On Some Special Finsler Spaces

Abstract

The present communication has mainly been divided into four sections of which the first section is introductory, the second section deals with R⁺- recurrent 𝐹𝑛 ∗ of order one. In this section we have derived results telling as to when a ⁺𝑅𝑗𝑘 𝑖 - ⨁ recurrent 𝐹𝑛 ∗ of order one will be R⁺ - ⨁ recurrent of order one, ⁺𝑅𝑗 𝑖 - ⨁ recurrent 𝐹𝑛 ∗ of order one will be a ⁺𝑅𝑗𝑘 𝑖 -⨁ recurrent of order one. In this section we have also derived the Bianchi’s identity and few more identities which hold in a R⁺- recurrent 𝐹𝑛 ∗ of order one. The third section deals with R⁺- recurrent 𝐹𝑛 ∗ of order two. In this section we have observed that the recurrence tensor field 𝑏𝑙𝑚(𝑥, 𝑥̇) is non-symmetric, few more relations and the Bianchi’s identity have been derived in a R⁺- recurrent 𝐹𝑛 ∗ of order two. In the fourth and the last section we have derived the conditions under which a Landsberg space in a 𝑃𝜆- Finsler space, a 𝑃𝜆 - Finsler space is semi - P2- like, a 𝑃 ∗ - Finsler space is a 𝑃𝜆- Finsler space, a 𝑃𝜆 – Finsler space is P- symmetric, a 𝑃𝜆 - Finsler space is P2 like.

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Keywords: Recurrent Finsler space, Bianchi’s identity, non-symmetric, Landsberg space, semi-P2- like, P* - Finsler space, 𝑃𝜆 – Finsler space, Psymmetric