Theory of Finsler spaces with (α, β)-metric

Projectively Flat Finsler Space of Douglas Type with Weakly-Berwald (<i>α</i>,<i>β</i>)-Metric

“…According to [17], we have the functions G i (x, y) of F n with the (α, β)-metric are written in the form,…”

“…And plugging (61) into (9), (17) and (20) in respective quantities, we have A = (n + 1)αβ 4 (2α + β) − α(α 2 b 2 − β 2 )(αβ + α 2 )(4α + 4β) β 3 ,…”

Homogeneous Finsler Square Metrics of Douglas Type.

“…In [15] shows an (α, β)-metric F = αφ( β α ) is a Berwald metric if and only if β is parallel with respect to α, i.e., b i|j = 0, regardless of the choice of a particular φ, and [12] the authors obtained a characterization of (α, β)-metrics of Douglas type.…”

“…A Finsler metric F on a manifold M is a homogeneous continuous function F : T M → [0; +∞) where F is smooth on the slit tangent bundle T M o satisfying nonnegativity (F (y) > 0 for any y = 0) and strong convexity (the fundamental tensor g ij := [ The notion of (α, β)-metric in Finsler spaces was introduced by M. Matsumoto [4] as a generalization of Randers metric L = α + β, where α is a regular Riemannian metric α = a ij (x)y i y j , i.e., det(a ij ) = 0 and β is a one-form β = b i (x)y i and studied by many authors ( [5], [6], [8], and [9]). A Finsler metric L(α, β) on a differentiable manifold M n is called an (α, β)-metric, if L is a positively homogeneous function of degree one in α and β.…”

On the geometry of randers manifolds

“…Randers metrics were seen in a more general class of metrics which emerged in the study of what are now called (α, β)-metrics. For a general survey of results and applications of Randers manifolds, we refer to [2] and [12].…”