Heat kernel on a finite isometric quotient ($K_{U\backslash N}(t,\bar{x},\bar{y})={\sum }_{u\in U}{K}_N(t,x,uy)$)

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.