Introduction

Summary

This book is intended as an introduction to the language of jet bundles, for the reader who is interested in mathematical physics, and who has a knowledge of modern differential geometry.

Several ways of applying geometric techniques to physics are now well established in the literature: two major examples are the study of tangent and cotangent bundles in mechanics, and the use of connections on principal fibre bundles in field theories. More recently, the language of jets has appeared as a concise way of describing phenomena associated with the derivatives of maps, particularly those associated with the calculus of variations. In fact, a jet is no more than a generalisation of a tangent vector, and the geometrical theory of jet bundles includes the theories mentioned earlier as special cases. Generalisation, of course, sometimes introduces complexity: for instance, the coordinate representation used for jets bears some resemblance to the traditional coordinate representation used in the tensor calculus, but differs in that the transformation rules are no longer linear. In addition, many of the coordinate formulæ are symmetric in their indices, as a consequence of the commutativity of repeated partial differentiation, and this also introduces a certain complexity. On the other hand, the geometric nature of the theory introduces simplicity: there is, for instance, a clear geometric interpretation of the reason why the curvature of a connection is the obstruction to the integrability of the system of partial differential equations represented by the connection.