Alice Chang’s Question: A very general question is to ask “What is the geometric content of Q-curvature?” For example, we know that one can associate the scalar curvature with the conformally invariant constant called the “Yamabe constant”. When this constant is positive, it describes the best constant (in a conformally invariant sense) of the Sobolev embedding of W 1,2 into L space; this in it...

A considerable interest has been shown to the study of physical properties of spacetimes which are conformal to certain well known gravitational fields. The general theory of relativity is believed by a number of unknown functions the ten components of g . Hence there is a little hope of finding physically interesting results without making reduction in their number. In conformally flat spaceti...

The objective of control generation in logic programming is to derive a computation rule for a program that is efficient and yet does not compromise program correctness. Progress in solving this fundamental problem in logic programming has been slow and, to date, only partial solutions have been proposed. Previously proposed schemes are either inefficient, incomplete (incorrect) or difficult to...

We discuss expansions for the Cotton-York tensor near infinity for arbitrary slices of stationary spacetimes. From these expansions it follows directly that a necessary condition for the existence of conformally flat slices in stationary solutions is the vanishing of a certain quantity of quadrupolar nature (obstruction). The obstruction is nonzero for the Kerr solution. Thus, the Kerr metric a...

We study the renormalized volume of a conformally compact Einstein manifold. In even dimenions, we derive the analogue of the Chern-Gauss-Bonnet formula incorporating the renormalized volume. When the dimension is odd, we relate the renormalized volume to the conformal primitive of the Q-curvature. 0. Introduction Recently, there is a series of work ([GZ],[FG-2] and [FH]) exploring the connecti...

Conformally equivariant quantization is a peculiar map between symbols of real weight δ and differential operators acting on tensor densities, whose real weights are designed by λ and λ + δ. The existence and uniqueness of such a map has been proved by Duval, Lecomte and Ovsienko for a generic weight δ. Later, Silhan has determined the critical values of δ for which unique existence is lost, an...

A massive scalar field in a curved spacetime can propagate along the light cone, a causal pathology, which can, in principle, be eliminated only if the scalar couples conformally to the Ricci curvature of spacetime. This property mandates conformal coupling for the field driving inflation in the early universe. During slow-roll inflation, this coupling can cause super-acceleration and, as a sig...

A new derivation is given of Branson’s factorization formula for the conformally invariant operator on the sphere whose principal part is the k-th power of the scalar Laplacian. The derivation deduces Branson’s formula from knowledge of the corresponding conformally invariant operator on Euclidean space (the k-th power of the Euclidean Laplacian) via conjugation by the stereographic projection ...

In this paper we prove that every Riemannian metric on a locally conformally flat manifold with umbilic boundary can be conformally deformed to a scalr flat metric having constant mean curvature. This result can be seen as a generalization to higher dimensions of the well known Riemann mapping Theorem in the plane.

We introduced a class of conformally invariant Ehresmann connections so–called L-horizontal endomorphism in [7]. Using this class, we define conformally invariant manifolds: Wagner–type manifold and locally Minkowski–type manifold as special generalized Berwald manifolds. Then a generalization of Hashiguchi–Ichijyō’s Theorems to Wagner–type manifolds is presented. Mathematics Subject Classifica...