List of manifolds

From Wikipedia, the free encyclopedia

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

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

[edit] 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
• Grassmann manifold
• Flag manifold
• Stiefel manifold

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

[edit] Manifolds of a specific dimension

[edit] 1-manifolds

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

[edit] 2-manifolds

• Sphere, S²
• Real projective plane, RP²
• Torus
• Klein bottle
• Cylinder
• Möbius strip
• Double torus

[edit] 3-manifolds

• 3-sphere, S³
• SO(3) ≅ RP³
• Poincaré sphere
• Whitehead manifold
• Weeks manifold
• Solid torus
• Solid Klein bottle

For more examples see 3-manifold.

[edit] 4-manifolds

• Exotic R⁴
• E8 manifold

For more examples see 4-manifold.

[edit] Special types of manifolds

[edit] Manifolds related to spheres

• Homology sphere
• Homotopy sphere
• Exotic sphere
• Milnor sphere
• Spherical 3-manifold
• Lens space

[edit] Special classes of Riemannian manifolds

• Einstein manifold
  • Ricci-flat manifold
• Kähler manifold
  • Calabi-Yau manifold
  • Hyperkähler manifold
• Quaternionic Kähler manifold
• G₂ manifold
• Spin(7) manifold

[edit] Categories of manifolds

[edit] Manifolds definable by a particular choice of atlas

• Topological manifold
• Piecewise linear manifold
• Differentiable manifold
• Smooth manifold
• Complex manifold

[edit] Manifolds with additional structure

• Riemannian manifold
• Pseudo-Riemannian manifold
• Finsler manifold
• Symplectic manifold
• Almost complex manifold
• Lie group

[edit] Infinite-dimensional manifolds

• Hilbert manifold
• Banach manifold
• Fréchet manifold