  We develop the recent proposal to use dimensional reduction from the
four-dimensional space-time D=(1+3) to the variant with a smaller number of
space dimensions D=(1+d), d < 3, at sufficiently small distances to construct a
renormalizable quantum field theory. We study the Klein-Gordon equation on a
few toy examples ("educational toys") of a space-time with variable special
geometry, including a transition to a dimensional reduction. The examples
considered contain a combination of two regions with a simple geometry
(two-dimensional cylindrical surfaces with different radii) connected by a
transition region. The new technique of transforming the study of solutions of
the Klein-Gordon problem on a space with variable geometry into solution of a
one-dimensional stationary Schr\"odinger-type equation with potential generated
by this variation is useful. We draw the following conclusions: (1) The signal
related to the degree of freedom specific to the higher-dimensional part does
not penetrate into the smaller-dimensional part because of an inertial force
inevitably arising in the transition region (this is the centrifugal force in
our models). (2) The specific spectrum of scalar excitations resembles the
spectrum of the real particles; it reflects the geometry of the transition
region and represents its "fingerprints". (3) The parity violation due to the
asymmetric character of the construction of our models could be related to
violation of the CP symmetry.
