Topological Manifold - Examples

Examples

• The real coordinate space Rn is the prototypical n-manifold.
• Any discrete space is a 0-dimensional manifold.
• A circle is a compact 1-manifold.
• A torus and a Klein bottle are compact 2-manifolds (or surfaces).
• The n-dimensional sphere Sn is a compact n-manifold.
• The n-dimensional torus Tn (the product of n circles) is a compact n-manifold.
• Projective spaces over the reals, complexes, or quaternions are compact manifolds.
  • Real projective space RPn is a n-dimensional manifold.
  • Complex projective space CPn is a 2n-dimensional manifold.
  • Quaternionic projective space HPn is a 4n-dimensional manifold.
• Manifolds related to projective space include Grassmannians, flag manifolds, and Stiefel manifolds.
• Lens spaces are a class of manifolds that are quotients of odd-dimensional spheres.
• Lie groups are manifolds endowed with a group structure.
• Any open subset of an n-manifold is a n-manifold with the subspace topology.
• If M is an m-manifold and N is an n-manifold, the product M × N is a (m+n)-manifold.
• The disjoint union of a family of n-manifolds is a n-manifold (the pieces must all have the same dimension).
• The connected sum of two n-manifolds results in another n-manifold.

See also: List of manifolds