  This rather technical paper presents some generalization of the results of
recent publications \cite{Shirkov2010, DVPF2010, PFDV2010} where toy models of
dimensional reduction of space-time were considered. Here we introduce and
consider a specific type of multidimensional space-times with nontrivial
topology and nontrivial Riemannian metric, which admit a reduction of the
dimension $d$ of the space to any lower one $d_{low} = 1, 2,..., d-1$. The
variable geometry is described by several variable radii of compactification of
part of space dimensions. We succeed once more in transforming the shape of the
variable geometry of the $d$-dimensional spaces under consideration to a
specific potential interaction, described by the potential $V$ in the
one-dimensional Schr\"odinger-like equation. This way one may hope to study the
possible physical signals going from both higher and lower dimensions into our
obviously four dimensional real world.
