Supergravity description of field theories on curved manifolds and a no go theorem

In the first part of this paper we find supergravity solutions corresponding to branes on worldvolumes of the form $R^d\times \mathrm{\Sigma }$ where $\mathrm{\Sigma }$ is a Riemann surface. These theories arise when we wrap branes on holomorphic Riemann surfaces inside $K3$ or CY manifolds. In some cases the theory at low energies is a conformal field theory with two less dimensions. We find some non-singular supersymmetric compactifications of M-theory down to $Ad{S}_5$. We also propose a criterion for permissible singularities in supergravity solutions. In the second part of this paper, which can be read independently of the first, we show that there are no non-singular Randall-Sundrum or de-Sitter compactifications for large class of gravity theories.