Shen's $L$-process on Berwald connection

• Datum: 2020
• Hauptverfasser: Faghfouri, M., Jazer , N., Faghfouri, Morteza
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Institute of Mathematics, NAS of Ukraine 2020
• Online Zugang: https://umj.imath.kiev.ua/index.php/umj/article/view/6001
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

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.