- #1
- 22,183
- 3,324
- Author: Serge Lang
- Title: Fundamentals of Differential Geometry
- Amazon Link: https://www.amazon.com/dp/038798593X/?tag=pfamazon01-20
- Prerequisities: Grad Analysis, Differential Geometry
- Level: Grad
Table of Contents:
Code:
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[*] Foreword
[*] Acknowledgments
[*] General Differential Theory
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[*] Differential Calculus
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[*] Categories
[*] Topological Vector Spaces
[*] Derivatives and Composition of Maps
[*] Integration and Taylor's Formula
[*] The Inverse Mapping Theorem
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[*] Manifolds
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[*] Atlases, Charts, Morphisms
[*] Submanifolds, Immersions, Submersions
[*] Partitions of Unity
[*] Manifolds with Boundary
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[*] Vector Bundles
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[*] Definition, Pull Backs
[*] The Tangent Bundle
[*] Exact Sequences of Bundles
[*] Operations on Vector Bundles
[*] Splitting of Vector Bundles
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[*] Vector Fields and Differential Equations
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[*] Existence Theorem for Differential Equations
[*] Vector Fields, Curves, and Flows
[*] Sprays
[*] The Flow of a Spray and the Exponential Map
[*] Existence of Tubular Neighborhoods
[*] Uniqueness of Tubular Neighborhoods
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[*] Operations on Vector Fields and Differential Forms
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[*] Vector Fields, Differential Operators, Brackets
[*] Lie Derivative
[*] Exterior Derivative
[*] The Poincare Lemma
[*] Contractions and Lie Derivative
[*] Vector Fields and 1-Forms Under Self Duality
[*] The Canonical 2-Form
[*] Darboux's Theorem
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[*] The Theorem of Frobenius
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[*] Statement of the Theorem
[*] Differential Equations Depending on a Parameter
[*] Proof of the Theorem
[*] The Global Formulation
[*] Lie Groups and Subgroups
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[*] Metrics, Covariant Derivatives, and Riemannian Geometry
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[*] Metrics
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[*] Definition and Functoriality
[*] The Hilbert Group
[*] Reduction to the Hilbert Group
[*] Hilbertian Tubular Neighborhoods
[*] The Morse-Palais Lemma
[*] The Riemannian Distance
[*] The Canonical Spray
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[*] Covariant Derivatives and Geodesies
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[*] Basic Properties
[*] Sprays and Covariant Derivatives
[*] Derivative Along a Curve and Parallelism
[*] The Metric Derivative
[*] More Local Results on the Exponential Map
[*] Riemannian Geodesic Length and Completeness
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[*] Curvature
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[*] The Riemann Tensor
[*] Jacobi Lifts
[*] Application of Jacobi Lifts to Texp_x
[*] Convexity Theorems
[*] Taylor Expansions
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[*] Jacobi Lifts and Tensorial Splitting of the Double Tangent Bundle
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[*] Convexity of Jacobi Lifts
[*] Global Tubular Neighborhood of a Totally Geodesic Submanifold
[*] More Convexity and Comparison Results
[*] Splitting of the Double Tangent Bundle
[*] Tensorial Derivative of a Curve in TX and of the Exponential Map
[*] The Flow and the Tensorial Derivative
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[*] Curvature and the Variation Formula
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[*] The Index Form, Variations, and the Second Variation Formula
[*] Growth of a Jacobi Lift
[*] The Semi Parallelogram Law and Negative Curvature
[*] Totally Geodesic Submanifolds
[*] Rauch Comparison Theorem
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[*] An Example of Seminegative Curvature
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[*] Pos_n(R) as a Riemannian Manifold
[*] The Metric Increasing Property of the Exponential Map
[*] Totally Geodesic and Symmetric Submanifolds
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[*] Automorphisms and Symmetries
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[*] The Tensorial Second Derivative
[*] Alternative Definitions of Killing Fields
[*] Metric Killing Fields
[*] Lie Algebra Properties of Killing Fields
[*] Symmetric Spaces
[*] Parallelism and the Riemann Tensor
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[*] Immersions and Submersions
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[*] The Covariant Derivative on a Submanifold
[*] The Hessian and Laplacian on a Submanifold
[*] The Covariant Derivative on a Riemannian Submersion
[*] The Hessian and Laplacian on a Riemannian Submersion
[*] The Riemann Tensor on Submanifolds
[*] The Riemann Tensor on a Riemannian Submersion
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[*] Volume Forms and Integration
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[*] Volume Forms
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[*] Volume Forms and the Divergence
[*] Covariant Derivatives
[*] The Jacobian Determinant of the Exponential Map
[*] The Hodge Star on Forms
[*] Hodge Decomposition of Differential Forms
[*] Volume Forms in a Submersion
[*] Volume Forms on Lie Groups and Homogeneous Spaces
[*] Homogeneously Fibered Submersions
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[*] Integration of Differential Forms
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[*] Sets of Measure 0
[*] Change of Variables Formula
[*] Orientation
[*] The Measure Associated with a Differential Form
[*] Homogeneous Spaces
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[*] Stokes' Theorem
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[*] Stokes' Theorem for a Rectangular Simplex
[*] Stokes' Theorem on a Manifold
[*] Stokes' Theorem with Singularities
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[*] Applications of Stokes' Theorem
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[*] The Maximal de Rham Cohomology
[*] Moser's Theorem
[*] The Divergence Theorem
[*] The Adjoint of d for Higher Degree Forms
[*] Cauchy's Theorem
[*] The Residue Theorem
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[*] Appendix: The Spectral Theorem
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[*] Hilbert Space
[*] Functionals and Operators
[*] Hermitian Operators
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[*] Bibliography
[*] Index
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