List of manifolds

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

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
• 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 ≅ RP³
• Solid Klein bottle
• Solid torus
• Whitehead manifold
• Meyerhoff manifold
• Weeks manifold

4-manifolds

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

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 manifold

  Categories of manifolds

Manifolds definable by a particular choice of atlas

• Affine manifold
• Analytic manifold
• Complex manifold
• Differentiable manifold
• Piecewise linear manifold
• Smooth 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