On F−flat Structures in Vector Bundles Over Foliated Manifolds

: We give the definition of the families of F − flat structures and F − flat connections in vector bundles over F − foliated manifolds. Essential: existence of a F − flat structure is equivalent to the existence of a F − flat connection. Let { λ ξ } be a family of subbundles of a vector bundle ξ . There exists a family of F − flat structure { λ Λ } in ξ , relative at λ ξ , if and only if exists a family of F − flat connections { λ ∇} in ξ (Theorem III.5). F − flat structures (Theorem III.1), and integrable F − flat structures (Theorem III.5), are considered. Finally, integrable Γ − structure and F − flat structures on total space of a vector bundle are presented (Theorem IV.1).