In Riemannian geometry, a branch of mathematics, the prescribed scalar curvature problem is as follows: given a closed, smooth manifold M and a smooth, real-valued function ÃÂ on M, construct a Riemannian metric on M whose scalar curvature equals ÃÂ. Due primarily to the work of Jerry Kazdan and Frank Wilson Warner in the 1970s, this problem is well understood.
If the dimension of M is three or greater, then any smooth function ÃÂ which takes on a negative value somewhere is the scalar curvature of some Riemannian metric. The assumption that ÃÂ be negative somewhere is needed in general, since not all manifolds admit metrics which have strictly positive scalar curvature. (For example, the three-dimensional torus is such a manifold.) However, Kazdan and Warner proved that if M does admit some metric with strictly positive scalar curvature, then any smooth function ÃÂ is the scalar curvature of some Riemannian metric.