Differential and Riemannian Manifolds: (160 Graduate Texts in Mathematics)

This is the third version of a book on Differential Manifolds; in this latest expansion three chapters have been added on Riemannian and pseudo-Riemannian geometry, and the section on sprays and Stokes' theorem have been rewritten.This text provides an introduction to basic concepts in differential topology, differential geometry and differential equations. In differential topology one studies classes of maps and the possibility of finding differentiable maps in them, and one uses differentiable structures on manifolds to determine their topological structure. In differential geometry one adds structures to the manifold (vector fields, sprays, a metric, and so forth) and studies their properties. In differential equations one studies vector fields and their integral curves, singular points, stable and unstable manifolds, and the like.

Table of Contents:
I Differential Calculus.- §1. Categories.- §2. Topological Vector Spaces.- §3. Derivatives and Composition of Maps.- §4. Integration and Taylor’s Formula.- §5. The Inverse Mapping Theorem.- II Manifolds.- §1. Atlases, Charts, Morphisms.- §2. Submanifolds, Immersions, Submersions.- §3. Partitions of Unity.- §4. Manifolds with Boundary.- III Vector Bundles.- §1. Definition, Pull Backs.- §2. The Tangent Bundle.- §3. Exact Sequences of Bundles.- §4. Operations on Vector Bundles.- §5. Splitting of Vector Bundles.- IV Vector Fields and Differential Equations.- §1. Existence Theorem for Differential Equations.- §2. Vector Fields, Curves, and Flows.- §3. Sprays.- §4. The Flow of a Spray and the Exponential Map.- §5. Existence of Tubular Neighborhoods.- §6. Uniqueness of Tubular Neighborhoods.- V Operations on Vector Fields and Differential Forms.- §1. Vector Fields, Differential Operators, Brackets.- §2. Lie Derivative.- $3. Exterior Derivative.- §4. The Poincaré Lemma.- §5. Contractions and Lie Derivative.- §6. Vector Fields and 1-Forms Under Self Duality.- §7. The Canonical 2-Form.- §8. Darboux’s Theorem.- VI The Theorem of Frobenius.- §1. Statement of the Theorem.- §2. Differential Equations Depending on a Parameter.- §3. Proof of the Theorem.- §4. The Global Formulation.- §5. Lie Groups and Subgroups.- VII Metrics.- §1. Definition and Functoriality.- §2. The Hilbert Group.- §3. Reduction to the Hilbert Group.- §4. Hilbertian Tubular Neighborhoods.- §5. The Morse—Palais Lemma.- §6. The Riemannian Distance.- §7. The Canonical Spray.- VIII Covariant Derivatives and Geodesics.- §1. Basic Properties.- §2. Sprays and Covariant Derivatives.- §3. Derivative Along a Curve and Parallelism.- §4. The Metric Derivative.- §5. More LocalResults on the Exponential Map.- §6. Riemannian Geodesic Length and Completeness.- IX Curvature.- §1. The Riemann Tensor.- §2. Jacobi Lifts.- §3. Application of Jacobi Lifts to dexpx.- §4. The Index Form, Variations, and the Second Variation Formula.- §5. Taylor Expansions.- X Volume Forms.- §1. The Riemannian Volume Form.- §2. Covariant Derivatives.- §3. The Jacobian Determinant of the Exponential Map.- §4. The Hodge Star on Forms.- §5. Hodge Decomposition of Differential Forms.- XI Integration of Differential Forms.- §1. Sets of Measure 0.- §2. Change of Variables Formula.- §3. Orientation.- §4. The Measure Associated with a Differential Form.- XII Stokes’ Theorem.- §1. Stokes’ Theorem for a Rectangular Simplex.- §2. Stokes’ Theorem on a Manifold.- §3. Stokes’ Theorem with Singularities.- XIII Applications of Stokes’ Theorem.- §1. The Maximal de Rham Cohomology.- §2. Moser’s Theorem.- §3. The Divergence Theorem.- §4. The Adjoint of d for Higher Degree Forms.- §5. Cauchy’s Theorem.- §6. The Residue Theorem.- Appendix The Spectral Theorem.- §1. Hilbert Space.- §2. Functionals and Operators.- §3. Hermitian Operators.

Review :
S. Lang Differential and Riemannian Manifolds "An introduction to differential geometry, starting from recalling differential calculus and going through all the basic topics such as manifolds, vector bundles, vector fields, the theorem of Frobenius, Riemannian metrics and curvature. Useful to the researcher wishing to learn about infinite-dimensional geometry." -MATHEMATICAL REVIEWS