A unimodular Kaluza-Klein theory

Unimodular gravity became an object of increasing interest in the late $80$-ties and was recently used in primordial Universe modeling with cosmological constant, in the context of the Brans-Dicke gravity including scalar field. In the present article we investigate the possibility of imposing the unimodular condition within the $5$-dimensional Kaluza-Klein theory including the scalar field. The variational principle is formulated in $5$ dimensions first, and dimensional reduction is applied to the resulting set of equations. A cosmological model based on these equations is then presented and discussed.