Duality Symmetry in Kaluza-Klein $n+D+d$ Dimensional Cosmological Model
  It is shown that, with the only exception of $n=2$, the Einstein-Hilbert action in $n+D+d$ dimensions, with $n$ times, is invariant under the duality transformation $a\to \frac{1}{a}$ and $b\to \frac{1}{b}$, where $a$ is a Friedmann-Robertson-Walker scale factor in $D$ dimensions and $b$ a Brans-Dicke scalar field in $d$ dimensions respectively. We investigate the $2+D+d$ dimensional cosmological model in some detail. 
J. A. Nieto, M. P. Ryan, O. Velarde, C. M. Yee