Abstract
In the present paper we established some interesting results on purely Hermitian R-complex Finsler space with (α, β)-metrics, Firstly we characterize the conditions for the (α, β)-metric F = p α2 + εβ2 to be a purely Hermitian. Then determined the fundamental metric tensor, its inverse and determinent of the above metric. Further obtained Chern-Finsler connection coefficients and analysed necessary conditions under which an purely Hermitian R-complex Finsler space with (α, β)-metric to be Berwald, Kahler and strongly Kahler also given some examples.
Highlights
In [1, 2, 9, 10, 12, 13, 14, 18, 20], many geometers contributed the field of complex Finsler geometry with reference to the notions of real Finsler geometry
In the very begining Rizza [17] extended the homogeneous property of real Finsler metric to the complex case by defining the function F : T M → R+ with the condition F (z, λη) = |λ|F (z, η), for any λ ∈ C, where (z, η) are complex coordinates
In [11, 15] authors reduced the definition of complex Finsler space [16, 19] was extended, reducing homogeneity property to the scalars λ ∈ R
Summary
In [1, 2, 9, 10, 12, 13, 14, 18, 20], many geometers contributed the field of complex Finsler geometry with reference to the notions of real Finsler geometry. In the very begining Rizza [17] extended the homogeneous property of real Finsler metric to the complex case by defining the function F : T M → R+ with the condition F (z, λη) = |λ|F (z, η), for any λ ∈ C, where (z, η) are complex coordinates. In [11, 15] authors reduced the definition of complex Finsler space [16, 19] was extended, reducing homogeneity property to the scalars λ ∈ R, . We will focus only on the study of the purely Hermitian complex Finsler spaces, (meaning gij is invertible). We show that any purely Hermitian R-complex Finsler spaces with (α, β)-metric is Berwald. We prove that any strongly Berwald space is strongly Kahler, by some explicit examples
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