- 8.5k Downloads
We are now ready to compute the curvature tensors on all of the examples constructed in chapter 1 After a few more general computations, we will exhibit Riemannian manifolds with constant sectional, Ricci, and scalar curvature. In particular, we shall look at the space forms S k n , products of spheres, and the Riemannian version of the Schwarzschild metric. We also offer a local characterization of certain warped products and rotationally symmetric constant curvature metrics in terms of the Hessian of certain modified distance functions.
KeywordsWarped Product Constant Curvature Metrics Modified Distance Function Schwarzschild Metric Scalar ﬂat Metrics
- 12.A.L. Besse, Einstein Manifolds (Springer, Berlin-Heidelberg, 1978)Google Scholar
- 47.S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry (Springer, Berlin-Heidelberg, 1987)Google Scholar
- 80.B. O’Neill, Semi-Riemannian Geometry (Academic Press, New York-London, 1983)Google Scholar
- 97.M. Spivak, A Comprehensive Introduction to Differential Geometry, vol. I-V (Publish or Perish, Wilmington, 1979)Google Scholar