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Differential and Riemannian Manifolds / Edition 3

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This text provides an introduction to basic concepts in differential topology, differential geometry, and differential equations, and some of the main basic theorems in all three areas: for instance, the existence, uniqueness, and smoothness theorems for differential equations and the flow of a vector field; the basic theory of vector bundles including the existence of tubular neighborhoods for a submanifold; the calculus of differential forms; basic notions of symplectic manifolds, including the canonical 2-form; sprays and covariant derivatives for Riemannian and pseudo-Riemannian manifolds; applications to the exponential map, including the Cartan-Hadamard theorem, and the first basic theorem of calculus of variations. These are all covered for infinite-dimensional manifolds, modeled on Banach and Hilbert spaces, at no cost in complications, and some gain in the elegance of the proofs. In the finite-dimensional case, differential forms of top degree are discussed, leading to Stokes' theorem even for manifolds with singular boundary, and several of its applications to the differential or Riemannian case.
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Editorial Reviews

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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."


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Product Details

  • ISBN-13: 9780387943381
  • Publisher: Springer New York
  • Publication date: 3/9/1995
  • Series: Graduate Texts in Mathematics Series , #160
  • Edition description: 3rd ed. 1995. Corr. 2nd printing 1996
  • Edition number: 3
  • Pages: 384
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.94 (d)

Table of Contents

Ch. I Differential Calculus 1
Ch. II Manifolds 20
Ch. III Vector Bundles 40
Ch. IV Vector Fields and Differential Equations 64
Ch. V Operations on Vector Fields and Differential Forms 114
Ch. VI The Theorem of Frobenius 153
Ch. VII Metrics 169
Ch. VIII Covariant Derivatives and Geodesics 191
Ch. IX Curvature 225
Ch. X Volume Forms 261
Ch. XI Integration of Differential Forms 284
Ch. XII Stokes' Theorem 307
Ch. XIII Applications of Stokes' Theorem 321
Appendix: The Spectral Theorem 343
Bibliography 355
Index 361
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