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. In differential equations, one studies vector fields and their in tegral curves, singular points, stable and unstable manifolds, etc. Formally, one may say that one studies properties invariant under the group of differentiable automorphisms which preserve the additional structure. ) and studies properties connected especially with these objects.
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. One may also use differentiable structures on topological manifolds to deter mine the topological structure of the manifold (for example, it la Smale ). 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. 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. The size of the book influenced where to stop, and there would be enough material for a second volume (this is not a threat). The present book aims to give a fairly comprehensive account of the fundamentals of differential manifolds and differential geometry.
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