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

Riemannian $\mathbf{(1+d)}$-Dim Space-Time Manifolds with Nonstandard  Topology which Admit Dimensional Reduction to Any Lower Dimension and  Transformation of the Klein-Gordon Equation to the $\mathbf{1}$-Dim  Schr\"odinger Like Equation