scalar curvature
Curvature can be described in finer and finer summaries: a number for each 2-plane (sectional), a number for each direction (Ricci), and finally a single number for the whole point. That last, coarsest summary is the scalar curvature. It boils all the curvature at a point down to one value, answering the bluntest possible question: at this point, is a small ball bigger or smaller than a ball of the same radius in flat space?
Precisely, the scalar curvature S (often written R) is the full contraction of the Ricci tensor with the metric: S = sum g^jk Ric_jk, equivalently the average of all the Ricci curvatures over all directions, scaled up by the dimension. The geometric meaning is concrete: in a space of dimension n, the volume of a small geodesic ball of radius r is the flat-space volume reduced by a factor proportional to S times r^2 — positive scalar curvature makes small balls smaller than Euclidean, negative makes them larger. In dimension 2 the scalar curvature is just twice the Gaussian curvature, tying the whole hierarchy back to Gauss.
Scalar curvature is the weakest curvature invariant — it averages away almost everything, so a space can have constant positive scalar curvature while looking very different in different directions. Yet it carries real weight: it is the integrand of the Einstein-Hilbert action whose variation yields Einstein's equations, and deep theorems (positive mass, the existence of positive-scalar-curvature metrics) hinge on it. The honest caveat is precisely its coarseness: knowing S tells you the total curvature budget at a point but nothing about how it is distributed among directions — many genuinely different geometries share the same scalar curvature.
On a sphere of radius a in 2 dimensions, the Gaussian curvature is 1/a^2, so the scalar curvature is S = 2/a^2 — a single positive number. It says small disks on the sphere have slightly less area than flat disks of equal radius, the precise amount of crowding you would measure if you tried to tile the sphere with flat paper.
Scalar curvature compresses all curvature at a point to one number — the fractional volume deficit of a tiny ball.
Scalar curvature is the coarsest curvature measure: it can stay constant while the geometry varies enormously direction by direction. Constant scalar curvature is much weaker than constant sectional curvature, and the two must not be conflated.