Extending Tangent Bundles by an Algebra Bundle

In this paper, by considering a bundle of algebras on a manifold, we construct a vector bundle that naturally contains the tangent bundle of that manifold and has the most important properties of tangent bundles. This vector bundle can be considered as an extended tangent bundle associated to that algebra bundle. To this end, we consider the algebra of the sections of that algebra bundle and find all derivations on this algebra. These derivations are related to the sections of a vector bundle that is the desired bundle. We also investigate the concepts of connection and metric on this extended tangent bundle and prove the existence of the Levi-Civita connection of a metric on the extended tangent bundle.