Acceleration of the Universe via $f(R)$ gravities and the stability of extra dimensions

We discuss the possibility of obtaining the present acceleration of the Universe via $f(R)$ gravity theories which recently attracted much attention. It is known that $f(R)$ theories generally have room for this. In this work we stress that the requirement for the stabilization of extra dimensions is naturally incorporated in such a generalization of Einstein gravity under rather orthodox assumptions. We have restricted our study to pure $f(R)$ gravity without additional matter sources partially in view of the fact that if the acceleration is to continue indefinitely any ordinary matter term is to redshift to irrelevancy and mostly for economy in understanding. The general conditions we find are that the manifold of the extra dimensional space is to have negative internal curvature and that the Ricci scalar of the full space-time manifold is also negative. The positive curvature case for an extra dimensional manifold is actually the most generic case. However, this necessitates a fine-tuning between the Hubble constant and the size of extra dimensions in the absence of matter sources.