The Schouten-Van Kampen connection on the \((1,1)\)-tensor bundle
Let \(M\) be a smooth manifold and \(T^1_1(M)\) the \((1,1)\)-tensor bundle on \(M\). Given the Levi-Civita connection \(\widetilde{\nabla}\) on \((T^1_1(M),\widetilde{g})\), the Schouten-Van Kampen connection \(\overline{\nabla}\) adapted to \(\widetilde{\nabla}\) is defined by \(\overline{\nabla}_XY=V\widetilde{\nabla}_{\widetilde{X}}\,V\widetilde{Y}+ H\widetilde{\nabla}_{\widetilde{X}}H\,\widetilde{Y}\), for all vector fields \(\widetilde{X},\widetilde{Y}\) on \(T^1_1(M)\), where \(V\) and \(H\) are the vertical and horizontal projectors associated with \(\widetilde{\nabla}\). The present paper deals with some problems related to the \((1,1)\)-tensor bundle \(T^1_1(M)\) endowed with the Schouten-Van Kampen connection \(\overline{\nabla}\) adapted to the Levi-Civita connection of the Riemannian metric \(\widetilde{g}\), which generalizes both Sasaki and Cheeger-Gromoll metrics on \(T^1_1(M)\). The authors give the coefficients of the Schouten-Van Kampen connection, present some of its properties and compute all non-vanishing coefficients of its curvature tensor. Then, they consider an almost product structure compatible with the Riemannian metric \(\widetilde{g}\) and find the conditions under which this structure is parallel with respect to \(\overline{\nabla}\). Finally, they define and study the deformation algebra on \(T^1_1(M)\) of the pair of connections \((\overline{\nabla},\widetilde{\nabla})\).
deformation algebra of a pair of connections
scientific article; zbMATH DE number 6992790