Scalar curvature - Knowlpedia

Definition

Let $(M,g)$ be a pseudo-Riemannian manifold. Its scalar curvature is the smooth function

\operatorname{Scal}_g=\operatorname{tr}_g(\operatorname{Ric}) =g^{ij}\operatorname{Ric}_{ij},

obtained by contracting the Ricci tensor with the inverse metric.

Scalar curvature is a single contraction of the curvature tensor and therefore does not determine the full curvature in dimensions greater than two. Its sign depends on the sign convention for the Riemann curvature tensor. The convention here agrees with the linked Ricci-curvature knowl.

Applications

Scalar curvature appears as the curvature term in conformal scalar coupling and in geometric operators such as the conformal Laplacian.

References

John M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Springer, 2018. Publisher record. Relevant: Chapter 7.