Fundamentals of Differential Geometry

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.

I General Differential Theory. - I Differential Calculus. - II Manifolds. - III Vector Bundles. - IV Vector Fields and Differential Equations. - V Operations on Vector Fields and Differential Forms. - VI The Theorem of Frobenius. - II Metrics, Covariant Derivatives, and Riemannian Geometry. - VII Metrics. - VIII Covariant Derivatives and Geodesics. - IX Curvature. - X Jacobi Lifts and Tensorial Splitting of the Double Tangent Bundle. - XI Curvature and the Variation Formula. - XII An Example of Seminegative Curvature. - XIII Automorphisms and Symmetries. - XIV Immersions and Submersions. - III Volume Forms and Integration. - XV Volume Forms. - XVI Integration of Differential Forms. - XVII Stokes Theorem. - XVIII Applications of Stokes Theorem.