General Natural α-Structures Parallel with Respect to the Schouten–Van Kampen Connection on the Tangent Bundle

We determine the Schouten–Van Kampen connection associated to the Levi-Civita connection of a general natural metric on the total space TM of the tangent bundle of a Riemannian manifold. We provide the necessary and sufficient conditions for the obtained Schouten–Van Kampen connection to be torsion free and then to coincide with the Levi-Civita connection. We characterize the general natural α -structures on TM , which are parallel with respect to the torsion free Schouten–Van Kampen connection. Finally, we obtain the (para-)Kähler structures of general natural lift type on TM , for which the α -structure is parallel with respect to both Levi-Civita and Schouten–Van Kampen connection.