List of manifolds explained

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

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

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