On Wagner Connections and Wagner Manifolds

Let (M, E) be a Finsler manifold. A triplet (¯D, ¯h, α) is said to be a Wagner connection on M if (¯D, ¯h) is a Finsler connection, α ∈ C∞ (M) and the axioms (W1)–(W4), formulated originally by M. Hashiguchi, are satisfied. Then ¯h is called a Wagner endomorphism on M. We establish an explicit relation between the (canonical) Barthel endomorphism of (M, E) and a Wagner endomorphism ¯h. We show that the second Cartan tensors ¯C′, ¯C′b belonging to ¯h are symmetric and totally symmetric, respectively. An explicit relation between the "canonical" tensors C′, C′b and the "Wagnerian" ones is also derived. We can conclude that the rules of calculation with respect to a Wagner connection are formally the same as those with respect to the classical Cartan connection. We establish some basic curvature identities concerning a Wagner connection, including Bianchi identities. Finally, we present a new, intrinsic definition as well as several tensorial characterizations of Wagner manifolds.