Kaluza-Klein reduction and Bergmann-Wagoner bi-scalar general action of scalar-tensor gravity

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.