List of manifolds

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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....
  

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....
  , RP¹ ≅ S¹
• 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....
  

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....
  

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....