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Branes typically have infinite volume. As an intuitive example of a brane, imagine an infinite plane in 3D, which is a 2-brane. The brane is specified by (for example), $z=0$ if $(x,y,z)$ are the usual Cartesian coordinates. An infinite line is a 1-brane and is specified by (again, for example) $y=z=0$. If the branes are Euclidean and wrapped up, then it is ...
Let $\Omega^2=\exp\phi/2$ so that $\gamma=\Omega^2 g$. Then we have the standard formula $$R_\gamma=\Omega^{-2}\big(R_g-2(n-1)\Delta\ln\Omega-(n-2)(n-1)[\nabla\ln\Omega]^2\big)$$ This is proven in any number of General Relativity texts, but the one in R.M. Wald, General Relativity (1984), is particularly easy to follow and is proved in the form shown here. ...