This note investigates the possibility of converses of the Weyl theorems that two conformally related metrics on a manifold have the same Weyl conformal tensor and that two projectively related connections on a manifold have the same Weyl projective tensor. It shows that, in all relevant cases, counterexamples to each of Weyl’s theorems exist except for his conformal theorem in the 4-dimensiona...

In this paper we study a Riemanian metric on the tangent bundle T (M) of a Riemannian manifold M which generalizes the Cheeger Gromoll metric and a compatible almost complex structure which together with the metric confers to T (M) a structure of locally conformal almost Kählerian manifold. We found conditions under which T (M) is almost Kählerian, locally conformal Kählerian or Kählerian or wh...

Recent work in even-dimensional conformal geometry [1],[3], [4], [6] has revealed the importance of conformally invariant powers of the Laplacian on a conformal manifold; that is, of operators Pk whose principal part is the same as ∆ k with respect to a representative of the conformal structure. These invariant powers of the Laplacian were first defined in [5] in terms of the Fefferman-Graham [...

Let M be a complete Einstein manifold of negative curvature, and assume that (as in the AdS/CFT correspondence) it has a Penrose compactification with a conformal boundary N of positive scalar curvature. We show that under these conditions, H n (M ; Z) = 0 and in particular N must be connected. These results resolve some puzzles concerning the Suppose that M is a complete Einstein manifold of n...

In this paper we formulate new curvature functions on Sn via integral operators. For certain even orders, these curvature functions are equivalent to the classic curvature functions defined via differential operators, but not for all even orders. Existence result for antipodally symmetric prescribed curvature functions on Sn is obtained. As a corollary, the existence of a conformal metric for a...

We develop a general theory of transport-limited aggregation phenomena occurring on curved surfaces, based on stochastic iterated conformal maps and conformal projections to the complex plane. To illustrate the theory, we use stereographic projections to simulate diffusionlimited-aggregation (DLA) on surfaces of constant Gaussian curvature, including the sphere (K > 0) and pseudo-sphere (K < 0)...

We show that sets of conformal data on closed manifolds with the metric in the positive or zero Yamabe class, and with the gradient of the mean curvature function sufficiently small, are mapped to solutions of the vacuum Einstein constraint equations. This result extends previous work which required the conformal metric to be in the negative Yamabe class, and required the mean curvature functio...

In the general Euclidean formulation of the AdS/CFT correspondence put forth in [25], one considers Riemannian manifolds of the form M × Y , where M is conformally compactifiable with conformal boundaryat-infinity N. It is then a problem of fundamental interest to determine, for a given compact manifold N with a given conformal structure, the complete Einstein manifolds of negative Ricci curvat...

In the context of the AdS/CFT correspondence, an explicit relation between the physical degrees of freedom of 2+1d gravity and the stress tensor of 1+1d conformal field theory is exhibited. Gravity encodes thermodynamic state variables of conformal field theory, but does not distinguish among different CFT states with the same expectation value for the stress tensor. Simply put, gravity is ther...

We present a new approach to constructing inflationary models in closed universes. Conformal embedding of closed-universe models in a de Sitter background suggests a quantisation condition on the available conformal time. This condition implies that the universe is closed at no greater than the 10% level. When a massive scalar field is introduced to drive an inflationary phase this figure is re...