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Differential and Riemannian Manifolds

by Serge Lang

  • ISBN: 9780387943381
  • ISBN10: 0387943382

Differential and Riemannian Manifolds

by Serge Lang

  • Binding: Hardcover
  • Edition: 3
  • Publisher: Springer Verlag
  • Publish date: 03/01/1995
  • ISBN: 9780387943381
  • ISBN10: 0387943382
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Description: 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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