Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...

, a branch of mathematics, the prescribed scalar curvature problem is as follows: given a closed,

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

''M'' and a smooth, real-valued function ''ƒ'' on ''M'', construct a

Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...

on ''M'' whose

scalar curvature In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometr ...

equals ''ƒ''. Due primarily to the work of J. Kazdan and F. Warner in the 1970s, this problem is well understood.

The solution in higher dimensions

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. (For example, the three-dimensional

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...

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 ''ƒ'' is the scalar curvature of some Riemannian metric.

References

*Aubin, Thierry. ''Some nonlinear problems in Riemannian geometry.'' Springer Monographs in Mathematics, 1998. *Kazdan, J., and Warner F. ''Scalar curvature and conformal deformation of Riemannian structure.'' Journal of Differential Geometry. 10 (1975). 113–134. Riemannian geometry Mathematical problems

Scalar curvature In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometr ...

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The solution in higher dimensions

See also

References