Introduction to Global Variational Geometry

Tentative Table of Contents:PrefaceList of Standard SymbolsChapter 1: Smooth Manifolds1. Smooth Manifolds1.1. Smooth Manifolds1.2. Smooth Mappings1.3. Contact of Smooth Mappings1.4. Jet of a Mapping1.5. Composition of Mappings1.6. Submanifolds1.7. Product of Manifolds2. The Tangent Bundle2.1. Tangent Vectors2.2. Tangent Bundle2.3. Tangent Mapping3. Tensor Bundles3.1. Cotangent Bundle3.2. Tensor Bundles3.3. Fibered Product of Tensor Bundles3.4. Induced Morphisms of Tensor Bundles3.5. Tensor Fields4. The Rank of a Mapping4.1. The Rank Theorem4.2. Immersions4.3. Submersions5. Fibered Manifolds5.1. Fibered Manifolds5.2. Morphisms of Fibered Manifolds5.3. Vertical Vectors, Horizontal Forms5.4. Fibrations5.5. Sections of Fibered Manifolds5.6. Vector Bundles5.7. Vector Bundle Morphisms5.8. Inverse Image of a Vector BundleNotes and Additional TopicsChapter 2: Analysis on Manifolds1. Vector Fields1.1. Vector Fields1.2. Local Flow1.3. Global Flow1.4. Differential Equations2. Differential Forms2.1. Differential Forms2.2. Exterior Derivative2.3. The Poincare Lemma 2.4. Contraction of a Form by a Vector Field2.5. The Lie Derivative3. Distributions3.1. Vector Distributions3.2. Distributions Generated by Forms3.3. Complete Integrability3.4. Differential Ideals4. Integration of Forms4.1. Orientation of a Vector Space4.2. Orientability of Manifolds, Volume Elements4.3. Piece of a Manifold4.4. Boundary of a Piece, Induced Volume Elements4.5. Integration of Forms4.6. Transformation of an Integral4.7. Differentiation of an Integral with Respect to a Parameter4.8. Integration on the Product of Manifolds4.9. Integration of Exact Forms5. Integration of Odd Forms5.1. Odd Scalars, Odd Forms, Volume Elements 5.2. Integration of Odd Forms5.3. Integration of Functions5.4. Transformation of an Integral5.5. Differentiation of an Integral with Respect to a Parameter5.6. Integration on the Product of Manifolds5.7. Restriction of a Volume Element to a Submanifold5.8. Integration of Exact FormsNotes and Additional TopicsChapter 3: Lie Transformation Groups1. Lie Groups1.1. Lie Groups1.2. Lie Group Morphisms1.3. Lie Subgroups1.4. Normal Subgroups and Quotient Groups1.5. Invariant Vector Fields1.6. Lie Algebra of a Lie Group2. Products of Lie Groups2.1. Product of Lie Groups2.2. Interior Semi-Direct Product2.3. Exterior Semi-Direct Product 3. Lie Group Actions3.1. Group Actions3.2. Equivariant Mappings3.3. Orbit Space3.4. Orbit Manifold3.5. Orbit Reduction3.6. Fundamental Vector Fields4. Principal Bundles4.1. Principal Group Actions4.2. Quotient Manifolds4.3. Morphisms of Principal Bundles5. Associated Bundles5.1. Frame Mappings5.2. Associated Bundles5.3. Morphisms of Associated BundlesNotes and Additional TopicsChapter 4: Lagrange Structures1. Jet Prolongations of Fibered Manifolds1.1. Jet Prolongations of Fibered Manifolds1.2. Jet Prolongations of Morphisms1.3. Horizontalization of Tangent Vectors2. Trace Decompositions of Tensor Spaces2.1. Trace Mappings2.2. Kronecker Tensors2.3. Duality in Tensor Spaces2.4. Trace Decomposition3. Differential Forms on Jet Prolongations of Fibered Manifolds3.1. Horizontalization of Forms3.2. Formal Derivative of a Function3.3. The contact ideal3.4. First Canonical Decomposition of Forms3.5. Second Canonical Decomposition of Forms3.6. Third Canonical Decomposition of Forms3.7. The Poincare Lemma4. Lagrangians, Variational Functionals4.1. Fundamental Forms, Lagrangians4.2. Lepage Forms4.3. First Variation Formula4.4. Extremals4.5. The Euler-Lagrange Mapping4.6. Higher Variations5. Invariant Variational Principles and Conservation Laws5.1. Invariant Lagrangians5.2. Invariant Euler-Lagrange Forms5.3. Symmetries of Extremals6. Regularity and Hamilton Equations6.1. Non-holonomic Jet