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The Ambient Metric (AM-178)

The Ambient Metric (AM-178)

Charles Fefferman
C. Robin Graham
Copyright Date: 2012
Pages: 128
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  • Book Info
    The Ambient Metric (AM-178)
    Book Description:

    This book develops and applies a theory of the ambient metric in conformal geometry. This is a Lorentz metric inn+2dimensions that encodes a conformal class of metrics inndimensions. The ambient metric has an alternate incarnation as the Poincaré metric, a metric inn+1dimensions having the conformal manifold as its conformal infinity. In this realization, the construction has played a central role in the AdS/CFT correspondence in physics.

    The existence and uniqueness of the ambient metric at the formal power series level is treated in detail. This includes the derivation of the ambient obstruction tensor and an explicit analysis of the special cases of conformally flat and conformally Einstein spaces. Poincaré metrics are introduced and shown to be equivalent to the ambient formulation. Self-dual Poincaré metrics in four dimensions are considered as a special case, leading to a formal power series proof of LeBrun's collar neighborhood theorem proved originally using twistor methods. Conformal curvature tensors are introduced and their fundamental properties are established. A jet isomorphism theorem is established for conformal geometry, resulting in a representation of the space of jets of conformal structures at a point in terms of conformal curvature tensors. The book concludes with a construction and characterization of scalar conformal invariants in terms of ambient curvature, applying results in parabolic invariant theory.

    eISBN: 978-1-4008-4058-8
    Subjects: Mathematics

Table of Contents

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  1. Front Matter
    (pp. [i]-[vi])
  2. Table of Contents
    (pp. [vii]-[x])
  3. Chapter One Introduction
    (pp. 1-8)

    Conformal geometry is the study of spaces in which one knows how to measure infinitesimal angles but not lengths. A conformal structure on a manifold is an equivalence class of Riemannian metrics, in which two metrics are identified if one is a positive smooth multiple of the other. The study of conformal geometry has a long and venerable history. From the beginning, conformal geometry has played an important role in physical theories.

    A striking historical difference between conformal geometry compared with Riemannian geometry is the scarcity of local invariants in the conformal case. Classically known conformally invariant tensors include the...

  4. Chapter Two Ambient Metrics
    (pp. 9-16)

    LetMbe a smooth manifold of dimensionn≥ 2 equipped with a conformal class [g]. Here,gis a smooth pseudo-Riemannian metric of signature (p,q) onMand [g] consists of all metrics\[\hat g = e^{2\Upsilon } g\]onM, where ϒ is any smooth real-valued function onM.

    The space 𝓖 consists of all pairs (h,x), wherexM, andhis a symmetric bilinear form onTxMsatisfyingh=s²gxfor some$s\in {{\mathbb{R}}_{+}}$. Here and below,gxdenotes the symmetric bilinear form onTxMinduced by the metricg. We write π : 𝓖 →...

  5. Chapter Three Formal Theory
    (pp. 17-41)

    The first goal of this chapter is to prove Theorem 2.9 forn> 2. We begin with the following lemma.

    Lemma 3.1.Let($\tilde{\cal{ G \cal} }, \tilde{ g }$)be a pre-ambient space for(M, [g]),where$\tilde{\cal{ G \cal}}$has the property that for each z ∈ 𝓖, the set of all ρ ∈ ℝ such that$(z, \rho ) \in \tilde{\cal{ G \cal}}$is an open interval Izcontaining0.Let g be a metric in the conformal class, with associated identification$\mathbb{R}_{ + }\times M \times \mathbb{R} \simeq \cal{ G \cal} \times \mathbb{R}$.Then($\tilde{\cal{ G \cal}}, \tilde{ g }$)is in normal form relative to g if and only if one has on$\tilde{ \cal{ G \cal}}$\[\tilde g_{0\infty } = t,\quad \tilde g_{i\infty } = 0,\quad \tilde g_{\infty \infty } = 0. \caption{(3.1)}\]

    Proof. Since a pre-ambient metric satisfies$\iota *\tilde g = g_0$, if$\tilde g$...

