1-integrability of nonholonomic differential-geometric structures. 1 (Q1099814)

[Parts 2, 3, and 4, see the review below.] Let \(\pi\) : \(E\to M\) be a locally trivial bundle of class \(C^{\infty}\) and let \(J_ kE\) be the differential extension of order k of E. Let then \(T^ v=T^ v(E)\), \(T^ v_ 1=T^ v(J_ 1E)\), \(T^ v_ 2=T^ v(J_ 2)\) be the vertical tangent bundles of the manifolds E, \(J_ 1E\), \(J_ 2E\). A nonholonomic differential-geometric structure on E is a distribution on the vertical tangent bundle of the manifold \(J_ kE\) and a nonholonomic differential-geometric connection is a distribution of a special type. \(\Gamma_ 1\) nd \(\Gamma_{1,2}\)-connections are lifts \(\Gamma_ 1: T^ v\to T^ v_ 1\), \(\Gamma_{1,2}: T^ v_ 1\to T^ v_ 2\) of canonical projections \(J_ 1E\to E\), \(J_ 2E\to J_ 1E\), generated by mappings \(T^ v_ 1\to T^ v\), \(T^ v_ 2\to T^ v_ 1\) and the corresponding nonholonomic structures are defined by \(b_ 1=T_ 1(T^ v)\), \(b_{1,2}=\Gamma_{1,2}(T^ v_ 1)\). to the connections \(\Gamma_ 1\) and \(\Gamma_{1,2}\) the author associates other types of connections, operators of covariant derivation and also tensors of deformation and of curvature for which he obtains different properties and gives geometric interpretations.