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List of manifolds

 

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List of manifolds



 
 
This is a list of particular manifold
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
s, by Wikipedia page. See also list of geometric topology topics
List of geometric topology topics

This is a list of geometric topology topics, by Wikipedia page....
. For categorical listings see :Category:Manifolds and its subcategories.








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This is a list of particular manifold
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
s, by Wikipedia page. See also list of geometric topology topics
List of geometric topology topics

This is a list of geometric topology topics, by Wikipedia page....
. For categorical listings see :Category:Manifolds and its subcategories.

Generic families of manifolds


  • Euclidean space
    Euclidean space

    Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
    , Rn
  • n-sphere, Sn
  • n-torus, Tn
  • Real projective space
    Real projective space

    In mathematics, real projective space, or RPn is the projective space of lines in Rn+1. It is a compact space, smooth manifold of dimension n, and a special case of a Grassmannian....
    , RPn
  • Complex projective space
    Complex projective space

    In mathematics, complex projective space, P, Pn or CPn, in fact preferablyis the projective space of line in Cn+1....
    , CPn
  • Quaternionic projective space
    Quaternionic projective space

    In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions H....
    , HPn
  • Grassmann manifold
  • Flag manifold
    Flag manifold

    In mathematics, a generalized flag variety is a homogeneous space whose points are flag in a finite-dimensional vector space V over a field F....
  • Stiefel manifold
    Stiefel manifold

    In mathematics, the Stiefel manifold Vk is the set of all orthonormal Frame of a vector space in R'n. That is, it is the set of ordered k-tuples of orthonormal vector in R'n....


Lie groups provide more interesting families. See Table of Lie groups
Table of Lie groups

This article gives a table of some common Lie groups and their associated Lie algebras.The following are noted: the topology properties of the group , as well as on their algebraic properties ....
 for examples. See also: List of simple Lie groups
List of simple Lie groups

In mathematics, the simple Lie groups were classified by ?lie Cartan.The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces....
 and List of Lie group topics
List of Lie group topics

This is a list of Lie group topics, by Wikipedia page....
.

Manifolds of a specific dimension


1-manifolds

  • Real line
    Real line

    In mathematics, the real line is simply the set R of singleton real numbers.However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space....
    , R
  • Circle
    Circle

    A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
    , S1
  • Real projective line
    Real projective line

    In real analysis, the real projective line , is the set , also denoted by and by .The symbol represents the point at infinity, an idealized point that bridges the two "ends" of the real line....
    , RP1S1
  • Long line
    Long line (topology)

    In topology, the long line is a topological space analogous to the real line, but much longer. Because it behaves locally just like the real line, but has different large-scale properties, it serves as one of the basic counterexamples of topology....


2-manifolds

  • Sphere
    Sphere

    A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
    , S2
  • Real projective plane
    Real projective plane

    In mathematics, the real projective plane is the space of lines in R3 passing through the origin. It is a non-Orientability two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedding in our usual three-dimensional space without intersecting itself....
    , RP2
  • Torus
    Torus

    In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle....
  • Klein bottle
    Klein bottle

    In mathematics, the Klein bottle is a certain non-orientability surface, i.e., a surface with no distinct "inner" and "outer" sides. Other related non-orientable objects include the M?bius strip and the real projective plane....
  • Cylinder
    Cylinder (geometry)

    A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder....
  • Möbius strip
    Möbius strip

    The M?bius strip or M?bius band is a surface with only one side and only one boundary component. The M?bius strip has the mathematical property of being orientability....
  • Double torus
    Double torus

    In mathematics, a genus-2 surface is a topology object formed by the connected sum of two torus. That is to say, from each of two torii the interior of a disk is removed, and the boundaries of the two disks are identified , forming a double torus....
  • Klein quartic
    Klein quartic

    In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact space Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168....
     (a genus 3 surface)


3-manifolds

  • 3-sphere
    3-sphere

    In mathematics, a '3-sphere' is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space....
    , S3
  • SO(3) ≅ RP3
  • Poincaré homology sphere
  • Whitehead manifold
    Whitehead manifold

    In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to R3. Henry Whitehead discovered this puzzling object while he was trying to prove the Poincar? conjecture....
  • Weeks manifold
    Weeks manifold

    In mathematics, the Weeks manifold, sometimes called the Fomenko-Matveev-Weeks manifold, is a closed hyperbolic 3-manifold obtained by and Dehn surgery on the Whitehead link....
  • Solid torus
    Solid torus

    In mathematics, a solid torus is a topological space homeomorphic to , i.e. the cartesian product of the circle with a two dimensional ball endowed with the product topology....
  • Solid Klein bottle
    Solid Klein bottle

    In mathematics, a solid Klein bottle is a 3-manifold homeomorphism to the quotient space obtained by gluing the top of to the bottom by a reflection, i.e....


For more examples see 3-manifold
3-manifold

In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is usually made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds....
.


4-manifolds

  • Exotic R4
    Exotic R4

    In mathematics, an exotic R4 is a differentiable manifold that is homeomorphic to the Euclidean space R4, but not diffeomorphism....
  • E8 manifold
    E8 manifold

    In mathematics, the E8 manifold is the unique compact space, simply connected topological 4-manifold with intersection form the E8 lattice....


