= Heat semigroup descent through a finite normal covering
{title2=$I\phi=|U|^{-1/2}\phi\circ p$}
The normalized pullback is a unitary map from $L^2$ of the quotient onto the deck-invariant subspace of $L^2$ of the cover. Local isometry and the sheet-counting integration formula preserve the Dirichlet quadratic form. Therefore the <Friedrichs extension> and its <heat semigroup> intertwine with this map. A finite sum of lifted <Riemannian heat kernels> gives the quotient kernel; independence of lifts follows by reindexing the deck group. This formulation avoids a uniqueness claim for unrestricted heat-equation solutions on a noncompact or incomplete manifold.
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