The Schouten–Van Kampen connection on the (1, 1)-tensor bundle

The problems in the present paper are related to the Schouten–Van Kampen connection ∇ ¯ associated with the Levi-Civita connection ∇ ~ of the Riemannian metric g~ which includes both Sasaki and Cheeger–Gromoll type metrics on the (1, 1)-tensor bundle. Firstly, we define the Schouten–Van Kampen connection ∇ ¯ on the (1, 1)-tensor bundle and give some applications concerning ∇ ¯. Secondly, we compute all forms of curvature tensors of ∇ ¯. Thirdly, we consider an almost product structure which is compatible with the Riemannian metric g~ and investigate the conditions under which this structure is parallel with respect to ∇ ¯. Lastly, we define and study the deformation algebra of the connections (∇ ¯ , ∇ ~). © 2018, Springer-Verlag Italia S.r.l., part of Springer Nature.