Conformal Symmetry Breaking Operators for Differential Forms on Spheres

This work is the first systematic study of all possible conformally covariant differential operators transforming differential forms on a Riemannian manifold X into those on a submanifold Y with focus on the model space (X, Y) = (Sn, Sn-1).The authors give a complete classification of all such confo...

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Bibliographic Details
Main Author Kobayashi, Toshiyuki
Format eBook Book
LanguageEnglish
Published Singapore Springer Nature 2016
Springer
Springer Singapore
Edition1
SeriesLecture Notes in Mathematics
Subjects
Broken symmetry (Physics)
Conformal geometry
Differential Geometry
Differential operators
Fourier Analysis
Geometry
Geometry, Differential
Mathematical Physics
Mathematics
Mathematics and Statistics
Partial Differential Equations
Topological Groups, Lie Groups
Online AccessGet full text
ISBN9789811026577
9811026572
9811026564
9789811026560
ISSN0075-8434
1617-9692
DOI10.1007/978-981-10-2657-7

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Table of Contents:
  • Intro -- Contents -- Summary -- Chapter 1 Introduction -- Chapter 2 Symmetry Breaking Operators and Principal Series Representations of G=O(n+1,1) -- 2.1 Principal Series Representations of G=O(n+1,1) -- 2.2 Conformal View on Principal Series Representations of O(n+1,1) -- 2.3 Representation Theoretic Properties of (ω(i)u,δ, Ei(Sn)) -- 2.4 Differential Symmetry Breaking Operators for Principal Series -- 2.5 Symmetry Breaking Operators for Connected Group SO0(n,1) -- 2.6 Branching Problems for Verma Modules -- Chapter 3 F-method for Matrix-Valued Differential Operators -- 3.1 Algebraic Fourier Transform -- 3.2 Differential Operators Between Two Manifolds -- 3.3 F-method for Principal Series Representations -- 3.4 Matrix-Valued Differential Operators in the F-method -- Chapter 4 Matrix-Valued F-method for O(n+1,1) -- 4.1 Strategy of Matrix-Valued F-method for (G,G') = (O(n+1,1), O(n,1)) -- 4.2 Harmonic Polynomials -- 4.3 Description of HomL'(V,W Pol(n+)) -- 4.4 Decomposition of the Equation (dπ(σ,λ)*(N+1)idw)Ψ=0 -- 4.5 Matrix Coefficients in the F-method -- Chapter 5 Application of Finite-Dimensional Representation Theory -- 5.1 Signatures in Index Sets -- 5.2 Action of O(N) on the Exterior Algebra *(CN) -- 5.3 Construction of Intertwining Operators -- 5.4 Application of Finite-Dimensional Representation Theory -- 5.5 Classification of HomO(n-1) (i(Cn), j(Cn-1)Hk(Cn-1)) -- 5.6 Descriptions of HomO(n-1) (i(Cn) ,j(Cn-1)Pol[ζ1,…, ζn]) -- 5.7 Proof of the Implication (i)(iii) in Theorem 2.8 -- Chapter 6 F-system for Symmetry Breaking Operators (j=i-1, i case) -- 6.1 Proof of Theorem 2.8 for j = i-1, i -- 6.2 Reduction Theorem -- 6.3 Step 2: Matrix Coefficients MIJ for (dπ(i,λ)*(N+1)Ψ -- 6.4 Step 3: Case-Reduction for MvectIJ -- 6.5 Step 4 - Part I: Formulæ for Saturated Differential Equations -- 6.6 Step 4 - Part II: Explicit Formulæ for MIJ
  • Hodge Star Operator and Branson's Operator T2(i) -- Chapter 13 Matrix-Valued Factorization Identities -- 13.1 Matrix-Valued Factorization Identities -- 13.2 Proof of Theorem 13.1 (1) -- 13.3 Proof of Theorem 13.1 (2) -- 13.4 Proof of Theorem 13.2 (1) -- 13.5 Proof of Theorem 13.2 (2) -- 13.6 Proof of Theorem 13.3 -- 13.7 Proof of Theorem 13.4 -- 13.8 Renormalized Factorization Identities -- Chapter 14 Appendix: Gegenbauer Polynomials -- 14.1 Normalized Gegenbauer Polynomials -- 14.2 Derivatives of Gegenbauer Polynomials -- 14.3 Three-Term Relations Among Renormalized Gegenbauer Polynomials -- 14.4 Duality of Gegenbauer Polynomials for Special Values -- 14.5 Proof of Theorem 6.7 -- References -- List of Symbols -- Index
  • 6.7 Step 5: Deduction from MIJ=0 to Lr(g0,g1, g2) = 0 -- Chapter 7 F-system for Symmetry Breaking Operators (j = i-2, i+1 case) -- 7.1 Proof of Theorem 7.1 -- Chapter 8 Basic Operators in Differential Geometry and Conformal Covariance -- 8.1 Twisted Pull-Back of Differential Forms by Conformal Transformations -- 8.2 Hodge Star Operator Under Conformal Transformations -- 8.3 Normal Derivatives Under Conformal Transformations -- 8.4 Basic Operators on Ei(Rn) -- 8.5 Transformation Rules Involving the Hodge Star Operator and Restxn=0. -- 8.6 Symbol Maps for Differential Operators Acting on Forms -- Chapter 9 Identities of Scalar-Valued Differential Operators Dul -- 9.1 Homogeneous Polynomial Inflation Ia -- 9.2 Identities Among Juhl's Conformally Covariant Differential Operators -- 9.3 Proof of Proposition 1.4 -- 9.4 Two Expressions of Di→i-1u,a -- Chapter 10 Construction of Differential Symmetry Breaking Operators -- 10.1 Proof of Theorem 2.9 in the Case j=i-1 -- 10.2 Proof of Theorem 2.9 in the Case j=i+1 -- 10.3 Application of the Duality Theorem for Symmetry Breaking Operators -- 10.4 Proof of Theorem 2.9 in the Case j=i -- 10.5 Proof of Theorem 2.9 in the Case j=i-2 -- Chapter 11 Solutions to Problems A and B for (Sn, Sn-1) -- 11.1 Problems A and B for Conformal Transformation Group Conf(X -- Y) -- 11.2 Model Space (X,Y)=(Sn,Sn-1) -- 11.3 Proof of Theorem 1.1 -- 11.4 Proof of Theorems 1.5-1.8 -- 11.5 Change of Coordinates in Symmetry Breaking Operators -- Chapter 12 Intertwining Operators -- 12.1 Classification of Differential Intertwining Operators Between Forms on Sn -- 12.2 Differential Symmetry Breaking Operators Between Principal Series Representations -- 12.3 Description of HomL(V,WPol(n+)) -- 12.4 Solving the F-system when j=i+1 -- 12.5 Solving the F-system when j=i -- 12.6 Solving the F-system when j=i-1 -- 12.7 Proof of Theorem 12.1