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**The geometry of jet bundles.**
*(English)*
Zbl 0665.58002

London Mathematical Society Lecture Note Series, 142. Cambridge etc.: Cambridge University Press. 293 p. £20.00; $ 29.95 (1989).

The author himself characterizes the book 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. From the mathematical point of view, the core of the book deals with the variational calculus in fibred manifolds and with the foundations of the geometric theory of systems of partial differential equations. The initial chapters represent a readable introduction into the geometry of the jet prolongations of fibred manifolds, so that the book will also be useful for the mathematicians interested in those subjects.

The main aim of the book induces the author to study the general bundles, i.e. the locally trivial fibred manifolds with no structure group taken into account. The first 3 chapters are devoted to the basic operations with general bundles and to the vector and affine bundles, which frequently appear as the derived objects. Given a general bundle \(\pi\) : \(E\to M\), the author develops an interesting theory of derivations \(\bigwedge M\to \bigwedge E\) along \(\pi\), where \(\bigwedge E\) or \(\bigwedge M\) means the exterior algebra of E or M, respectively. This enables him to introduce a generalization of the Frölicher-Nijenhuis bracket, which is a very useful new tool for several constructions related with jet bundles. The fourth chapter is devoted to a detailed study of the geometry of the first jet prolongation \(J^ 1\pi\) of a general bundle. Special attention is paid to the prolongations of vector fields and to the first order Lagrangians. (A first order Lagrangian density is defined as a function on \(J^ 1\pi\) and it is multiplied by a volume form on M to obtain a suitable object for the variational calculus. It seems to be more geometrical to define the Lagrangian as a special exterior form, but the author overcomes with success all technical problems arising from a non-canonical decomposition of a Lagrangian form into a function and a volume form.)

The fifth chapter deals with the second order jets. This is not only an instructive special case of the higher order theories, but it also provides the geometric framework for the study of the Euler-Lagrange equations and of the Poincaré-Cartan form of a first order Lagrangian. It is remarkable that the author describes the curvature of general connections in terms of the semi-holonomic 2-jets. In the sixth chapter, the k-th order jet prolongations of fibred manifolds and the higher order variational calculus are studied from similar points of view. The last chapter presents the foundations of the theory of infinite order jet bundles. The variational bicomplex is defined and its local exactness is proved. A consequence of this result is the Helmholtz condition in the inverse problem of the calculus of variations.

The main aim of the book induces the author to study the general bundles, i.e. the locally trivial fibred manifolds with no structure group taken into account. The first 3 chapters are devoted to the basic operations with general bundles and to the vector and affine bundles, which frequently appear as the derived objects. Given a general bundle \(\pi\) : \(E\to M\), the author develops an interesting theory of derivations \(\bigwedge M\to \bigwedge E\) along \(\pi\), where \(\bigwedge E\) or \(\bigwedge M\) means the exterior algebra of E or M, respectively. This enables him to introduce a generalization of the Frölicher-Nijenhuis bracket, which is a very useful new tool for several constructions related with jet bundles. The fourth chapter is devoted to a detailed study of the geometry of the first jet prolongation \(J^ 1\pi\) of a general bundle. Special attention is paid to the prolongations of vector fields and to the first order Lagrangians. (A first order Lagrangian density is defined as a function on \(J^ 1\pi\) and it is multiplied by a volume form on M to obtain a suitable object for the variational calculus. It seems to be more geometrical to define the Lagrangian as a special exterior form, but the author overcomes with success all technical problems arising from a non-canonical decomposition of a Lagrangian form into a function and a volume form.)

The fifth chapter deals with the second order jets. This is not only an instructive special case of the higher order theories, but it also provides the geometric framework for the study of the Euler-Lagrange equations and of the Poincaré-Cartan form of a first order Lagrangian. It is remarkable that the author describes the curvature of general connections in terms of the semi-holonomic 2-jets. In the sixth chapter, the k-th order jet prolongations of fibred manifolds and the higher order variational calculus are studied from similar points of view. The last chapter presents the foundations of the theory of infinite order jet bundles. The variational bicomplex is defined and its local exactness is proved. A consequence of this result is the Helmholtz condition in the inverse problem of the calculus of variations.

Reviewer: I.Kolář

### MSC:

58A20 | Jets in global analysis |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

49Q99 | Manifolds and measure-geometric topics |

58A15 | Exterior differential systems (Cartan theory) |

49S05 | Variational principles of physics |