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 ƒ on M, construct a Riemannian metric on M whose scalar curvature equals ƒ.
Due primarily to the work of J. Kazdan and F. Warner in the 1970s, this problem is well understood.
If the dimension of M is three or greater, then any smooth function ƒ which takes on a negative value somewhere is the scalar curvature of some Riemannian metric.
The assumption that ƒ be negative somewhere is needed in general, since not all manifolds admit metrics which have strictly positive scalar curvature.
However, Kazdan and Warner proved that if M does admit some metric with strictly positive scalar curvature, then any smooth function ƒ is the scalar curvature of some Riemannian metric.