Fundamental of Differential Geometry by Lang

[micromass]

• 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:

[LIST]
[*] Foreword
[*] Acknowledgments
[*] General Differential Theory
[LIST]
[*] Differential Calculus
[LIST]
[*] Categories
[*] Topological Vector Spaces
[*] Derivatives and Composition of Maps
[*] Integration and Taylor's Formula
[*] The Inverse Mapping Theorem
[/LIST]
[*] Manifolds
[LIST]
[*] Atlases, Charts, Morphisms
[*] Submanifolds, Immersions, Submersions
[*] Partitions of Unity
[*] Manifolds with Boundary
[/LIST]
[*] Vector Bundles
[LIST]
[*] Definition, Pull Backs
[*] The Tangent Bundle
[*] Exact Sequences of Bundles
[*] Operations on Vector Bundles
[*] Splitting of Vector Bundles
[/LIST]
[*] Vector Fields and Differential Equations
[LIST]
[*] 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
[/LIST]
[*] Operations on Vector Fields and Differential Forms
[LIST]
[*] 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
[/LIST]
[*] The Theorem of Frobenius
[LIST]
[*] Statement of the Theorem
[*] Differential Equations Depending on a Parameter
[*] Proof of the Theorem
[*] The Global Formulation
[*] Lie Groups and Subgroups
[/LIST]
[/LIST]
[*] Metrics, Covariant Derivatives, and Riemannian Geometry
[LIST]
[*] Metrics
[LIST]
[*] Definition and Functoriality
[*] The Hilbert Group
[*] Reduction to the Hilbert Group
[*] Hilbertian Tubular Neighborhoods
[*] The Morse-Palais Lemma
[*] The Riemannian Distance
[*] The Canonical Spray
[/LIST]
[*] Covariant Derivatives and Geodesies
[LIST]
[*] 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
[/LIST]
[*] Curvature
[LIST]
[*] The Riemann Tensor
[*] Jacobi Lifts
[*] Application of Jacobi Lifts to Texp_x
[*] Convexity Theorems
[*] Taylor Expansions
[/LIST]
[*] Jacobi Lifts and Tensorial Splitting of the Double Tangent Bundle
[LIST]
[*] 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
[/LIST]
[*] Curvature and the Variation Formula
[LIST]
[*] 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
[/LIST]
[*] An Example of Seminegative Curvature
[LIST]
[*] Pos_n(R) as a Riemannian Manifold
[*] The Metric Increasing Property of the Exponential Map
[*] Totally Geodesic and Symmetric Submanifolds
[/LIST]
[*] Automorphisms and Symmetries
[LIST]
[*] 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
[/LIST]
[*] Immersions and Submersions
[LIST]
[*] 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
[/LIST]
[/LIST]
[*] Volume Forms and Integration
[LIST]
[*] Volume Forms
[LIST]
[*] 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
[/LIST]
[*] Integration of Differential Forms
[LIST]
[*] Sets of Measure 0
[*] Change of Variables Formula
[*] Orientation
[*] The Measure Associated with a Differential Form
[*] Homogeneous Spaces
[/LIST]
[*] Stokes' Theorem
[LIST]
[*] Stokes' Theorem for a Rectangular Simplex
[*] Stokes' Theorem on a Manifold
[*] Stokes' Theorem with Singularities
[/LIST]
[*] Applications of Stokes' Theorem
[LIST]
[*] 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
[/LIST]
[/LIST]
[*] Appendix: The Spectral Theorem
[LIST]
[*] Hilbert Space
[*] Functionals and Operators
[*] Hermitian Operators
[/LIST]
[*] Bibliography
[*] Index
[/LIST]

 

[mathwonk]

lang of course was a phenomenally productive writer in the 60's who is now dead. this book is apparently a great expansion of the 100 page or so book i bought as a grad student in about 1965, "foundations of differential manifolds". that book was too abstract and almost useless. but this version is 3 or 4 times as long and has got there by being augmented apparently by various chapters of his other books as well as extra chapters?

this way of writing books piecemeal does not always work so well, and i cannot say how well this one works. i am skeptical though. we need an opinion from someone who has actually used this version to learn the subject. anyone?