The Deformation of an $(\alpha, \beta)$-Metric

Abstract

In this paper, we will continue our investigation on the new recently introduced; where α is a Riemannian metric; β is a 1-form, and a ∈ 1 4 , +∞ is a real positive scalar. We will investigate the deformation of this metric, and we will investigate its properties.

Nonholonomic Frame for a Deformed $(\alpha,\beta )$-metric

“…and from this inequality, the conclusion follows easily. In the view of Lemma 1.1, the link between the spray coefficients G i of the metric (1.1) and the G i α of the metric α, is presented in [14]. Next, we will compute the following Finsler invariants, attached for the metric (1.1)…”

“…where α = a ij y i y j is a Riemannian metric; β = b i y i is a 1-form, | | < 2 √ a + 1 is a real parameter and a ∈ 1 4 , +∞ is a real positive scalar. This metric was introduced in [14]. The perturbed (α, β)-metrics was first introduced by Matsumoto in [7] and since then, the theory of this kind of metrics in Finsler geometry was developed in a lot of papers (for example please see [17], [18]).…”

Some Aspects on a Special Type of $(\alpha,\beta )$-metric

“…We introduced a very special type of (α, β)-metrics in some previous papers [14], [15], [16], [17] and we analyzed some results regarding the properties for this metric. This special (α, β)-metric is given in the following form:…”