= Heat kernel trace formula
{title2=$\operatorname{Tr}e^{-t\Delta}=\int_M K_t(x,x)\,d\mathrm{vol}_g(x)$}
On a <closed manifold>, the <positive Laplace-Beltrami operator> has a smooth <heat kernel> and its <heat semigroup> is a <trace-class operator> for positive time. Expanding in a smooth <orthonormal eigenbasis> gives $K_t(x,y)=\sum_je^{-t\lambda_j}\phi_j(x)\overline{\phi_j(y)}$. <Elliptic regularity> bounds each derivative of $\phi_j$ polynomially in $\lambda_j$, while the exponential dominates those powers, so the expansion converges with all derivatives. Integrating the diagonal and using the unit norm of each <eigenfunction> proves the formula. For a bundle operator, insert the fibre <matrix trace> inside the integral.
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