Induced worldvolume metric ($h_{\mu \nu }={\partial }_{\mu }{X}^a{\partial }_{\nu }{X}^b{g}_{ab}$)

The induced worldvolume metric is the pullback of the target metric tensor by a brane embedding. For a nondegenerate timelike embedding, $\sqrt{-\det h}$ is its invariant worldvolume volume density. The $p=1$ specialization is the induced worldsheet metric.