Models of Finsler Geometry on Lie algebroids

Abstract

Horizontal endomorphisms, almost complex structures, vertical, horizontal and complete lifts on prolongation of a Lie algebroid are considered. Then using exact sequences, semisprays are constructed. Moreover, important geometrical objects such as classical distinguished connections, torsions and partial curvatures are studied on prolongation of Lie algebroids. Considering pullback bundle, covariant derivatives are scrutinized based on anchor map. Several Finsler geometry models on Lie algebroid structures, will organized via recent arguments. Finally, it will be overhaulled some special Finsler Lie algebroid spaces.

Curvature collineations on Lie algebroid structure

“…Then the elements of v£ π E are of the form (0, z) ∈ E × T E such that π * (z) = 0, these elements are called vertical. Since π * (z) = 0 and ker π * = vE (π * : T E → T M ), then we deduce that if (0, z) is vertical then z is a vertical vector on E [9].…”

“…Then ξ ∨ is a vertical vector field on E. Thus it follows that if ξ = ξ α e α ∈ Γ(E), then the vertical lift ξ ∨ has the locally expression ξ ∨ = (ξ α • π) ∂ ∂y α . If ξ, η are sections of E and f ∈ C ∞ (M ), then using the local expressions of them, we obtain [9] (ξ + η)…”

“…where η = η α e α ∈ Γ(E) [9]. Note that the first components of η V and η C are free of indeterminacy.…”

Vanishing Theorems on Holomorphic Lie Algebroids

“…On the other hand, E. Martinez [8,9] developed another approach using the notion of prolongation of Lie algebroid, for that fundamental notions on tangent bundle geometry, such as spray theory and nonlinear connection, can be transferred to this prolongation. Many recently results are obtained on Lie algebroids ( [2,3,12,13,11] etc. ).…”

Vertical and complete lifts of sections of a (dual) vector bundle and Legendre duality

“…Several authors presented and studied the lift of geometrical objects of a Lie algebroid ((F, ν, N ), [, ] F , (ρ, Id N )) to the Lie algebroid prolongation. Using the vertical and complete lifts of sections of a Lie algebroid, the first author presented important results about Lie simetry and horizontal lifts in the general framework of prolongation Lie algebroid [10].…”