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

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 FνW\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}^{\nu } {\mathscr {W}}$$\end{document}.