Μαθηματικά Πολύπτυχα

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.

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.

Manifolds of a specific dimension

1-manifolds

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

2-manifolds

• Sphere, S²
• Real projective plane, RP²
• Torus
• Klein bottle
• Cylinder
• Möbius strip
• Double torus
• Klein quartic (a genus 3 surface)

3-manifolds

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

For more examples see 3-manifold.

4-manifolds

• Exotic R⁴
• E8 manifold

For more examples see 4-manifold.

Special types of manifolds

Manifolds related to spheres

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

Special classes of Riemannian manifolds

• Riemannian symmetric space
• Einstein manifold
  • Ricci-flat manifold
• Kahler manifold
  • Calabi-Yau manifold
  • Hyperkahler manifold
• Quaternionic Kähler manifold
• G₂ manifold
• Spin(7) manifold

Categories of manifolds

Manifolds definable by a particular choice of atlas

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

Manifolds with additional structure

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

Infinite-dimensional manifolds

• Hilbert manifold
• Banach manifold
• Frechet manifold