Riemannian Geometry, Fiber Bundles, Kaluza-Klein Theories and All That…

The following sections are included:

• Acknowledgments

• CONTENTS

The following sections are included:

• Introduction
  • Physical motivations
  • Content of the book
  • New results

• Differentiable manifolds

• Riemannian manifolds
  • Metrics, connections and curvatures
    • Metrics
    • Connections
    • Curvatures
    • Lie derivatives
    • Riemannian spaces with transitive group of isometries
  • Particular spaces
    • Spaces of constant curvature
    • Einstein spaces
    • Spaces of constant scalar curvature
    • Einstein metrics as saddle points

The following sections are included:

• Summary

• The case of SU(2)

• Left and right fundamental fields

• Principal fibration of a group with respect to a subgroup

• Bi-invariant metrics

• Left and Right invariant metrics

• Metrics with isometry group H×K where H.K are subgroups of G

• G×K invariant metrics related to a fibration of G

• Dimensionally reducible metrics (action of a bundle of groups)

• Einstein metrics on groups

• On the classification of compact simple Lie groups

• Standard normalizations, indices etc.
  • -Conventional normalization of generators in Physics
  • -Indices of representations
  • -Coupling constants in Yang-Mills theories
  • -Casimir operators
  • -Embedding coefficient and Dynkin index
  • -Relation between normalization of volumes, scalar curvature etc.
  • -Standard volumes on spheres and projective spaces

• Pointers to the literature

The following sections are included:

• Summary

• The example of S²

• The role of the normalizer N of H in G
  • Spheres
  • Projective spaces

• Fundamental fields

• G-invariant metrics on H\G
  • Comments

• Invariant metrics on H\G related to the fibration of H\G

• Einstein metrics on homogeneous spaces

• Dimensionally reducible metrics

• Pointers to the literature

The following sections are included:

• Summary section

• Example of M×SU(2)

• Fundamental fields on P
  • Definition
  • Properties

• Local product representation of P
  • Local trivialization of P
  • Definition of invariant fields (via the choice of a local section)
  • Relation between fundamental fields and (local) invariant fields

• G-invariant metrics and the reduction theorem
  • The reduction theorem
  • Proof of the reduction theorem
  • Comments on scalar fields
  • Adapted moving frame
  • Several expressions for the metric and for the connection
  • Examples

• Curvature of G-invariant metrics on a principal bundle P
  • Levi-Civita connection in the adapted moving frame
  • Ricci tensor and scalar curvature
  • Comments

• Action principle and the consistency requirement
  • The (dimensionnally reduced) scalar curvature as a Lagrangian?
  • Validity of the consistency requirement

• Conformal rescaling and the effective Lagrangian
  • Conformal rescaling
  • The particular case G=U(1)

• On the Interpretation of the scalar fields. Physical units
  • Comment on scalar fields
  • Cosmological constant and units

• Color charges, scalar charges and the particule trajectories
  • Geodesics in P
  • Projection in M: equations of motion

• Generalised Kaluza-Klein metrics (action of a bundle of groups)
  • Motivations
  • G×G invariant metrics on the tautological bundle P×G
  • Dimensional reduction and inconsistency of this ansatz

• The group of automorphisms of a principal bundle P
  • The structure group and associated bundles
  • The group Aut P of automorphisms of P
  • The group Int P of vertical automorphism P (the gauge group)
  • The pointed gauge group
  • The infinite-dimensional Lie algebras: aut P =

• Pointers to the literature

The following sections are included:

• Summary

• The structure of a simple G-space
  • Summary
  • Bundle structure of E
  • The role of the normalizer
  • From P(M,N|H) to E(M,H\G)
  • Global action of G on E(M,H\G)
  • Embedding of P in E
  • The “other” fibration of E

• Examples
  • Trivial examples
  • Less trivial examples
  • Even less trivial examples
  • Remark
  • Unexpected examples

• Fundamental and invariant vector fields
  • Fundamental vector fields
  • Invariant vector fields

• G-invariant metrics on E
  • The reduction theorem
  • Proof of the reduction theorem
  • Comments on scalar fields
    • Constraints on the fields h_αβ(x)
    • Counting the number of scalar fields
    • Remark
  • Adapted moving frame and explicit expression for the metric
    • The moving frame
    • Expressions for the metric
  • Structure functions of the adapted moving frame

• Curvature tensors for G-invariant metrics
  • Levi-Civita connection in the adapted moving frame
  • Ricci tensor and scalar curvature
  • Comments

