Integrability of the Metallic Structures on the Frame Bundle
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Earlier investigators have made detailed studies of geometric properties such as integrability, partial integrability, and invariants, such as the fundamental 2-form, of some canonical f-structures, such as f 3 ± f = 0, on the frame bundle FM. Our aim is to study metallic structures on the frame bundle: polynomial structures of degree 2 satisfying F2 = pF +qI where p, q are positive integers. We introduce a tensor field Fα, α = 1, 2..., n on FM show that it is a metallic structure. Theorems on Nijenhuis tensor and integrability of metallic structure Fα on FM are also proved. Furthermore, the diagonal lifts g D and the fundamental 2-form Ωα of a metallic structure Fα on FM are established. Moreover, the integrability condition for horizontal lift F H α of a metallic structure Fα on FM is determined as an application. Finally, the golden structure that is a particular case of a metallic structure on FM is discussed as an example.
Abstract
Earlier investigators have made detailed studies of geometric properties such as integrability, partial integrability, and invariants, such as the fundamental 2-form, of some canonical f-structures, such as f 3 ± f = 0, on the frame bundle FM. Our aim is to study metallic structures on the frame bundle: polynomial structures of degree 2 satisfying F2 = pF +qI where p, q are positive integers. We introduce a tensor field Fα, α = 1, 2..., n on FM show that it is a metallic structure. Theorems on Nijenhuis tensor and integrability of metallic structure Fα on FM are also proved. Furthermore, the diagonal lifts g D and the fundamental 2-form Ωα of a metallic structure Fα on FM are established. Moreover, the integrability condition for horizontal lift F H α of a metallic structure Fα on FM is determined as an application. Finally, the golden structure that is a particular case of a metallic structure on FM is discussed as an example.
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