Obstructions to conformally Einstein metrics in n dimensions

Gover, A. Rod; Nurowski, Paweł

doi:10.1016/j.geomphys.2005.03.001

Cited by 95 publications

(134 citation statements)

References 18 publications

Supporting

Mentioning

133

Contrasting

Order By: Relevance

“…It is only a slight generalisation of the Einstein condition, and in fact it follows easily from the Theorem above, that on an almost Einstein manifold σ is non-zero on an open dense set [30]. Note that the zero locus Z(σ) of the "scale" σ is a conformal infinity, if non-empty.…”

Section: Elements Of Tractor Calculusmentioning

confidence: 95%

Boundary calculus for conformally compact manifolds

Gover¹,

Waldron²

2014

Indiana Univ. Math. J.

View full text Add to dashboard Cite

Abstract. On conformally compact manifolds of arbitrary signature, we use conformal geometry to identify a natural (and very general) class of canonical boundary problems. It turns out that these encompass and extend aspects of already known holographic bulk-boundary problems, the conformal scattering description of boundary conformal invariants, and corresponding questions surrounding a range of physical bulk wave equations. These problems are then simultaneously solved asymptotically to all orders by a single universal calculus of operators that yields what may be described as a solution generating algebra. The operators involved are canonically determined by the bulk (i.e. interior) conformal structure along with a field which captures the singular scale of the boundary; in particular the calculus is canonical to the structure and involves no coordinate choices. The generic solutions are also produced without recourse to coordinate or other choices, and in all cases we obtain explicit universal formulae for the solutions that apply in all signatures and to a range of fields. A specialisation of this calculus yields holographic formulae for GJMS operators and Branson's Q-curvature.

show abstract

Section: Elements Of Tractor Calculusmentioning

confidence: 95%

Boundary calculus for conformally compact manifolds

Gover¹,

Waldron²

2014

Indiana Univ. Math. J.

View full text Add to dashboard Cite

show abstract

“…To obtain its correct tractor transformation law, one must first transform σ as a weight one scalar and subsequently evaluate the derivatives in D M . 10 The space of solutions to the requirement of parallel I M can be enhanced to include almost Einstein structures by allowing zeroes in σ. at conformal infinities [43,44].…”

Section: Scalarsmentioning

confidence: 99%

Tractors, mass, and Weyl invariance

2009

View full text Add to dashboard Cite

Deser and Nepomechie established a relationship between masslessness and rigid conformal invariance by coupling to a background metric and demanding local Weyl invariance, a method which applies neither to massive theories nor theories which rely upon gauge invariances for masslessness. We extend this method to describe massive and gauge invariant theories using Weyl invariance. The key idea is to introduce a new scalar field which is constant when evaluated at the scale corresponding to the metric of physical interest. This technique relies on being able to efficiently construct Weyl invariant theories. This is achieved using tractor calculus-a mathematical machinery designed for the study of conformal geometry. From a physics standpoint, this amounts to arranging fields in multiplets with respect to the conformal group but with novel Weyl transformation laws. Our approach gives a mechanism for generating masses from Weyl weights. BreitenlohnerFreedman stability bounds for Anti de Sitter theories arise naturally as do direct derivations of the novel Weyl invariant theories given by Deser and Nepomechie. In constant curvature spaces, partially massless theories-which rely on the interplay between mass and gauge invariance-are also generated by our method. Another simple consequence is conformal invariance of the maximal depth partially massless theories. Detailed examples for spins s ≤ 2 are given including tractor and component actions, on-shell and off-shell approaches and gauge invariances. For all spins s ≥ 2 we give tractor equations of motion unifying massive, massless, and partially massless theories.

show abstract

“…Since any conformally Einstein manifold is necessarily Bach flat, Theorem 2.4 shows that the analysis of the conformally Einstein equation (42) 2 Hes ϕ +ϕ ρ = 1 4 {2∆ϕ + ϕ τ }g must be carried out only for the following cases: It is important to emphasize that although any locally conformally Einstein metric is Bach flat, there are examples of strictly Bach flat manifolds, i.e., they are neither half conformally flat nor locally conformally Einstein (see, for example [1,10,18] and references therein). Indeed, one has the following necessary conditions for any solution of (42) (see also [14]). Recall that the solutions ϕ of the conformally Einstein equation (42) and the functions σ in Proposition 4.1-(1) are related by σ = −2 log(ϕ).…”

Section: Conformally Einstein Non-reductive Homogeneous Spacesmentioning

confidence: 99%

“…Hence, the first non-trivial case to study is dimension four, where harmonicity of the Weyl tensor is a necessary condition to be conformally Einstein. Gover and Nurowski [14] obtained some tensorial obstructions for a metric to be conformally Einstein under some non-degeneracy conditions for the conformal Weyl tensor. It is still an open question to characterize conformally Einstein manifolds by tensorial equations.…”

Section: Introductionmentioning

confidence: 99%