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The geometry of conformally-embedded hypersurfaces in general compact manifolds is important for the study of boundary value PDEs. As part of a larger program studying the conformal geometry of such embeddings, I will construct a sequence of conformally invariant tensors that generalize the second fundamental form. Like the second fundamental form, these tensors encode the local extrinsic curvatures of such an embedding. In particular, our main result shows that these tensors characterize the failure of a conformally compact manifold to have an asymptotic Poincare-Einstein structure. The frequency with which these tensors appear in the calculus of conformally-embedded hypersurfaces suggests a deeper picture which is yet to be fully understood.