Yano-ledger connection and induced connection on vector bundles

Abstract

K. Yano and A. J. Ledger [13] const ructed f rom a linear connection V on a manifold B a torsion-free linear connection on T B (called the Yano-Ledger connection). M. Ma t sumoto [7] proved tha t : a) V determines a Finsler connection (75, V) in the space V T B of Finsler vectors; b) the symmetr iza t ion of the extension V ~ of (75, V) to T T B is exactly the Yano-Ledger connect ion on TB; c) the Levi-Civi ta connection of the Riemannian metr ic on T B derived f rom a Riemannian metric g on B coincides with the Yano-Ledger connect ion derived f rom the Levi-Civi ta connection V of g iff the Riemannian curvature tensor of g vanishes (see also [2]). On the other hand R. Miron [8] developed a theory of Finsler connections on vector bundles. The purpose of this paper is to construct a Yano-Ledger connect ion on vector bundles, and then to prove the analogues of Matsumoto ' s theorems a), b), and c) for vector bundles and vector bundle Finsler connections. In our considerations and construct ions we apply pullback of pseudoconnections. They are developed and invest igated in §§1, 2, and 3. §4 yields the Yano-Ledger connect ion for vector bundles, and §§4, 5, and 6 present the ment ioned theorems analogous to those of Matsumoto . Concerning no ta t ion and terminology we refer to the monographs [1],