Local holonomy groups of induced connections

There are two naturally induced connections on the tangent bundle, the so called Jacobi connection and the Sasaki connection. By using the elementary theory of systems of linear differential equations, we completely determine the local holonomy group of these two induced connections, and find some relation to the local holonomy group of the manifold itself. There is an induced connection on the vector bundle of linear maps of the fibers. We also investigate the properties of the holonomy group of this bundle.