• Examples
  • Counting the number of scalar fields
  • A class of almost trivial examples
  • Model Building

• Action principle and the consistency requirement
  • The (dimensionally reduced) scalar curvature as a Lagrangian?
  • Validity of the consistency requirement

• Conformal rescaling and the effective Lagrangian

• Normalization and units, the potential for scalar fields
  • Behaviour of the potential for scalar fields
  • Example of M×S³

• Color charges, scalar charges and the particle trajectories

• Generalized Kaluza-Klein metrics (action of a bundle of groups)
  • Motivations
  • G×N|H-invariant metrics on the tautological bundle E × G
  • Groups of automorphisms

• Some complements on G-spaces
  • The principal orbit theorem and stratification
  • Generalization of the reduction theorem for G-invariant metrics

• Pointers to the literature

The following sections are included:

• Summary section

• Description of matter fields

• Covariant derivative and curvature

• The case of M endowed with affine connection and / or metric
  • Covariant derivative of vector fields on M
  • Covariant derivative of arbitrary tensor fields (tensor-valued 0-forms) on M
  • Covariant derivative of tensor-valued r-forms on M

• The case of a bundle P = P(M.G) endowed with a principal connection ω and whose base is endowed with affine connection

• Spin structures and spinors
  • Spin structures on an orientable manifold E
  • The bundle of spinors
  • The spin-connection
  • Covariant derivative on spinors. The Dirac operator
  • Spinors before metric?

• Generalized spin structures

• An example: Einstein-Cartan theory with spinor fields

• Miscellaneous

• Pointers to the literature

The following sections are included:

• Summary Section

• A particular case of 7.2: non-abelian harmonic analysis

• Harmonic expansion (generalized Peter-Weyl theorem)

• A particular case of 7.4: induced representations

• Harmonic expansion -2- (generalized Frobenius theorem)

• Harmonic expansion and dimensional reduction

• Generalised homogeneous differential operators and the consistency problem for matter fields

• Pointers to the literature

The following sections are included:

• Summary

• The space of adapted orthonormal frames
  • Adapted orthonormal frames
  • The effective gauge group
  • The affine connection in an adapted orthonormal frame
  • The effective gauge fields

• Dimensionally reduced Laplace-Beltrami operators
  • Laplace operator on scalar fields
  • Laplace operator on 1-forms

• Dimensional reduction of spinor fields

• The spectrum of Laplace operator for G invariant metrics on groups and homogeneous spaces
  • Generalities
  • The Laplacian on a homogeneous 2 sphere
  • The Laplacian on S³ with a SU(2)×SU(2) invariant metric
  • The Laplacian on S³ with a SU(2)_L × U(1)_R invariant metric
  • The Lapiacian on S³ with an SU(2) - invariant metric
  • Laplacian on G with a bi -invariant metric
  • Laplacian on G for a metric invariant under G×H
  • Laplacian on S⁷ with an SO(8) - invariant metric
  • S⁷ with a metric invariant under U(2, IH)×SU(2)

• The spectrum of the Dirac operator for G-invariant metrics on groups and homogeneous spaces
  • The Dirac operator on the two-sphere
  • The Dirac operator on S³ for a bi-invariant metric under SU(2)
  • The Dirac operator on S³ with a metric invariant under SU(2)×U(1)

• Examples of dimensional reduction of spinor fields

• Pointers to the literature

The following sections are included:

• Summary section
  • Introduction
  • Content of the chapter
  • Summary of the results
  • The Space of Higgs Fields
  • Extrema of Potentials

• Symmetries of a Principal Bundle
  • Symmetries of Yang-Mills Fields (global description)
  • Symmetries of Yang-Mills fields (local description)
  • Several Bundle Structures of the Principle Bundle U
  • Lie Algebra Decomposition and the Vielbein

• Reduction of the Einstein Yang-Mills Action
  • Outline of the Method
  • The Link No.2
  • The Link No. 3
  • The Link No. 4
  • The Link No 5
  • The Link No. 1

• Examples and Comments

• Pointers to the literature

The following sections are included:

• Motivations

• Examples of dimensionally reducible metrics (but not G-invariant)
  • Remark
  • The case where G acts globally on E=E(M,H\G)
  • The case where G does not act globally on E=E(M,H\G)
  • Double cosets
  • The Klein-bottle

• An extended Kaluza-Klein scheme

The following sections are included:

• BIBLIOGRAPHY AND REFERENCES