Shen's $L$-process on Berwald connection
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 Cher...
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| Date: | 2020 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2020
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/6001 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | 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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| DOI: | 10.37863/umzh.v72i8.6001 |