= Curvature coefficient of the scalar heat kernel
{title2=$K(t;x,x)=(4\pi t)^{-d/2}(1+tR(x)/6+O(t^2))$}
For $L=-\nabla^a\nabla_a$ on scalar functions, the leading coefficient of the <heat parametrix> in <normal coordinates> is the inverse square root of the metric volume density: $u_0=1+R_{ab}\xi^a\xi^b/12+O(|\xi|^3)$. The <heat-kernel transport equations> give $u_1(x,x)=\nabla^a\nabla_a u_0(x,x)=R(x)/6$. This gives the displayed diagonal expansion. The coefficient is specific to the minimally coupled scalar operator; a bundle connection or a curvature-dependent potential changes the general Laplace-type coefficients.
Back to article page