For more examples see 4-manifold
4-manifold

In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different....
.


Special types of manifolds


Manifolds related to spheres

  • Homology sphere
    Homology sphere

    In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n = 1. That is,...
  • Homotopy sphere
    Homotopy sphere

    In algebraic topology, a branch of mathematics, a homotopy sphere is an n-manifold Homotopy#Homotopy_equivalence_of_spaces to the n-Sphere#Topology....
  • Exotic sphere
    Exotic sphere

    In mathematics, an exotic sphere is a differentiable manifold that is homeomorphic to the standard Euclidean n-sphere, but not diffeomorphic....
  • Milnor sphere
  • Spherical 3-manifold
    Spherical 3-manifold

    In mathematics, a spherical 3-manifold M is a 3-manifold of the formwhere Γ is a Finite group subgroup of Special orthogonal group Group action by rotations on the 3-sphere ....
  • Lens space
    Lens space

    A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions....


Special classes of Riemannian manifolds
  • Riemannian symmetric space
    Riemannian symmetric space

    In differential geometry, representation theory and harmonic analysis, a symmetric space is a smooth manifold whose group of symmetries contains an "inversion symmetry" about every point....
  • Einstein manifold
    Einstein manifold

    In differential geometry and mathematical physics, an Einstein manifold is a Riemannian manifold or pseudo-Riemannian manifold whose Ricci tensor is proportional to the metric tensor....
    • Ricci-flat manifold
      Ricci-flat manifold

      In mathematics, Ricci-flat manifolds are Riemannian manifolds whose Ricci curvature vanishes. In physics, they represent vacuum solutions to the analogues of Einstein's equations for Riemannian manifolds of any dimension, with vanishing cosmological constant....
  • Kähler manifold
    Kähler manifold

    In mathematics, a K?hler manifold is a manifold with unitary group structure satisfying an integrability condition.In particular, it is a complex manifold, a Riemannian manifold, and a symplectic manifold, with these three structures all mutually compatible....
    • Calabi-Yau manifold
      Calabi-Yau manifold

      In mathematics, Calabi–Yau manifolds are sometimes defined as compact K?hler manifolds whose canonical bundle is trivial, though many other similar but inequivalent definitions are sometimes used....
    • Hyperkähler manifold
      Hyperkähler manifold

      In differential geometry, a hyperk?hler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Symplectic group ....
  • Quaternionic Kähler manifold
  • G2 manifold
    G2 manifold

    A G2 manifold is a seven-dimensional Riemannian manifold with holonomy group G2 . The group is one of the five exceptional simple Lie groups....
  • Spin(7) manifold


Categories of manifolds


Manifolds definable by a particular choice of atlas
  • Topological manifold
    Topological manifold

    In mathematics, a topological manifold is a Hausdorff space topological space which looks locally like Euclidean space in a sense defined below....
  • Piecewise linear manifold
    Piecewise linear manifold

    In mathematics, a piecewise linear manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas , such that one can pass from chart to chart in it by piecewise linear functions....
  • Differentiable manifold
    Differentiable manifold

    A differentiable manifold is a type of manifold that is locally similar enough to Euclidean space to allow one to do calculus. This article deals with differentiability in different contexts including: smooth function, k times differentiable, and holomorphic function....
  • Smooth manifold
  • Analytic manifold
    Analytic manifold

    In mathematics, an analytic manifold is a topological manifold with analytic function transition maps. Every complex manifold is an analytic manifold....
  • Complex manifold
    Complex manifold

    In differential geometry, a complex manifold is a manifold with an atlas of chart to the open unit disk in Cn, such that the transition maps are holomorphic....


Manifolds with additional structure
  • Riemannian manifold
    Riemannian manifold

    In Riemannian geometry, a Riemannian manifold is a real differentiable manifold M in which each tangent space is equipped with an Inner product space g in a manner which varies smoothly from point to point....
  • Pseudo-Riemannian manifold
    Pseudo-Riemannian manifold

    In differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many things named after Bernhard Riemann....
  • Finsler manifold
    Finsler manifold

    In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold M with a Banach norm defined over each tangent space, smooth function depending on position, and assumed to satisfy the following condition:...
  • Symplectic manifold
    Symplectic manifold

    In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a Closed and exact differential forms, nondegenerate form, differential form, ?, called the symplectic form....
  • Almost complex manifold
    Almost complex manifold

    In mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. The existence of this structure is a necessary, but not sufficient, condition for a manifold to be a complex manifold....
  • Lie group
    Lie group

    In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....


Infinite-dimensional manifolds
  • Hilbert manifold
    Hilbert manifold

    In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable space Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space....
  • Banach manifold
    Banach manifold

    In mathematics, a Banach manifold is a manifold modeled on Banach spaces. Thus it is a topological space in which each point has a Neighbourhood homeomorphic to an open set in a Banach space ....
  • Fréchet manifold
    Fréchet manifold

    In mathematics, in particular in nonlinear analysis, a Fr?chet manifold is a topological space modeled on a Fr?chet space in much the same way as a manifold is modeled on a Euclidean space....