05294nam 2201033Ia 450 991081332620332120210520003344.01-280-49429-897866135895211-4008-4272-710.1515/9781400842728(CKB)2550000001273118(EBL)870005(OCoLC)780425982(SSID)ssj0000623887(PQKBManifestationID)11388715(PQKBTitleCode)TC0000623887(PQKBWorkID)10656329(PQKB)11538876(StDuBDS)EDZ0000406948(DE-B1597)447835(OCoLC)979624183(DE-B1597)9781400842728(Au-PeEL)EBL870005(CaPaEBR)ebr10539569(CaONFJC)MIL358952(PPN)199244308(PPN)18795965X(FR-PaCSA)88837999(MiAaPQ)EBC870005(EXLCZ)99255000000127311820111018d2012 uy 0engurun#---|u||utxtccrThe decomposition of global conformal invariants /Spyros AlexakisCourse BookPrinceton Princeton University Press20121 online resource (460 p.)Annals of mathematics studies ;no. 182Description based upon print version of record.0-691-15348-5 0-691-15347-7 Includes bibliographical references and index.Front matter --Contents --Acknowledgments --1. Introduction --2. An Iterative Decomposition of Global Conformal Invariants: The First Step --3. The Second Step: The Fefferman-Graham Ambient Metric and the Nature of the Decomposition --4. A Result on the Structure of Local Riemannian Invariants: The Fundamental Proposition --5. The Inductive Step of the Fundamental Proposition: The Simpler Cases --6. The Inductive Step of the Fundamental Proposition: The Hard Cases, Part I --7. The Inductive Step of the Fundamental Proposition: The Hard Cases, Part II --A. Appendix --Bibliography --Index of Authors and Terms --Index of SymbolsThis book addresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies. The question concerns conformally invariant functionals on the space of Riemannian metrics over a given manifold. These functionals act on a metric by first constructing a Riemannian scalar out of it, and then integrating this scalar over the manifold. Suppose this integral remains invariant under conformal re-scalings of the underlying metric. What information can one then deduce about the Riemannian scalar? Deser and Schwimmer asserted that the Riemannian scalar must be a linear combination of three obvious candidates, each of which clearly satisfies the required property: a local conformal invariant, a divergence of a Riemannian vector field, and the Chern-Gauss-Bonnet integrand. This book provides a proof of this conjecture. The result itself sheds light on the algebraic structure of conformal anomalies, which appear in many settings in theoretical physics. It also clarifies the geometric significance of the renormalized volume of asymptotically hyperbolic Einstein manifolds. The methods introduced here make an interesting connection between algebraic properties of local invariants--such as the classical Riemannian invariants and the more recently studied conformal invariants--and the study of global invariants, in this case conformally invariant integrals. Key tools used to establish this connection include the Fefferman-Graham ambient metric and the author's super divergence formula.Annals of mathematics studies ;no. 182.Conformal invariantsDecomposition (Mathematics)CauchyВiemann geometry.DeserГchwimmer conjecture.Khler geometry.Riemannian invariants.Riemannian metrics.Riemannian scalar.Schouten tensor.Weyl tensor.algebraic propositions.ambient metrics.conformal anomalies.conformal invariant.conformal invariants.conformally invariant functionals.curvature tensor.decomposition.differential geometry.global conformal invariant.global invariants.grand conclusion.index theory.induction.iterative decomposition.lemma.lemmas.manifold.theoretical physics.Conformal invariants.Decomposition (Mathematics)518Alexakis Spyros1978-1629867MiAaPQMiAaPQMiAaPQBOOK9910813326203321The decomposition of global conformal invariants3967866UNINA