Manifolds With Constant Sectional Curvature
Riemannian manifolds with constant sectional curvature are the most simple. These are called space forms. By rescaling the metric there are three possible cases
- negative curvature −1, hyperbolic geometry
- zero curvature, Euclidean geometry
- positive curvature +1, elliptic geometry
The model manifolds for the three geometries are hyperbolic space, Euclidean space and a unit sphere. They are the only complete, simply connected Riemannian manifolds of given sectional curvature. All other connected complete constant curvature manifolds are quotients of those by some group of isometries.
If for each point in a connected Riemannian manifold (of dimension three or greater) the sectional curvature is independent of the tangent 2-plane, then the sectional curvature is in fact constant on the whole manifold.
Read more about this topic: Sectional Curvature
Famous quotes containing the words constant and/or sectional:
“But when with moving accents thou
Shalt constant faith and service vow,
Thy Celia shall receive those charms
With open ears, and with unfolded arms.”
—Thomas Carew (1589–1639)
“It is to be lamented that the principle of national has had very little nourishment in our country, and, instead, has given place to sectional or state partialities. What more promising method for remedying this defect than by uniting American women of every state and every section in a common effort for our whole country.”
—Catherine E. Beecher (1800–1878)