| Fundamentals of Differential Geometry 1999. Corr. 2nd Edition Contributor(s): Lang, Serge (Author) |
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ISBN: 038798593X ISBN-13: 9780387985930 Publisher: Springer OUR PRICE: $104.49 Product Type: Hardcover Published: December 1998 Annotation: 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. Although the book grew out of the author's earlier book "Differential and Riemannian Manifolds," the focus has now changed from the general theory of manifolds to general differential geometry, and includes new chapters on Jacobi lifts, tensorial splitting of the double tangent bundle, curvature and the variation formula, a generalization of the Cartan-Hadamard theorem, the semiparallelogram law of Bruhat-Tits and its equivalence with seminegative curvature and the exponential map distance increasing property, a major example of seminegative curvature (the space of positive definite symmetric real matrices), automorphisms and symmetries, and immersions and submersions. 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 withsingular boundary), and several of its applications to the differential or Riemannian case. Basic formulas concerning the Laplacian are given, exhibiting several of its features in immersions and submersions. |
| Additional Information |
| BISAC Categories: - Mathematics | Geometry - Differential - Mathematics | Topology - General - Mathematics | Mathematical Analysis |
| Dewey: 516.36 |
| LCCN: 98029993 |
| Series: Fundamentals of Differential Geometry |
| Physical Information: 1.18" H x 6.37" W x 9.51" (2.08 lbs) 540 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: The present book aims to give a fairly comprehensive account of the fundamentals of differential manifolds and differential geometry. The size of the book influenced where to stop, and there would be enough material for a second volume (this is not a threat). 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 differen- tiable maps in them (immersions, embeddings, isomorphisms, etc. ). One may also use differentiable structures on topological manifolds to deter- mine the topological structure of the manifold (for example, it la Smale [Sm 67]). In differential geometry, one puts an additional structure on the differentiable manifold (a vector field, a spray, a 2-form, a Riemannian metric, ad lib. ) and studies properties connected especially with these objects. Formally, one may say that one studies properties invariant under the group of differentiable automorphisms which preserve the additional structure. In differential equations, one studies vector fields and their in- tegral curves, singular points, stable and unstable manifolds, etc. A certain number of concepts are essential for all three, and are so basic and elementary that it is worthwhile to collect them together so that more advanced expositions can be given without having to start from the very beginnings. |