= Sunada orbital heat-trace formula
{c}
{title2=$Z_{U\backslash N}(t)=|U|^{-1}\sum_{u\in U}I_t(u)$}
For a finite <subgroup> $U$ of a <group> $T$ of <Riemannian isometries> of a <closed manifold> $N$, assume $U$ acts freely and define $I_t(g)=\int_N H_N(t,x,gx)\,dV(x)$. Invariance of the <Riemannian heat kernel> and change of variables show that $I_t$ is a <class function> on $T$. For $U\le T$, the <heat kernel on a finite isometric quotient> and integration over a <fundamental domain> give the displayed <heat trace> formula. Grouping by <conjugacy classes> proves equality for <Gassmann equivalent> <subgroups>. More generally the same argument allows fixed points of $T$ outside the freely acting <subgroups>; only the quotient under consideration must be smooth.
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