List of manifolds

Short description: none

## Contents

* 1 Generic families of manifolds
* 2 Manifolds of a specific dimension
* 2.1 1-manifolds
* 2.2 2-manifolds
* 2.3 3-manifolds
* 2.4 4-manifolds
* 3 Special types of manifolds
* 3.1 Manifolds related to spheres
* 3.2 Special classes of Riemannian manifolds
* 4 Categories of manifolds
* 4.1 Manifolds definable by a particular choice of atlas
* 4.2 Manifolds with additional structure
* 4.3 Infinite-dimensional manifolds
* 5 See also
* 6 References

## Generic families of manifolds

* Euclidean space, Rn
* n-sphere, Sn
* n-torus, Tn
* Real projective space, RPn
* Complex projective space, CPn
* Quaternionic projective space, HPn
* Flag manifold
* Grassmann manifold
* Stiefel manifold

Lie groups provide several interesting families. See Table of Lie groups for examples. See also: List of simple Lie groups and List of Lie group topics.

## Manifolds of a specific dimension

### 1-manifolds

* Circle, S1
* Long line
* Real line, R
* Real projective line, RP1 ≅ S1

### 2-manifolds

* Cylinder, S1 × R
* Klein bottle, RP2 # RP2
* Klein quartic (a genus 3 surface)
* Möbius strip
* Real projective plane, RP2
* Sphere, S2
* Surface of genus g
* Torus
* Double torus

### 3-manifolds

* 3-sphere, S3
* 3-torus, T3
* Poincaré homology sphere
* SO(3) ≅ RP3
* Solid Klein bottle
* Solid torus
* Whitehead manifold
* Meyerhoff manifold
* Weeks manifold

For more examples see 3-manifold.

### 4-manifolds

* Complex projective plane
* Del Pezzo surface
* E8 manifold
* Enriques surface
* Exotic R4
* Hirzebruch surface
* K3 surface

For more examples see 4-manifold.

## Special types of manifolds

### Manifolds related to spheres

* Brieskorn manifold
* Exotic sphere
* Homology sphere
* Homotopy sphere
* Lens space
* Spherical 3-manifold

### Special classes of Riemannian manifolds

* Einstein manifold
* Ricci-flat manifold
* G2 manifold
* Kähler manifold
* Calabi–Yau manifold
* Hyperkähler manifold
* Quaternionic Kähler manifold
* Riemannian symmetric space
* Spin(7) manifold

## Categories of manifolds

| The Wikibook Topology has a page on the topic of: Categories of Manifolds

|

### Manifolds definable by a particular choice of atlas

* Affine manifold
* Analytic manifold
* Complex manifold
* Differentiable (smooth) manifold
* Piecewise linear manifold
* Lipschitz manifold
* Topological manifold

### Manifolds with additional structure

* Almost complex manifold
* Almost symplectic manifold
* Calibrated manifold
* Complex manifold
* Contact manifold
* CR manifold
* Finsler manifold
* Hermitian manifold
* Hyperkähler manifold
* Kähler manifold
* Lie group
* Pseudo-Riemannian manifold
* Riemannian manifold
* Sasakian manifold
* Spin manifold
* Symplectic manifold

### Infinite-dimensional manifolds

* Banach manifold
* Fréchet manifold
* Hilbert manifold

## See also

* List of topological spaces - List of concrete topologies and topological spaces

## References

* v
* t
* e

Manifolds (Glossary, List, Category)

Basic concepts|

* Topological manifold
* Atlas
* Differentiable/Smooth manifold
* Differential structure
* Smooth atlas
* Submanifold
* Riemannian manifold
* Smooth map
* Submersion
* Pushforward
* Tangent space
* Differential form
* Vector field

Main theorems (list)|

* Atiyah–Singer index
* Darboux's
* De Rham's
* Frobenius
* Generalized Stokes
* Hopf–Rinow
* Noether's
* Sard's
* Whitney embedding

Maps|

* Curve
* Diffeomorphism
* Local
* Geodesic
* Exponential map
* in Lie theory
* Foliation
* Immersion
* Integral curve
* Lie derivative
* Section
* Submersion

Types of
manifolds|

* Calabi–Yau
* Closed
* Collapsing
* Complete
* (Almost) Complex
* (Almost) Contact
* Einstein
* Fibered
* Finsler
* (Almost, Ricci-) Flat
* G-structure
* Hadamard
* Hermitian
* Hyperbolic
* (Hyper) Kähler
* Kenmotsu
* Lie group
* Lie algebra
* Manifold with boundary
* Nilmanifold
* Oriented
* Parallelizable
* Poisson
* Prime
* Quaternionic
* Hypercomplex
* (Pseudo-, Sub-) Riemannian
* Rizza
* Sasakian
* Stein
* (Almost) Symplectic
* Tame

Tensors|

| Vectors|

* Distribution
* Lie bracket
* Pushforward
* Tangent space
* bundle
* Torsion
* Vector field
* Vector flow

|

Covectors|

* Closed/Exact
* Covariant derivative
* Cotangent space
* bundle
* De Rham cohomology
* Differential form
* Complex
* Vector-valued
* One-form
* Exterior derivative
* Interior product
* Pullback
* Ricci curvature
* flow
* Riemann curvature tensor
* Tensor field
* density
* Volume form
* Wedge product

Bundles|

* Adjoint
* Affine
* Associated
* Cotangent
* Dual
* Fiber
* (Co-) Fibration
* Jet
* Lie algebra
* (Stable) Normal
* Principal
* Spinor
* Subbundle
* Tangent
* Tensor
* Vector

Connections|

* Affine
* Cartan
* Ehresmann
* Form
* Generalized
* Koszul
* Levi-Civita
* Principal
* Vector
* Parallel transport

Related|

* Classification of manifolds
* Gauge theory
* History
* Morse theory
* Moving frame
* Singularity theory

Generalizations|

* Banach
* Diffeology
* Diffiety
* Fréchet
* Hilbert
* K-theory
* Non-Hausdorff
* Orbifold
* Secondary calculus
* over commutative algebras
* Sheaf
* Stratifold
* Supermanifold
* Stratified space

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