eBook

The ambient metric

Graham, C. Robin, Fefferman, Charles

2012,2011

Overview

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.

• Publisher: Princeton University Press
• Subject: Ambient space / Analytic function / Asymptote / Calculation / Change of variables / Coefficient / Computation / Conformal geometry / Conformal group / Conformal invariants / Cotton tensor / Covariant derivative / Curvature / Curvature tensor / Derivative / Diffeomorphism / Differentiable manifold / Equation / Equivalence class / Existential quantification / Explicit formula / Fiber bundle / Formal power series / Hyperbolic space / Hypersurface / Initial condition / Interval (mathematics) / Invariant theory / Isomorphism theorem / Levi-Civita connection / Linear combination / Linearization / Local diffeomorphism / Mathematical induction / MATHEMATICS / MATHEMATICS / Geometry / Analytic / Metric tensor / Monomial / Neighbourhood (mathematics) / Order by / Orthogonal group / Pairing / Partial differential equation / Permutation / Polynomial / Pseudo-Riemannian manifold / Quadratic form / Requirement / Ricci curvature / Riemann surface / Riemannian geometry / Scalar curvature / Scientific notation / Sign convention / Smoothness / Submanifold / Subset / Summation / Symmetrization / Tangent bundle / Tangent space / Tangent vector / Taylor series / Tensor / Tensor algebra / Theorem / Trace (linear algebra) / Triangular matrix / Uniformization theorem / Vector field / Vector space / Volume form / Weyl tensor
• ISBN: 9780691153148, 0691153140, 0691153132, 9780691153131, 1400840589, 9781400840588