Formal Geometry of Differential Equations

In this chapter we lay the geometric foundations of the formal theory of

differential equations

. The name “formal theory” stems from the fact that it is, at least indirectly, concerned with the analysis of formal power series solutions, i. e. one ignores the question whether the series actually converge. Another interpretation of the name is that one tries to extract as much information as possible on the solution space by purely formal operations like algebraic manipulations of the equations or their differentiations without actually solving the given equation.

The basic tool for a geometric approach to differential equations is the jet bundle formalism and the first two sections give an introduction to it. We do this first in a more pedestrian way considering jets as a differential geometric approach to (truncated) power series. For most computational purposes this simple point of view is sufficient. In order to obtain a deeper understanding of certain structural properties which will later be of importance, we redevelop the theory in the second section in a more abstract but intrinsic way which does not require power series. In both approaches, special emphasis is put on the contact structure as the key to the geometry of jet bundles. Because of its great importance, we consider different geometric realisations of it, each having its advantages in certain applications.