Local $AdS_2\times S^2$Topology, Perfect Fluids and Conformal Weyl Gravity*

We study the solution space of Conformal Weyl Gravity with local $AdS_2\times S^2$ topology. We find a vacuum solution exhibiting a Petrov Type $O$ (conformally flat) and Segre Type $\{(111),1\}$ (perfect fluid) classification. We compute the perfect fluid energy momentum tensor within the Einstein Gravity frame work and also compute the conformal diffeomorphism, mapping the solution to flat spacetime modulo a conformal factor. We comment on future work concerning the quantum gravity of this solution within the Conformal Weyl Gravity paradigm and the $AdS/CFT$ correspondence.