Scalar curvature

Scalar curvature is the simplest curvature invariant of a Riemannian manifold: at each point, it is the trace of the Ricci curvature tensor, a single number measuring whether the volume of a small geodesic ball exceeds or falls short of the volume of a Euclidean ball of the same radius. Positive scalar curvature means the manifold is locally volume-deflating; negative scalar curvature means it is volume-inflating. The scalar curvature is weak enough that it does not control topology — there are manifolds with positive scalar curvature and arbitrarily complicated topology — but it is strong enough to appear in the Einstein field equations and in the Yamabe invariant, which asks whether every conformal class of metrics contains one of constant scalar curvature. The persistent mystery is why scalar curvature, the weakest of the classical curvatures, should appear in the deepest physical equations; the answer may be that it is not curvature at all but a Lagrange multiplier for volume constraint.