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.
KeywordsBianchi Identity Explicit Relation Finsler Manifold Curvature Identity Cartan Connection
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