Zakrzewski. Parallel transport along a space curve and related phases. , 22:L461–L466, 1989.

Bozhidar Z. Iliev. Transports along maps in fibre bundles. JINR Communication E5-97-2, Dubna, 1997. (LANL xxx archive server, E-print No. dg-ga/9709016).

James D. Stasheff. “[P]{}arallel” transport in fibre spaces. Reimpreso del boletin de la sociedad matematica mexicana, [[S]{}ociedad]{} [[M]{}atematica]{} [[M]{}exicana]{}, Printed in USA, 1966. pages 68–84.

Bozhidar Z. Iliev. Fibre bundle formulation of nonrelativistic quantum mechanics. [III]{}. [Pictures]{} and integrals of motion. (LANL xxx archive server, E-print No. quant-ph/9806046) Submitted to J. Physics A: Math. & Gen., ?? 1998.

Sze-Tsen Hu. . Academic Press, New York-London, 1959.

Bozhidar Z. Iliev. Transports along paths in fibre bundles. [III]{}. [Consistency]{} with bundle morphisms. JINR Communication E5-94-41, Dubna, 1994. (LANL xxx archive server, E-print No. dg-ga/9704004).

Introduction {#introduction-I}
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Usually the standard nonrelativistic quantum mechanics of pure states is formulated in terms of vectors and operators in a Hilbert space . This is in discrepancy and not in harmony with the new trends in (mathematical) physics [@Schutz; @Coquereaux; @Konopleva-Popov] in which the theory of fibre bundles [@Husemoller; @R_Hermann/Geom-phys-systems], in particular vector bundles [@R_Hermann-I; @R_Hermann-II], is essentially used. This paper (and its further continuation(s)) is intended to incorporate the quantum theory 