Differential geometric connection of finite order on composite manifold and nonlinear product integral (Q1340767)

First the nonlinear generalization of the product integral is introduced, its basic properties and its power series representations are established. Then this, as a main tool, is applied to deduce some general formulae for the parallel transport along an infinitesimal curvilinear polygon in the base space \(B_ n\) of the composite manifold \(L_ m(B_ n)\) endowed with a differential geometric connection of finite order. Investigation of the commutator relations leads the author to the anholonomic generalization of the Wagner curvature tensor. Finally the particular case of the linear connection is investigated. The work is related to the author's paper in Rep. Math. Phys. 30, No. 1, 107-118 (1991; Zbl 0763.53020).