactificationTeichmüller metric
This provides a starting point for one of the approaches to Teichmüller theory, which provides a finer classification of Riemann surfaces than the topological one by Euler characteristic alone.
In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism.

Mapping class group

Torelli groupMumford ConjectureStable mapping class group
The unique compact orientable surface of genus g and with k boundary components is often denoted for example in the study of the mapping class group.
The mapping class groups of surfaces have been heavily studied, and are sometimes called Teichmüller modular groups (note the special case of MCG(T 2 ) above), since they act on Teichmüller space and the quotient is the moduli space of Riemann surfaces homeomorphic to the surface.

Diffeomorphism

diffeomorphicdiffeomorphismsdiffeomorphism group
It is also possible to define smooth surfaces, in which each point has a neighborhood diffeomorphic to some open set in E 2.
A diffeomorphism f : U → V between two surfaces U and V has a Jacobian matrix Df that is an invertible matrix.

Euler characteristic

Euler's formulaEuler–Poincaré characteristicElements
The Euler characteristic \chi of M # N is the sum of the Euler characteristics of the summands, minus two:
The Euler characteristic of a closed orientable surface can be calculated from its genus g (the number of tori in a connected sum decomposition of the surface; intuitively, the number of "handles") as

Walther von Dyck

Walther Franz Anton von Dyckvon Dyck, WaltherWalther Dyck
The identity is the sphere, while the real projective plane and the torus generate this monoid, with a single relation P # P # P = P # T, which may also be written P # K = P # T, since K = P # P. This relation is sometimes known as ' after Walther von Dyck, who proved it in, and the triple cross surface P # P # P is accordingly called '.
The Dyck language in formal language theory is named after him, as are Dyck's theorem and Dyck's surface in the theory of surfaces, together with the von Dyck groups, the Dyck tessellations, Dyck paths, and the Dyck graph.