Development of an irregular curve in a space with affine connection (Q910008)

In two papers of the author [Sib. Math. J. 13(1972), 739-755 (1973); translation from Sib. Mat. Zh. 13, 1067-1090 (1972; Zbl 0248.53026), Sib. Math. J. 16(1976), 453-460 (1976); translation from Sib. Mat. Zh. 16, No.3, 588-598 (1975; Zbl 0333.53034)] the notion of parallel transfer and of lift is extended to certain classes of paths in a fibre bundle, which contain irregular curves (for example all curves of the class \(C^{\alpha}\) with \(\alpha >1/2).\) In the given paper the notion of the development is extended to the mentioned classes of paths in an affinely connected space. To this end it is first shown that the construction of the development of a curve in an affinely connected space is reduced to the construction of the lift of this curve in a suitable fibre bundle with a connection. Then the required result is obtained by the application of a theorem from the second paper of the author cited above. Similar results are established in this paper for curves in a Lie group with an invariant Cartan connection and in the paper of \textit{I. F. Mainik} [Sov. Math., Dokl. 18(1977), 972-974 (1978); translation from Dokl. Akad. Nauk SSSR 235, No.3, 531-533 (1977; Zbl 0378.53028)] for curves in a reductive space.