Constant curvature

Classification

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

  Properties

• Every space of constant curvature is locally symmetric, i.e. its curvature tensor is parallel.
• Every space of constant curvature is locally maximally symmetric, i.e. it has 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 isometries, has constant curvature.
• The universal cover of a manifold of constant sectional curvature is one of the model spaces:
• * sphere
• * plane
• * hyperbolic manifold
• 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.
• 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.