Fundamentals of Differential Geometry

Serge Lang (auth.)038798593X, 9780387985930

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.

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Table of contents :
Front Matter….Pages i-xvii
Front Matter….Pages 1-1
Differential Calculus….Pages 3-21
Manifolds….Pages 22-42
Vector Bundles….Pages 43-65
Vector Fields and Differential Equations….Pages 66-115
Operations on Vector Fields and Differential Forms….Pages 116-154
The Theorem of Frobenius….Pages 155-170
Front Matter….Pages 171-171
Metrics….Pages 173-195
Covariant Derivatives and Geodesics….Pages 196-230
Curvature….Pages 231-266
Jacobi Lifts and Tensorial Splitting of the Double Tangent Bundle….Pages 267-293
Curvature and the Variation Formula….Pages 294-321
An Example of Seminegative Curvature….Pages 322-338
Automorphisms and Symmetries….Pages 339-368
Immersions and Submersions….Pages 369-394
Front Matter….Pages 395-395
Volume Forms….Pages 397-447
Integration of Differential Forms….Pages 448-474
Stokes’ Theorem….Pages 475-488
Applications of Stokes’ Theorem….Pages 489-510
Back Matter….Pages 523-540