Spaces6.2. Hamilton Equations6.3. Legendre Coordinates6.4. Regular Variational PrinciplesNotes and Additional TopicsChapter 5: Elementary Sheaf Theory1. Sheaves1.1. Local Homeomorphisms1.2. Products and Fiber Products1.3. Sheaf Spaces1.4. The Quotient Sheaf Space1.5. Prolongations of Continuous Sections1.6. Sheaves1.7. Presheaves1.8. Complete Presheaves1.9. Sheaf Associated with a Presheaf2. Sheaf Cohomology2.1. Exact Sequences of Abelian Groups2.2. Complexes2.3. The Connecting Morphism2.4. Exact Sequences of Sheaves2.5. The Canonical Resolution of a Sheaf2.6. Cohomology Groups of a Sheaf3. Sheaves over Paracompact Hausdorff Spaces3.1. Soft Sheaves3.2. Fine Sheaves3.3. The Connecting Morphism3.4. Acyclic ResolutionsNotes and Additional TopicsChapter 6: Variational Sequences on Fibered Manifolds1. The Variational Sequence1.1. DeRham Sequence on Jet Bundles1.2. Contact Subsequence1.3. The Variational Sequence1.4. Cohomology of the Variational Sequence1.5. Interior Euler-Lagrange Operator1.6. Lepage forms2. The Structure of the Euler-Lagrange Mapping2.1. Trivial Lagrangians2.2. Dynamical Forms 2.3. The Inverse Problem of the Calculus of Variations2.4. Local Variational Principles, First Variation Formula2.5. Energy-Momentum Tensors for Differential Equations2.6. Order Reducibility of Lagrangians3. Variational Sequence and Symmetries3.1. Lie Derivatives of Classes3.2. Invariance of the Helmholtz ClassNotes and Additional TopicsChapter 7: Invariant Variational Functionals on Principal Bundles1. Prolongations of Principal Lie Group Actions1.1. Jet Prolongations of a Principal Bundle1.2. Bundles of Connections1.3. Invariant Differential Forms2. Invariant Variational Principles2.1. Invariant Differential Forms2.2. Invariant Vector Fields3. The First Variation Formula3.1. Euler-Lagrange Form3.2. Noether’s Currents4. Euler-Lagrange Equations4.1. Extremals and Conservation LawsNotes and Additional TopicsChapter 8: Differential Invariants 1. Differential Invariants1.1. Differential Groups1.2. Normal Subgroup Structure1.3. Differential Invariants1.4. Orbit Reduction 1.5. Orbit Manifolds2. Invariant Tensors3.1. Absolute Invariant Tensors 3.2. Relative Invariant Tensors 3.3. Multilinear Invariants3. Realizations of Differential Invariants3.1. Higher Order Frame Bundles3.2. Prolongations of Associated Bundles3.3. Associated Lifting Functors3.4. Natural Transformations3.5. Differential Invariants and Lie DerivativesNotes and Additional TopicsChapter 9: Natural Variational Principles 1. Odd Base Forms on Fibered Manifolds1.1. Orientation of a Vector Space and Odd Scalars1.2. Orientation of a Manifold1.3. Odd Scalars and Odd Forms on a Manifold1.4. Odd Base Scalars, Odd Base Forms1.5. The Pull-Back1.6. Exterior Product of a Form and an Odd Base Form1.7. Contraction by a Vector Field1.8. Exterior Derivative1.9. The Lie Derivative2. Jet Prolongations of Frame Bundles2.1. Prolongations of Group Action2.2. Invariant Differential Forms3. Natural Variational Principles on Frame Bundles3.1. Invariant Differential Forms3.2. Noether Currents3.3. Euler-Lagrange Equations4. Invariant Variational Principles on Associated Bundles4.1. Tensor Bundles4.2. First Variation Formula4.3. Week Conservation Laws4.4. Generalized Bianchi Identities4.5. Euler-Lagrange Equations and Conservation Laws5. Invariant Variational Principles in Tensor Bundles5.1. Differential Invariants of a Metric Field5.2. Invariant Variational Principles For the Metric Field5.3. Differential Invariants of a Linear Connection5.4. Invariant Variational Principles for Linear Connections6. The Energy Functional and Harmonic Mappings6.1. Energy of an Immersion6.2. Extremals of the Energy FunctionalNotes and Additional TopicsAppendices1. Multi-Linear Algebra2. Topology3. Analysis4. CategoriesBibliographyIndex