Geometry of deformations of branes in warped backgrounds
  The `braneworld' (described by the usual worldvolume action) is a D dimensional timelike surface embedded in a N dimensional ($N>D$) warped, nonfactorisable 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. Generalisations of the above formalism for the cases of (i) curved branes (ii) asymmetrical warping and (iii) higher co-dimension braneworlds are then presented alongwith some typical examples for each. Finally, we summarize our results and provide perspectives for future work along these lines. 
Supratik Pal, Sayan Kar (IIT Kharagpur)