scalar curvature

If Ricci curvature still carries directional information, scalar curvature is the single number you get by averaging away even the direction: one function on the manifold summarizing how curved the space is, overall, near each point. It is the crudest curvature invariant, and remarkably it is exactly the term that tells you whether small balls hold more or less volume than they would in flat space.

Scalar curvature S (often denoted R or Scal) is the trace of the Ricci tensor with respect to the metric: S = g^ij Ric_ij. Equivalently it is the sum, over an orthonormal basis e_i, of the sectional curvatures of all the coordinate 2-planes (counted with the right combinatorial factor), so it averages sectional curvature twice over. Its cleanest geometric meaning: the volume of a small geodesic ball of radius r satisfies Vol(B_r) = omega_n r^n ( 1 - (S/(6(n+2))) r^2 + ... ), where omega_n is the Euclidean ball volume. Positive S means small balls are slightly smaller than Euclidean; negative S means slightly larger.

Where it shows up and the honest hierarchy. Scalar curvature is the integrand of the Einstein-Hilbert action in general relativity, the quantity whose total over a surface is controlled by Gauss-Bonnet in dimension 2 (where S = 2K). It is also the WEAKEST of the curvatures: it is two traces below the full Riemann tensor, so a scalar bound says far less than a Ricci bound, which says far less than a sectional bound. Positive scalar curvature in particular is a famously weak condition — it permits topology that positive Ricci forbids — so never read 'positive scalar curvature' as 'positively curved' in the sectional sense.