  We examine the Kaluza-Klein (KK) dimensional reduction from
higher-dimensional space-time and the properties of the resultant
Bergmann-Wagoner general action of scalar-tensor theories. With the analysis of
the perturbations, we also investigate the stability of the anti-de Sitter
(AdS) space-time in the $D\in\mathcal{N}$-dimensional Einstein gravity with the
negative cosmological constant. Furthermore, we derive the conditions for the
dimensional reduction to successfully be executed and present the KK
compactification mechanism.
