= Integrated second scalar heat coefficient
{title2=$a_2=\frac1{360}\int_M(5R^2-2|\operatorname{Ric}|^2+2|\operatorname{Rm}|^2)\,dV$}
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, $R=2K$, $|\operatorname{Ric}|^2=2K^2$, $|\operatorname{Rm}|^2=4K^2$, giving $a_2=\frac1{15}\int_MK^2\,dA$. Together with volume and total <Gaussian curvature>, it determines the variance of curvature and detects constant curvature.
Back to article page
