Scalar Curvature - Special Cases - Space Forms

Space Forms

A space form is by definition a Riemannian manifold with constant sectional curvature. Space forms are locally isometric to one of the following types:

  • Euclidean space: The Riemann tensor of an n-dimensional Euclidean space vanishes identically, so the scalar curvature does as well.
  • n-spheres: The sectional curvature of an n-sphere of radius r is K = 1/r2. Hence the scalar curvature is S = n(n−1)/r2.
  • Hyperbolic spaces: By the hyperboloid model, an n dimensional hyperbolic space can be identified with the subset of (n+1)-dimensional Minkowski space
The parameter r is a geometrical invariant of the hyperbolic space, and the sectional curvature is K = −1/r2. The scalar curvature is thus S = −n(n−1)/r2.

Read more about this topic:  Scalar Curvature, Special Cases

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