  A manifestly covariant equation is derived to describe the second order
perturbations in topological defects and membranes on arbitrary curved
background spacetimes. This, on one hand, generalizes work on macroscopic
strings in Minkowski spacetime and introduces a framework for studing in a
precise manner membranes behavior near the black hole horizon and on the other
hand, introduces a more general framework for examining the stability of
topological defects in curved spacetimes.
