Differential Geometry

Abstract

As in this book we are much concerned with a geometric approach to differential equations, we need some basic notions in differential geometry which are collected in this chapter. We start with a section introducing (differentiable) manifolds and some spaces associated with them: the tangent and the cotangent bundle. Special emphasis is given to fibred manifolds (bundles are only briefly introduced, as we hardly need their additional structures). The second section studies in some more detail vector fields and differential forms and the basic operations with them. Distributions of vector fields (or dual codistributions of one-forms, respectively) and the Frobenius theorem are the topic of the third section.

A fundamental object in differential geometry are connections. In our context, it should be noted that ordinary differential equations and certain partial differential equations correspond geometrically to connections (see Remark 2.3.6). The fourth section introduces connections on arbitrary fibred manifolds. At a few places we need some elementary results about Lie groups and algebras which are collected in the fifth section. The final section is concerned with (co)symplectic geometry as the basis of the theory of Hamiltonian systems.

Standard references on differential geometry are [3, 49, 81, 256, 282, 474], but they go much further than we need. For most of our purposes a working knowledge of some basic notions around manifolds is sufficient. Much of the material we need can also be found in a very accessible form in [342, Chapt. 1].