Generic families of manifolds

• Euclidean space
  Euclidean space
  In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
  
  , Rⁿ
• n-sphere, Sⁿ
• n-torus, Tⁿ
• Real projective space
  Real projective space
  In mathematics, real projective space, or RPn, is the topological space of lines through 0 in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian.-Construction:...
  
  , RPⁿ
• Complex projective space
  Complex projective space
  In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...
  
  , 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. Quaternionic projective space of dimension n is usually denoted byand is a closed manifold of dimension 4n...
  
  , HPⁿ
• Grassmann manifold
• Flag manifold
  Flag manifold
  In mathematics, a generalized flag variety is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold...
  
  
• Stiefel manifold
  Stiefel manifold
  In mathematics, the Stiefel manifold Vk is the set of all orthonormal k-frames in Rn. That is, it is the set of ordered k-tuples of orthonormal vectors in Rn. It is named after Swiss mathematician Eduard Stiefel...
  
  

1-manifolds

• Real line
  Real line
  In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
  
  , R
• Circle
  Circle
  A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
  
  , S¹
• Real projective line, RP¹ ≅ S¹
• Long line
  Long line (topology)
  In topology, the long line is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties. Therefore it serves as one of the basic counterexamples of topology...
  
  

2-manifolds

• Sphere
  Sphere
  A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
  
  , S²
• Real projective plane
  Real projective plane
  In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded 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...
  
  
• Klein bottle
  Klein bottle
  In mathematics, the Klein bottle is a non-orientable surface, informally, a surface in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a...
  
  
• Cylinder
  Cylinder (geometry)
  A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
  
  
• Möbius strip
  Möbius strip
  The Möbius strip or Möbius band is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface...
  
  
• Double torus
  Double torus
  In mathematics, a genus-2 surface is a surface formed by the connected sum of two tori. That is to say, from each of two tori the interior of a disk is removed, and the boundaries of the two disks are identified , forming a double torus.This is the simplest case of the connected sum of n tori...
  
  
• Klein quartic
  Klein quartic
  In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 336 automorphisms if orientation may be reversed...
  
   (a genus 3 surface)
• Surface of genus g

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 Poincaré 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 surgeries on the Whitehead link. It has volume approximately equal to 0.9427... and showed that it has the smallest volume of any closed orientable...
  
  
• Solid torus
  Solid torus
  In mathematics, a solid torus is a topological space homeomorphic to S^1 \times D^2, i.e. the cartesian product of the circle with a two dimensional disc endowed with the product topology. The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary...
  
  
• Solid Klein bottle
  Solid Klein bottle
  In mathematics, a solid Klein bottle is a 3-manifold homeomorphic to the quotient space obtained by gluing the top of \scriptstyle D^2 \times I 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 diffeomorphic.The first examples were found by Robion Kirby and Michael Freedman, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's...
  
  
• E8 manifold
  E8 manifold
  In mathematics, the E8 manifold is the unique compact, simply connected topological 4-manifold with intersection form the E8 lattice.The E8 manifold was discovered by Michael Freedman in 1982...
  
  

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,andTherefore X is a connected space, with one non-zero higher Betti number: bn...
  
  
• Homotopy sphere
  Homotopy sphere
  In algebraic topology, a branch of mathematics, a homotopy sphere is an n-manifold homotopy equivalent to the n-sphere. It thus has the same homotopy groups and the same homology groups, as the n-sphere...
  
  
• Exotic sphere
  Exotic sphere
  In differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere...
  
  
• Milnor sphere
• Spherical 3-manifold
  Spherical 3-manifold
  In mathematics, a spherical 3-manifold M is a 3-manifold of the formM=S^3/\Gammawhere \Gamma is a finite subgroup of SO acting freely by rotations on the 3-sphere S^3. All such manifolds are prime, orientable, and closed...
  
  
• 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. There are two ways to formulate the inversion symmetry, via Riemannian geometry or via Lie theory...
  
  
• Einstein manifold
  Einstein manifold
  In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian manifold whose Ricci tensor is proportional to the metric...
  
  
  • 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 Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
  
  
  • Calabi–Yau manifold
  • Hyperkähler manifold
    Hyperkähler manifold
    In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp(k) In differential geometry, a hyperkähler...
    
    
• Quaternionic Kähler manifold
• G₂ manifold
  G2 manifold
  In differential geometry, a G2 manifold is a seven-dimensional Riemannian manifold with holonomy group G2. The group G_2 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 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.An isomorphism of PL manifolds is called a PL...
  
  
• Differentiable manifold
  Differentiable manifold
  A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
  
  
• Smooth manifold
• Analytic manifold
  Analytic manifold
  In mathematics, an analytic manifold is a topological manifold with analytic 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 charts 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 and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, 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 mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...
  
  
• Finsler manifold
  Finsler manifold
  In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold together with the structure of an intrinsic quasimetric space in which the length of any rectifiable curve is given by the length functional...
  
  
• Symplectic manifold
  Symplectic manifold
  In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...
  
  
• 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. That is, every complex manifold is an almost...
  
  
• 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 smooth 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 Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space. The concept of a Hilbert manifold provides a possibility of extending the theory of...
  
  
• 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 Fréchet manifold is a topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space....