Scalar curvature

Remarks

The value does not depend on the coordinates; it is the trace of the linear map $A_p \colon T_pM \to T_pM$ with $g(A_p u, v) = \operatorname{Ric}(u, v)$. In the Riemannian case, for an orthonormal basis $e_1, \ldots, e_n$ of $T_pM$, $S(p) = \sum_i \operatorname{Ric}(e_i, e_i) = \sum_{i \ne j} K(e_i, e_j)$.

Example: on the unit sphere $S^n$, $\operatorname{Ric} = (n - 1)g$, so $S = \sum_{i,j} g^{ij} (n - 1) g_{ij} = (n - 1) n$.

Non-example: the plain sum $\sum_i R_{ii}$, without the factor $g^{ij}$, depends on the coordinates and is not the scalar curvature. On the sphere of radius $r$ in $\R^3$, $\operatorname{Ric} = r^{-2} g$; in coordinates at $p$ with $g_{ij}(p) = r^2 \delta_{ij}$, one gets $\sum_i R_{ii}(p) = 2$ for every $r$, while $S = 2 / r^2$.

For a surface, $S = 2K$, where $K$ is the Gaussian curvature.