Shen's $L$-process on Berwald connection
Abstract
The Shen connection cannot be obtained by using Matsumoto's processes from the other well-known connections. Hence Tayebi–Najafi introduced two new processes called Shen's $C$ and $L$-processes and showed that the Shen connection is obtained from the Chern connection by Shen's $C$-process. In this paper, we study the Shen's $C$- and $L$-process on Berwald connection and introduce two new torsion-free connections in Finsler geometry. Then, we obtain all of Riemannian and non-Riemannian curvatures of these connections. Using it, we find the explicit form of $hv$-curvatures of these connections and prove that $hv$-curvatures of these connections are vanishing if and only if the Finsler structures reduce to Berwaldian or Riemannian structures. As an application, we consider compact Finsler manifolds and obtain ODEs.
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