Commun. Korean Math. Soc. 2024; 39(1): 187-199
Online first article January 25, 2024 Printed January 31, 2024
https://doi.org/10.4134/CKMS.c230087
Copyright © The Korean Mathematical Society.
Salah Gomaa Elgendi, Amr Soleiman
Islamic University of Madinah; Benha University
In the present paper, following the pullback approach to Finsler geometry, we study intrinsically the $C^v$-reducible and generalized $C^v$-reducible Finsler spaces. Precisely, we introduce a coordinate-free formulation of these manifolds. Then, we prove that a Finsler manifold is $C^v$-reducible if and only if it is $C$-reducible and satisfies the $\mathbb{T}$-condition. We study the generalized $C^v$-reducible Finsler manifold with a scalar $\pi$-form $\mathbb{A}$. We show that a Finsler manifold $(M,L)$ is generalized $C^v$-reducible with $\mathbb{A}$ if and only if it is $C$-reducible and $\mathbb{T}=\mathbb{A}$. Moreover, we prove that a Landsberg generalized $C^v$-reducible Finsler manifold with a scalar $\pi$-form $\mathbb{A}$ is Berwaldian. Finally, we consider a special $C^v$-reducible Finsler manifold and conclude that a Finsler manifold is a special $C^v$-reducible if and only if it is special semi-$C$-reducible with vanishing $\mathbb{T}$-tensor.
Keywords: $\mathbb{T}$-tensor, Cartan connection, $C$-reducible, $C^v$-reducible
MSC numbers: Primary 53C60, 53B40, 58B20
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