This article is a survey of recent work [15, 6, 7, 13] developing a new approach to quantization based on the equivariance with respect to some Lie group of symmetries. Examples are provided by conformal and projective differential geometry: given a smooth manifold M endowed with a flat conformal/projective structure, we establish a canonical isomorphism between the space of symmetric contravar...

A theorem of Escobar asserts that, on a positive three dimensional smooth compact Riemannian manifold with boundary which is not conformally equivalent to the standard three dimensional ball, a necessary and sufficient condition for a C2 function H to be the mean curvature of some conformal flat metric is that H is positive somewhere. We show that all such metrics stay in a compact set with res...

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...

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...

The aim of this paper is to extend the notion all known quasi-Einstein manifolds like generalized quasi-Einstein, mixed manifold, pseudo manifold and many more name it comprehensive quasi Einstein C(QE)$_{n}$. We investigate some geometric physical properties C(QE)$_{n}$ under certain conditions. study conformal conharmonic mappings between manifolds. Then we examine with harmonic Weyl tensor. ...

In this paper, we prove that a complete noncompact non-flat conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton. 1. The result A complete Riemannian metric gij on a smooth manifold M n is called a gradient steady Ricci soliton if there exists a smooth function F on M such that the Ricci tensor Rij of the metric gij is given by the Hessian of F : Rij = ∇i∇jF. (1....