A Lie Algebroid Structure for Vector Bundles of Finite Rank Isomorphic to Tangent Bundle of Their Base Space

Abstract We define a Lie algebroid structure for a class of vector bundles of rank k over a k -dimensional smooth manifold W , which are isomorphic to the tangent bundle TW . We construct an exciting example of these types of vector bundles. This example is constructed based on partial Caputo fractional derivatives. We call this vector bundle a fractional vector bundle and denote it by $${\mathscr {F}}^{\nu } {\mathscr {W}}$$ F ν W .