 Integrated second scalar heat coefficient

For the positive scalar Laplace-Beltrami operator on a closed manifold with a Riemannian metric, the coefficient of $t^2$ in the normalized heat trace is the displayed integral. The local coefficient additionally has a scalar-curvature divergence term whose integral vanishes. In dimension two, $R=2K$, $|\mathrm{Ric}{|}^2=2K^2$, $|\mathrm{Rm}{|}^2=4K^2$, giving $a_2=\frac{1}{15}{\int }_M{K}^{2}dA$. Together with volume and total Gaussian curvature, it determines the variance of curvature and detects constant curvature.