Some Identities for Generalized Curvature Tensors in 𝓑-Recurrent Finsler Space

Some Identities for Generalized Curvature Tensors in ๐“‘-Recurrent Finsler Space

The purpose of the present paper is to consider and study a certain identities for some generalized curvature tensors in โ„ฌ-recurrent Finsler space ๐น๐‘› in which Cartanโ€™s second curvature tensor ๐‘ƒ๐‘—๐‘˜โ„Ž๐‘– satisfies the generalized of recurrence condition with respect to Berwaldโ€™s connection parameters G๐‘˜โ„Ž๐‘–...

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Bibliographic Details
Main Author: Adel Mohammed Ali Al-qashbari
Format: Article
Language:English
Published: Naim ร‡aฤŸman 2020-09-01
Series:Journal of New Theory
Subjects:
finsler space
cartanโ€™s second curvature tensor ๐‘ƒ๐‘—๐‘˜โ„Ž๐‘–
generalized โ„ฌ๐‘ƒ-recurrent space
weylโ€™s projective curvature tensor ๐‘Š๐‘—๐‘˜โ„Ž๐‘–
cartanโ€™s fourth curvature tensor ๐‘…๐‘—๐‘˜โ„Ž๐‘–
Online Access:https://dergipark.org.tr/en/download/article-file/1317656
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Summary:The purpose of the present paper is to consider and study a certain identities for some generalized curvature tensors in โ„ฌ-recurrent Finsler space ๐น๐‘› in which Cartanโ€™s second curvature tensor ๐‘ƒ๐‘—๐‘˜โ„Ž๐‘– satisfies the generalized of recurrence condition with respect to Berwaldโ€™s connection parameters G๐‘˜โ„Ž๐‘– which given by the condition โ„ฌ๐‘š๐‘ƒ๐‘—๐‘˜โ„Ž๐‘–=๐œ†๐‘š ๐‘ƒ๐‘—๐‘˜โ„Ž๐‘–+๐œ‡๐‘š( ๐›ฟโ„Ž๐‘– ๐‘”๐‘—๐‘˜โˆ’ ๐›ฟ๐‘˜๐‘– ๐‘”๐‘—โ„Ž), where โ„ฌ๐‘š is covariant derivative of first order (Berwaldโ€™s covariant differential operator ) with respect to ๐‘ฅ๐‘š, itโ€™s called a generalized โ„ฌP-recurrent space. We shall denote it briefly by ๐บโ„ฌ๐‘ƒ-๐‘…๐น๐‘›. We have obtained Berwaldโ€™s covariant derivative of first order for the h(v)-torsion tensor ๐‘ƒ๐‘˜โ„Ž ๐‘–, the deviation tensor ๐‘ƒโ„Ž ๐‘– and the covariant derivative of the tensor ๐ป๐‘˜๐‘.โ„Ž (in the sense of Berwald), also we find some theorems of the R-Ricci tensor ๐‘…๐‘—๐‘˜ and the curvature vector ๐‘…๐‘—in our space. We obtained the necessary and sufficient condition for Berwaldโ€™s covariant derivative of Weylโ€™s projective curvature tensor ๐‘Š๐‘—๐‘˜โ„Ž๐‘– and its torsion tensor ๐‘Š๐‘˜โ„Ž๐‘– in our space. Also, we have proved that in ๐บโ„ฌ๐‘ƒ-๐‘…๐น๐‘›, Cartanโ€™s second curvature tensor ๐‘ƒ๐‘—๐‘˜โ„Ž๐‘– and the v(hv)-torsion tensor ๐‘ƒ๐‘˜โ„Ž๐‘– for ๐‘›=4 .
ISSN:2149-1402

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