Effective gravitational field equations on m-brane embedded in n-dimensional bulk of Einstein and f (R) gravity

We have derived effective gravitational field equations on a lower dimensional hypersurface (known as a brane), placed in a higher dimensional bulk spacetime for both Einstein and f(R) gravity theories. We have started our analysis on the n-dimensional bulk from which the effective field equations on a (n - 1)-dimensional brane have been obtained by imposing Z(2) symmetry. Subsequently, we have arrived at the effective equations in (n - 2) dimensions starting from the effective equations for (n - 1)-dimensional brane. This analysis has been carried out and is used to obtain the effective field equations for an (n - m)-dimensional brane, embedded in a n- dimensional bulk. Having obtained the effective field equations in Einstein gravity, we have subsequently generalized the effective field equation for an (n - m)-dimensional brane which is embedded in the n-dimensional bulk spacetime endowed with f (R) gravity. We have also presented applications of our results in the context of Einstein and f (R) gravity. In both cases we have discussed static spherically symmetric vacuum solutions as well as solutions in a cosmological context. Implications are also discussed.