Fibre Metrics for Jet Bundles

Abstract

One of the principal devices we use in the monograph is convenient seminorms for the various topologies we use for spaces of sections of vector bundles. Since such topologies rely on placing suitable norms on derivatives of sections, i.e., on jet bundles of vector bundles, in this chapter we present a means for defining such norms, using as our starting point a pair of connections, one for the base manifold, and one for the vector bundle. These allow us to provide a direct sum decomposition of the jet bundle into its component “derivatives”, and so then a natural means of defining a fibre metric for jet bundles using metrics on the tangent bundle of the base manifold and the fibres of the vector bundle. As we shall see, in the smooth case these constructions are a convenience, whereas in the real analytic case they provide a crucial ingredient in our global, coordinate-free description of seminorms for the topology of the space of real analytic sections of a vector bundle. For this reason, in this chapter we shall also consider the existence of, and some properties of, real analytic connections in vector bundles.KeywordsMetal FibersCoordinate-free DescriptionPrincipal DeviceBase ManifoldReal Analytic Vector BundleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.