= Euler characteristic from the heat trace of a bordered surface
{title2=$[t^0]\operatorname{Tr}(e^{-tP})=\chi(M)/6$}
For a compact smooth <Riemannian surface> with smooth boundary and a pure <Dirichlet boundary condition> or pure <Neumann boundary condition>, the constant coefficient in the <heat trace> is
$$
\frac1{12\pi}\left(\int_MK\,dA+\int_{\partial M}\kappa\,ds\right)=\frac{\chi(M)}6.
$$
The equality is the <Gauss-Bonnet theorem>. Thus such <isospectral manifolds> have the same <Euler characteristic>. A <simply connected> compact connected bordered surface is a disk with $\chi=1$, whereas a nonorientable surface with $c\geq1$ crosscaps and $b\geq1$ boundary components has $\chi=2-c-b\leq0$. These cannot be isospectral in this setting. Corners and mixed <boundary conditions> contribute additional heat coefficients, so the smooth pure-boundary-condition hypothesis is essential.
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