Fundamentals of Differential Geometry

1. Front Matter

   Pages i-xvii

   PDF

2. General Differential Theory

   1. Front Matter

      Pages 1-1

      PDF

   2. Differential Calculus

      • Serge Lang

      Pages 3-21

   3. Manifolds

      • Serge Lang

      Pages 22-42

   4. Vector Bundles

      • Serge Lang

      Pages 43-65

   5. Vector Fields and Differential Equations

      • Serge Lang

      Pages 66-115

   6. Operations on Vector Fields and Differential Forms

      • Serge Lang

      Pages 116-154

   7. The Theorem of Frobenius

      • Serge Lang

      Pages 155-170

3. Metrics, Covariant Derivatives, and Riemannian Geometry

   1. Front Matter

      Pages 171-171

      PDF

   2. Metrics

      • Serge Lang

      Pages 173-195

   3. Covariant Derivatives and Geodesics

      • Serge Lang

      Pages 196-230

   4. Curvature

      • Serge Lang

      Pages 231-266

   5. Jacobi Lifts and Tensorial Splitting of the Double Tangent Bundle

      • Serge Lang

      Pages 267-293

   6. Curvature and the Variation Formula

      • Serge Lang

      Pages 294-321

   7. An Example of Seminegative Curvature

      • Serge Lang

      Pages 322-338

   8. Automorphisms and Symmetries

      • Serge Lang

      Pages 339-368

   9. Immersions and Submersions

      • Serge Lang

      Pages 369-394

4. Volume Forms and Integration

   1. Front Matter

      Pages 395-395

      PDF

   2. Volume Forms

      • Serge Lang

      Pages 397-447

   3. Integration of Differential Forms

      • Serge Lang

      Pages 448-474

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.

• Derivative
• Riemannian geometry
• Smooth function
• calculus
• curvature
• differential equation
• differential geometry
• manifold
• spectral theorem
• vector bundle

"There are many books on the fundamentals of differential geometry, but this one is quite exceptional; this is not surprising for those who know Serge Lang's books. ...
It can be warmly recommended to a wide audience."
EMS Newsletter, Issue 41, September 2001

"The text provides a valuable introduction to basic concepts and fundamental results in differential geometry. A special feature of the book is that it deals with infinite-dimensional manifolds, modeled on a Banach space in general, and a Hilbert space for Riemannian geometry. The set-up works well on basic theorems such as the existence, uniqueness and smoothness theorem for differential equations and the flow of a vector field, existence of tubular neighborhoods for a submanifold, and the Cartan-Hadamard theorem. A major exception is the Hopf-Rinow theorem. Curvature and basic comparison theorems are discussed. In the finite-dimensional case, volume forms, the Hodge star operator, and integration of differentialforms are expounded. The book ends with the Stokes theorem and some of its applications."-- MATHEMATICAL REVIEWS

• Department of Mathematics, Yale University, New Haven, USA

  Serge Lang

Policies and ethics