AdS geometry from CFT on a general conformally flat manifold

We construct an anti-de-Sitter (AdS) geometry from a conformal field theory (CFT) defined on a general conformally flat manifold via a flow equation associated with the curved manifold, which we refer to as the primary flow equation. We explicitly show that the induced metric associated with the primary flow equation becomes AdS whose boundary is the curved manifold. Interestingly, it turns out that such an AdS metric with conformally flat boundary is obtained from the usual Poincare AdS by a simple bulk finite diffeomorphism. We also demonstrate that the emergence of such an AdS space is guaranteed only by the conformal symmetry at boundary, which converts to the AdS isometry after quantum averaging, as in the case of the flat boundary. As a side remark we show that a geometry with one warped direction becomes an Einstein manifold if and only if so is its boundary at the warped direction, and briefly discuss a possibility of a little extension beyond AdS/CFT correspondence by using a genuine Einstein geometry.