General Finsler connections on a Finsler vector bundle (Q1103206)

A Finsler vector bundle on a differentiable manifold M is a vector bundle E over the tangent bundle TM of M. Then a general Finsler connection on E is a triple (N,P,\(\nabla)\), where N is a nonlinear connection on TM, P is a vector bundle morphism on E and \(\nabla\) is a general connection [see the second author, Math. J. Okayama Univ. 9, 99-164 (1960; Zbl 0202.211) and \textit{N. Abe}, Kodai Math. J. 8, 322-329 (1985; Zbl 0594.53030)]. If in particular P is just the identity on E then we find the concept of vectorial Finsler connections which is the main tool in studying Finsler subspaces [see the first author, An. Stiint. Univ. Al. I. Cuza Iaşi, N. Ser., Secţ. Ia 32, No.2, 69-83 (1986; Zbl 0622.53018)]. In case E is the vertical vector bundle on M we obtain a general Finsler connection on a Finsler space. In the present paper we give the main properties of general Finsler connections on a Finsler vector bundle and study their interrelations with some other geometric objects from Finsler geometry. It is important to note that starting with a Finsler connection on a manifold we obtain two general Finsler connections whose torsions and curvatures give all torsions and curvatures of the initial Finsler connection. We consider the paper as the starting point to a coordinate-free theory of Finsler geometry.