The Berwald-type connection associated to time-dependent second-order differential equations
نویسندگان
چکیده
We investigate the notions of a connection of Finsler type and of Berwald type on the first jet bundle J1π of a manifold E which is fibred over IR. Such connections are associated to a given horizontal distribution on the bundle π0 1 : J 1π → E, which in particular may come from a time-dependent system of second-order ordinary differential equations. In order to accomodate three existing constructions of a Berwald-type connection for a second-order system, we first introduce equivalence classes of connections of Finsler and Berwald type. By exploring the differences between the existing models in more depth, we come to a new construction which in many respects can be regarded as giving an optimal representative of the class of Berwald-type connections. We briefly enter into two related matters: one is the definition of connections of the type of Cartan, Chern-Rund and Hashiguchi when a metric tensor field is given; the other one is the potential effect of the newly acquired insights on the theory of derivations on forms along the projection π0 1. AMS-classification: 58B20, 53C60 Suggested running head: ‘Berwald-type connections – time-dependent case’
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