  The AdS/Ricci-flat (AdS/RF) correspondence is a map between families of
asymptotically locally AdS solutions on a torus and families of asymptotically
flat spacetimes on a sphere. The aim of this work is to perturbatively extend
this map to general AdS and asymptotically flat solutions. A prime application
for such map would be the development of holography for Minkowski spacetime. In
this paper we perform a Kaluza-Klein (KK) reduction of AdS on a torus and of
Minkowski on a sphere, keeping all massive KK modes. Such computation is
interesting on its own, as there are relatively few examples of such explicit
KK reductions in the literature. We perform both KK reductions in parallel to
illustrate their similarity. In particular, we show how to construct gauge
invariant variables, find the field equations they satisfy, and construct a
corresponding effective action. We further diagonalize all equations and find
their general solution in closed form. Surprisingly, in the limit of large
dimension of the compact manifolds (torus and sphere), the AdS/RF
correspondence maps individual KK modes from one side to the other. In a sequel
of this paper we will discuss how the AdS/RF maps acts when the dimension of
the compact space is finite.
