11. The Yamabe problem is the following : Given a smooth, conformal to for which the scalar curvature of is constant? 12. This operator often makes an appearance when studying how the scalar curvature behaves under a conformal change of a Riemannian metric. 13. Here R ( g ) is the scalar curvature constructed from the metric g _ { \ mu \ nu }. 14. Where Rm is the full Riemann curvature tensor, Rc is the Ricci curvature tensor, and R is the scalar curvature . 15. Thus every manifold of dimension at least 3 has a metric with negative scalar curvature , in fact of constant negative scalar curvature. 16. Thus every manifold of dimension at least 3 has a metric with negative scalar curvature, in fact of constant negative scalar curvature . 17. Diego L . Rapoport, on the other hand, associates the relativistic quantum potential with the metric scalar curvature ( Riemann curvature ). 18. Where ? is the Laplace-Beltrami operator ( of negative spectrum ), and " R " is the scalar curvature . 19. It is also exactly half the scalar curvature of the 2-manifold, while the Ricci curvature tensor of the surface is simply given by 20. Yau and Schoen continued their work on manifolds with positive scalar curvature , which led to Schoen's final solution of the Yamabe problem.
