  We consider a theory of modified gravity possessing d extra spatial
dimensions with a maximally symmetric metric and a scale factor, whose
(4+d)-dimensional gravitational action contains terms proportional to quadratic
curvature scalars. Constructing the 4D effective field theory by dimensional
reduction, we find that a special case of our action where the additional terms
appear in the well-known Gauss-Bonnet combination is of special interest as it
uniquely produces a Horndeski scalar-tensor theory in the 4D effective action.
We further consider the possibility of achieving stabilised extra dimensions in
this scenario, as a function of the number and curvature of extra dimensions,
as well as the strength of the Gauss-Bonnet coupling. Further questions that
remain to be answered such as the influence of matter-coupling are briefly
discussed.
