= Heat kernel on a finite isometric quotient
{title2=$K_{U\backslash N}(t,\bar x,\bar y)=\sum_{u\in U}K_N(t,x,uy)$}
For a finite free action by <Riemannian isometries>, the <Riemannian heat kernel> on the quotient is the image sum shown above. Integrating over a fundamental domain combines all images into the integral on the covering manifold, proving the initial condition and showing that there is no averaging factor in the kernel. The <heat trace> does have an averaging factor $1/|U|$, because the integral of a quotient function over the cover is $|U|$ times its quotient integral. Mere freeness of a general nondiscrete group action is not enough for this covering formula.
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