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TitleDifferential and Riemannian Manifolds [electronic resource] / edited by Serge Lang
ImprintNew York, NY : Springer New York, 1995
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Descript XIV, 364 p. online resource


This is the third version of a book on differential manifolds. The first version appeared in 1962, and was written at the very beginning of a period of great expansion of the subject. At the time, I found no satisfactory book for the foundations of the subject, for multiple reasons. I expanded the book in 1971, and I expand it still further today. Specifically, I have added three chapters on Riemannian and pseudo Riemannian geometry, that is, covariant derivatives, curvature, and some applications up to the Hopf-Rinow and Hadamard-Cartan theorems, as well as some calculus of variations and applications to volume forms. I have rewritten the sections on sprays, and I have given more examples of the use of Stokes' theorem. I have also given many more references to the literature, all of this to broaden the perspective of the book, which I hope can be used among things for a general course leading into many directions. The present book still meets the old needs, but fulfills new ones. At the most basic level, the book gives an introduction to the basic concepts which are used in differential topology, differential geometry, and differential equations. In differential topology, one studies for instance homotopy classes of maps and the possibility of finding suitable differentiable maps in them (immersions, embeddings, isomorphisms, etc.)


I Differential Calculus -- ยง1. Categories -- ยง2. Topological Vector Spaces -- ยง3. Derivatives and Composition of Maps -- ยง4. Integration and Taylorโ{128}{153}s Formula -- ยง5. The Inverse Mapping Theorem -- II Manifolds -- ยง1. Atlases, Charts, Morphisms -- ยง2. Submanifolds, Immersions, Submersions -- ยง3. Partitions of Unity -- ยง4. Manifolds with Boundary -- III Vector Bundles -- ยง1. Definition, Pull Backs -- ยง2. The Tangent Bundle -- ยง3. Exact Sequences of Bundles -- ยง4. Operations on Vector Bundles -- ยง5. Splitting of Vector Bundles -- IV Vector Fields and Differential Equations -- ยง1. Existence Theorem for Differential Equations -- ยง2. Vector Fields, Curves, and Flows -- ยง3. Sprays -- ยง4. The Flow of a Spray and the Exponential Map -- ยง5. Existence of Tubular Neighborhoods -- ยง6. Uniqueness of Tubular Neighborhoods -- V Operations on Vector Fields and Differential Forms -- ยง1. Vector Fields, Differential Operators, Brackets -- ยง2. Lie Derivative -- $3. Exterior Derivative -- ยง4. The Poincarรฉ Lemma -- ยง5. Contractions and Lie Derivative -- ยง6. Vector Fields and 1-Forms Under Self Duality -- ยง7. The Canonical 2-Form -- ยง8. Darbouxโ{128}{153}s Theorem -- VI The Theorem of Frobenius -- ยง1. Statement of the Theorem -- ยง2. Differential Equations Depending on a Parameter -- ยง3. Proof of the Theorem -- ยง4. The Global Formulation -- ยง5. Lie Groups and Subgroups -- VII Metrics -- ยง1. Definition and Functoriality -- ยง2. The Hilbert Group -- ยง3. Reduction to the Hilbert Group -- ยง4. Hilbertian Tubular Neighborhoods -- ยง5. The Morseโ{128}{148}Palais Lemma -- ยง6. The Riemannian Distance -- ยง7. The Canonical Spray -- VIII Covariant Derivatives and Geodesics -- ยง1. Basic Properties -- ยง2. Sprays and Covariant Derivatives -- ยง3. Derivative Along a Curve and Parallelism -- ยง4. The Metric Derivative -- ยง5. More Local Results on the Exponential Map -- ยง6. Riemannian Geodesic Length and Completeness -- IX Curvature -- ยง1. The Riemann Tensor -- ยง2. Jacobi Lifts -- ยง3. Application of Jacobi Lifts to dexpx -- ยง4. The Index Form, Variations, and the Second Variation Formula -- ยง5. Taylor Expansions -- X Volume Forms -- ยง1. The Riemannian Volume Form -- ยง2. Covariant Derivatives -- ยง3. The Jacobian Determinant of the Exponential Map -- ยง4. The Hodge Star on Forms -- ยง5. Hodge Decomposition of Differential Forms -- XI Integration of Differential Forms -- ยง1. Sets of Measure 0 -- ยง2. Change of Variables Formula -- ยง3. Orientation -- ยง4. The Measure Associated with a Differential Form -- XII Stokesโ{128}{153} Theorem -- ยง1. Stokesโ{128}{153} Theorem for a Rectangular Simplex -- ยง2. Stokesโ{128}{153} Theorem on a Manifold -- ยง3. Stokesโ{128}{153} Theorem with Singularities -- XIII Applications of Stokesโ{128}{153} Theorem -- ยง1. The Maximal de Rham Cohomology -- ยง2. Moserโ{128}{153}s Theorem -- ยง3. The Divergence Theorem -- ยง4. The Adjoint of d for Higher Degree Forms -- ยง5. Cauchyโ{128}{153}s Theorem -- ยง6. The Residue Theorem -- Appendix The Spectral Theorem -- ยง1. Hilbert Space -- ยง2. Functionals and Operators -- ยง3. Hermitian Operators

Mathematics Mathematical analysis Analysis (Mathematics) Differential geometry Algebraic topology Mathematics Differential Geometry Analysis Algebraic Topology


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