Geometry of deformations of branes in warped backgrounds

The 'braneworld' (described by the usual worldvolume action) is a D-dimensional timelike surface embedded in an N-dimensional (N > D) warped, nonfactorizable spacetime. We first address the conditions on the warp factor required to have an extremal flat brane in a five-dimensional background. Subsequently, we deal with normal deformations of such extremal branes. The ensuing Jacobi equations are analysed to obtain the stability condition. It turns out that to have a stable brane, the warp factor should have a minimum at the location of the brane in the given background spacetime. To illustrate our results, we explicitly check the extremality and stability criteria for a few known co-dimension one braneworld models. Generalizations of the above formalism for the cases of (i) curved branes (ii) asymmetrical warping and (iii) higher co-dimension braneworlds are then presented along with some typical examples for each. Finally, we summarize our results and provide perspectives for future work along these lines