Heat semigroup descent through a finite normal covering ($I\varphi =|U{|}^{-1/2}\varphi \circ p$)

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.