4 - First-order Jet Bundles
Published online by Cambridge University Press: 03 November 2009
Summary
In basic differential geometry, a tangent vector to a manifold may be defined as an equivalence class of curves passing through a given point, where two curves are equivalent if they have the same derivative at that point: indeed, this is the definition we have used in earlier chapters. (There are other definitions which may be used instead, but the definition in terms of curves is perhaps the most intuitive.) A first-order jet is a generalisation of this idea to the case of families of higher-dimensional manifolds passing through a point, where the embedding maps have the same first derivatives at that point. We shall, however, choose to consider the graphs of these embeddings rather than the embeddings themselves, in line with our previous policy of considering bundles and sections rather than pairs of manifolds and maps. In the first section of this chapter we shall give a formal definition of a first-order jet, and show that the collection of all such jets is a differentiable manifold. We shall also see that this manifold is the appropriate setting for a general description of a first-order partial differential equation. In subsequent sections, we shall examine some of the properties of this jet manifold which arise when it is regarded as the total space of an affine bundle, and we shall introduce the idea of prolongation whereby a bundle morphism may be extended to act upon the jet manifold.
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- Information
- The Geometry of Jet Bundles , pp. 92 - 159Publisher: Cambridge University PressPrint publication year: 1989