WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 22, 2023
On F−flat Structures in Vector Bundles Over Foliated Manifolds
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Abstract: We give the definition of the families of $$\mathscr{F}$$−flat structures and $$\mathscr{F}$$−flat connections in vector bundles over $$\mathscr{F}$$−foliated manifolds. Essential: existence of a $$\mathscr{F}$$−flat structure is equivalent to the existence of a $$\mathscr{F}$$−flat connection. Let $$ \underset{\lbrace ξ \rbrace}{λ}$$ be a family of subbundles of a vector bundle $$ ξ$$. here exists a family of $$\mathscr{F}$$−flat structure Let $$ \underset{\lbrace Λ \rbrace}{λ}$$ in $$ ξ$$, relative at $$ \underset{ξ}{λ}$$, if and only if exists a family of $$\mathscr{F}$$−flat connections $$\underset{\lbrace\bigtriangledown\rbrace}{λ}$$ in $$ \lbrace ξ \rbrace$$ (Theorem III.5). $$\mathscr{F}$$−flat structures (Theorem III.1), and integrable $$\mathscr{F}$$−flat structures (Theorem III.5), are considered. Finally, integrable
$$Γ$$−structure and $$\mathscr{F}$$−flat structures on total space of a vector bundle are presented (Theorem IV.1).
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Keywords: $$\mathscr{F}$$−flat structure, $$\mathscr{F}$$−flat connection, integrable $$\mathscr{F}$$−flat structure.
Pages: 570-576
DOI: 10.37394/23206.2023.22.63