Fibered spaces, jet spaces and connections for field theories.

*(English)*Zbl 0539.53026
Geometry and physics, Proc. int. Meet., Florence/Italy 1982, 135-165 (1983).

Summary: [For the entire collection see Zbl 0536.00006.]

The geometrical framework for field theories, constituted by fibered spaces, jet spaces and connections, is analysed together with some new results. The fibered spaces constitute a weak structure, but sufficient enough to describe many facts. Further structures can be introduced in different ways such as bundles, bundles with symmetries and S-fibered spaces. However they might not always play a real role in physical applications. The natural structures of the tangent and jet spaces are the theoretical basis for all the differential operations. In particular a bracket on the projectable vector valued forms and a canonical differential operator d are presented. A theory of connections is developed in detail, in the weak framework of fibered spaces. In particular the curvature is introduced both via the bracket and the operator d. The usual theory of differential forms results as a particular case. The concept of structure of connections is introduced and a universal connection and curvature are found. The contact and symplectic forms turn out to be a particular case. The previous results lead naturally to writing Maxwell-type equations on a fibered space, by requiring only a metric on the base space.

The geometrical framework for field theories, constituted by fibered spaces, jet spaces and connections, is analysed together with some new results. The fibered spaces constitute a weak structure, but sufficient enough to describe many facts. Further structures can be introduced in different ways such as bundles, bundles with symmetries and S-fibered spaces. However they might not always play a real role in physical applications. The natural structures of the tangent and jet spaces are the theoretical basis for all the differential operations. In particular a bracket on the projectable vector valued forms and a canonical differential operator d are presented. A theory of connections is developed in detail, in the weak framework of fibered spaces. In particular the curvature is introduced both via the bracket and the operator d. The usual theory of differential forms results as a particular case. The concept of structure of connections is introduced and a universal connection and curvature are found. The contact and symplectic forms turn out to be a particular case. The previous results lead naturally to writing Maxwell-type equations on a fibered space, by requiring only a metric on the base space.

##### MSC:

53C05 | Connections (general theory) |

55R05 | Fiber spaces in algebraic topology |

58A99 | General theory of differentiable manifolds |

53C80 | Applications of global differential geometry to the sciences |