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

Abstract

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.