Intrinsic theory of $C^v$-reducibility in Finsler Geometry
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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.
Abstract
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.
참고문헌 (25)
- P. L. Antonelli, "The theory of sprays and Finsler spaces with applications in physics and biology" (1993), Kluwer Acad. Publ
- G. S. Asanov, "Finsleroid-Finsler spaces of positive-definite and relativistic types", Rep. Math. Phys, 58(2), pp.275~300 (2006)[DOI]
- S. G. Elgendi, "(α, β)-metrics satisfying the T-condition or the σTcondition", J. Geom. Anal, 31(8), pp.7866~7884 (2021)[DOI]
- S. G. Elgendi, "An intrinsic proof of Numata’s theorem on Landsberg spaces", J. Korean Math. Soc
- J. Grifone, "Structure presque-tangente et connexions. I", Ann. Inst. Fourier(Grenoble), 22(1), pp.287~334 (1972)
- F. Ikeda, "On special semi C-reducible Finsler spaces", Tensor(N. S. ), 66(1), pp.95~99 (2005)
- F. Ikeda, "On Finsler spaces with a special (α, β)-metric", Tensor(N. S. ), 69, pp.150~154 (2008)
- H. Izumi, "On Finsler spaces of perpendicular scalar curvature", Tensor(N. S. ), 32(2), pp.219~224 (1978)
- S. Kikuchi, "On the condition that a Finsler space be conformally flat", Tensor(N. S. ), 55(1), pp.97~100 (1994)
- M. Matsumoto, "On Finsler spaces with curvature tensors of some special forms", Tensor(N. S. ), 22, pp.201~204 (1971)
- M. Matsumoto, "On C-reducible Finsler spaces", Tensor(N. S. ), 24, pp.29~37 (1972)
- M. Matsumoto, "Foundations of Finsler Geometry and Special Finsler Spaces" (1986), Kaiseisha Press
- M. Matsumoto, "A conclusive theorem on C-reducible Finsler spaces", Tensor(N. S. ), 32(2), pp.225~230 (1978)
- R. Miron, "The geometry of Lagrange spaces: theory and applications" (1994), Kluwer Acad. Publ
- T. N. Pandey, "On generalized Cv-reducible Finsler spaces", Int. J. Pure Appl. Math, 80(4), pp.467~476 (2012)
- T. N. Pandey, "Cv-reducible Finsler spaces", Tensor(N. S. ), 61(3), pp.212~215 (1999)
- A. Soleiman, "Recurrent Finsler manifolds under projective change", Int. J. Geom. Methods Mod. Phys, 13(10), pp.10~ (2016)[DOI]
- A. Soleiman, "The geometry of Finsler spaces of Hp-scalar curvature", Differ. Geom. Dyn. Syst, 22, pp.254~268 (2020)
- Z. Szabo, "Positive definite Finsler spaces satisfying the T-condition are Riemannian", Tensor(N. S. ), 35(3), pp.247~248 (1981)
- A. A. Tamim, "Special Finsler manifolds", J. Egyptian Math. Soc, 10(2), pp.149~177 (2002)
- B. Tiwari, "On special reducible Finsler spaces", Differ. Geom. Dyn. Syst, 14, pp.132~137 (2012)
- N. L. Youssef, "A global approach to the theory of special Finsler manifolds", J. Math. Kyoto Univ, 48(4), pp.857~893 (2008)[DOI]
- N. L. Youssef, "A global approach to the theory of connections in Finsler geometry", Tensor(N. S. ), 71(3), pp.187~208 (2009)
- N. L. Youssef, "Geometric objects associated with the fundamental connections in Finsler geometry", J. Egyptian Math. Soc, 18(1), pp.67~90 (2010)
- N. L. Youssef, "Characterization of Finsler spaces of scalar curvature", J. Finsler Geom. Appl, 1(1), pp.15~25 (2020)[DOI]