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 Z₂ 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. (orig.)