  6. Chapter Four Poincaré Metrics
    (pp. 42-49)

    In this chapter we consider the formal theory for Poincaré metrics associated to a conformal manifold (M, [g]). We will see that even Poincaré metrics are in one-to-one correspondence with straight ambient metrics, if both are in normal form. Thus the formal theory for Poincaré metrics is a consequence of the results of Chapter 3. The derivation of a Poincaré metric from an ambient metric was described in [FG], and the inverse construction of an ambient metric as the cone metric over a Poincaré metric was given in §5 of [GrL].

    The definition of Poincaré metrics is motivated by the...

  7. Chapter Five Self-dual Poincaré Metrics
    (pp. 50-55)

    In [LeB], LeBrun showed using twistor methods that ifgis a real-analytic metric on an oriented real-analytic 3-manifoldM, then [g] is the conformal infinity of a real-analytic self-dual Einstein metric on a deleted collar neighborhood ofM× {0} inM× [0,∞), uniquely determined up to real-analytic diffeomorphism. As mentioned in [FG], LeBrun’s result can be proved as an application of our formal theory of Poincaré metrics. In this chapter we show that the corresponding formal power series statement is a consequence of Theorem 4.8. The self-duality condition can be viewed as providing a conformally invariant specification...

  8. Chapter Six Conformal Curvature Tensors
    (pp. 56-66)

    In this chapter we study conformal curvature tensors of a pseudo-Riemannian metricg. These are defined in terms of the covariant derivatives of the curvature tensor of an ambient metric in normal form relative tog. Their transformation laws under conformal change are given in terms of the action of a subgroup of the conformal groupO(p+ 1,q+ 1) on tensors. We assume throughout this chapter thatn≥ 3.

    Letgbe a metric on a manifoldM. By Theorem 2.9, there is an ambient metric in normal form relative tog, which by Proposition 2.6...

  9. Chapter Seven Conformally Flat and Conformally Einstein Spaces
    (pp. 67-81)

    Ifnis odd, an ambient metric in normal form is uniquely determined to infinite order by (M,g). Theorem 3.9 shows that within the family of all formal solutions, this one is distinguished by the vanishing of theρn/2coefficient, or equivalently by the condition that it be smooth. In the Poincaré realization, the corresponding condition is that the expansion have nornterm, or equivalently that it be even. The fact that there is such a normalization giving rise to a unique diffeomorphism class of ambient metrics associated to the conformal manifold (M, [g]) is crucial for the...

  10. Chapter Eight Jet Isomorphism
    (pp. 82-96)

    A fundamental result in Riemannian geometry is the jet isomorphism theorem which asserts that at the origin in geodesic normal coordinates, the full Taylor expansion of the metric may be recovered from the iterated covariant derivatives of curvature. As a consequence, one deduces that any local invariant of Riemannian metrics has a universal expression in terms of the curvature tensor and its covariant derivatives. Geodesic normal coordinates are determined up to the orthogonal group, so problems involving local invariants are reduced to purely algebraic questions concerning invariants of the orthogonal group on tensors.

    Our goal in this chapter is to...

  11. Chapter Nine Scalar Invariants
    (pp. 97-106)

    The jet isomorphism theorem 8.10 reduces the study of local invariants of conformal structures to the study ofP-invariants of$\tilde{\cal{ R \cal}}$(we must of course impose the usual finite-order truncation forneven). An invariant theory for scalarP-invariants of$T\tilde{\cal{ R \cal}}$was developed in [BEGr]. In this chapter we show how to derive a characterization of scalar invariants of conformal structures by reduction to the relevant results of [BEGr].

    Recall that a scalar invariantI(g) of metrics of signature (p,q) is a polynomial in the variables${{\left( {{\partial }^{\alpha }}{{g}_{ij}} \right)}_{\left| \alpha\right|\ge0}}$and${{\left| \det \,{{g}_{ij}} \right|}^{-1/2}}$, which is coordinate-free in the sense that its value...

  12. Bibliography
    (pp. 107-112)
  13. Index
    (pp. 113-113)