Constant curvature

The Riemannian manifolds of constant curvature can be classified into the following three cases:

• elliptic geometry – constant positive sectional curvature
• Euclidean geometry – constant vanishing sectional curvature
• hyperbolic geometry – constant negative sectional curvature.

• Every space of constant curvature is locally symmetric, i.e. its curvature tensor is parallel ${\displaystyle \nabla \mathrm {R} =0}$.
• Every space of constant curvature is locally maximally symmetric, i.e. it has ${\displaystyle {\frac {1}{2}}n(n+1)}$ number of local isometries, where n is its dimension.
• Conversely, there exists a similar but stronger statement: every maximally symmetric space, i.e. a space which has ${\displaystyle {\frac {1}{2}}n(n+1)}$ (global) isometries, has constant curvature.
• (Killing–Hopf theorem) The universal cover of a manifold of constant sectional curvature is one of the model spaces:
  • sphere (sectional curvature positive)
  • plane (sectional curvature zero)
  • hyperbolic manifold (sectional curvature negative)
• A space of constant curvature which is geodesically complete is called space form and the study of space forms is intimately related to generalized crystallography (see the article on space form for more details).
• Two space forms are isomorphic if and only if they have the same dimension, their metrics possess the same signature and their sectional curvatures are equal.