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 (global) isometries, has constant curvature.
- 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.
Read more about this topic: Constant Curvature
Famous quotes containing the word properties:
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—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)