The Ambient Metric (eBook)

Charles Fefferman, C. Robin Graham (Autoren)

eBook Download: EPUB

2011
128 Seiten
Princeton University Press (Verlag)
978-1-4008-4058-8 (ISBN)

Infos

This book develops and applies a theory of the ambient metric in conformal geometry. This is a Lorentz metric in n+2 dimensions that encodes a conformal class of metrics in n dimensions. The ambient metric has an alternate incarnation as the Poincare metric, a metric in n+1 dimensions 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. Poincare metrics are introduced and shown to be equivalent to the ambient formulation. Self-dual Poincare 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.

Autor

Charles Fefferman is the Herbert E. Jones, Jr., '43 University Professor of Mathematics at Princeton University. C. Robin Graham is professor of mathematics at the University of Washington.

Produkt-Details

• Erscheint lt. Verlag: 14.11.2011
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• Schlagworte: Addition • ambient curvature • ambient metric • ambient metric forms • Ambient space • analytic function • Asymptote • asymptotic expansion • Big O notation • bijection • Calculation • Change of variables • Christoffel Symbols • Codimension • coefficient • combination • Computation • conformal curvature tensors • Conformal geometry • Conformal group • conformal infinity • conformal invariants • conformal manifold • coordinate system • Cotton tensor • covariant derivative • Curvature • Curvature tensor • Derivative • diffeomorphism • differentiable manifold • Differential operator • Dirac Operator • Eistein metric • Endomorphism • Equation • equivalence class • Error Term • existential quantification • Explicit formula • fiber bundle • flat manifolds • Formal power series • Fuchsian problems • geodesic normal coordinates • GJMS operator • Homogeneous polynomial • hyperbolic space • hypersurface • infinite-order formal theory • Infinitesimal generator (stochastic processes) • Initial Condition • Interval (mathematics) • Invariant theory • isometry • Isomorphism Theorem • Iteration • jet isomorphism theorem • Levi-Civita connection • Lie derivative • Linear combination • Linearization • local diffeomorphism • Lorentz metric • manifold • Mathematical Induction • matrix group • Metric tensor • Minkowski space • Monomial • n+2 dimensions • Natural number • Neighbourhood (mathematics) • Open set • ORDER BY • Orthogonal complement • orthogonal group • Pairing • parabolic invariant theory • Parametrix • partial differential equation • Permutation • Poincar metric • Poincar metrics • polynomial • power series • Pseudo-Riemannian manifold • Pseudo-Riemannian metric • pullback • quadratic form • quadratic function • Requirement • Ricci curvature • Riemann curvature tensor • Riemannian Geometry • Riemannian metrics • Riemann surface • scalar curvature • scalar invariants • scientific notation • self-dual Einstein metric • Sign convention • Smoothness • submanifold • Subset • Summation • symmetrization • tangent bundle • Tangent Space • Tangent vector • Taylor expansion • Taylor series • Tensor • Tensor Algebra • Theorem • Trace (linear algebra) • Triangular Matrix • two-dimensional space • uniformization theorem • Vector field • Vector Space • Volume form • Weyl invariants • Weyl's theorem • Weyl tensor
• ISBN-10: 1-4008-4058-9 / 1400840589
• ISBN-13: 978-1-4008-4058-8 / 9781400840588