Abstract
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.
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A.L. Besse, Einstein Manifolds (Springer, Berlin-Heidelberg, 1978)
S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry (Springer, Berlin-Heidelberg, 1987)
B. O’Neill, Semi-Riemannian Geometry (Academic Press, New York-London, 1983)
M. Spivak, A Comprehensive Introduction to Differential Geometry, vol. I-V (Publish or Perish, Wilmington, 1979)
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Petersen, P. (2016). Examples. In: Riemannian Geometry. Graduate Texts in Mathematics, vol 171 . Springer, Cham. https://doi.org/10.1007/978-3-319-26654-1_4
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DOI: https://doi.org/10.1007/978-3-319-26654-1_4
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Online ISBN: 978-3-319-26654-1
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