arXiv:math/0304042v1 [math.DG] 3 Apr 2003 ON THE CURVATURE OF TENSOR PRODUCT CONNECTIONS AND COVARIANT DIFFERENTIALS JOSEF JANY ˇ SKA Abstract. We give coordinate formula and geometric description of the curva- ture of the tensor product connection of linear connections on vector bundles with the same base manifold. We define the covariant differential of geometric fields of certain types with respect to a pair of a linear connection on a vector bundle and a linear symmetric connection on the base manifold. We prove the generalized Bianchi identity for linear connections and we prove that the antisymmetrization of the second order covariant differential is expressed via the curvature tensors of both connections. Introduction In the theory of linear symmetric (classical) connections on a manifold there are many very well known identities of the curvature tensor (see for instance [1, 4]). Some of these identities can be generalized for any linear connection on a vector bundle. In this paper we give the coordinate formula for the curvature of the tensor product connection K ⊗ K ′ of two linear connections K or K ′ on vector bundles E → M or E ′ → M , respectively, and we give also the geometric description of this curvature. We prove that the curvature of K ⊗ K ′ is determined by the curvatures of K and K ′ . The above results are used in the case if one of linear connections is a classical (linear and symmetric) connection on the base manifold. We introduce the covari- ant differential of sections of tensor products (over the base manifold) of a vector bundle, its dual vector bundle, the tangent and the cotangent bundles of the base manifold. We prove that such (first order) covariant differential of the curvature tensor of a linear connection satisfies the generalized Bianchi identity and that the antisymmetrization of the second order covariant differential is expressed through the curvatures of linear and classical connections. All manifolds and maps are supposed to be smooth. 1. Linear connections on vector bundles Let p : E → M be a vector bundle. Local linear fiber coordinate charts on E will be denoted by (x λ ,y i ). The corresponding base of local sections of E or E ∗ will be denoted by b i or b i , respectively. 1991 Mathematics Subject Classification. 53C05. Key words and phrases. Linear connection, curvature, covariant differential. This paper is in final form and no version of it will be submitted for publication elsewhere. This paper has been supported by the Grant agency of the Czech Republic under the project number GA 201/02/0225. 1