International J.Math. Combin. Vol.1(2018), 61-67
Nonholonomic Frames for Finsler Space with Deformed Special (α, β) Metric Brijesh Kumar Tripathi (Department of Mathematics, L.E. College Morbi, Morbi-2,(Gujarat), 363642, India) E-mail: brijeshkumartripathi4@gmail.com
Abstract: The purpose of present paper to determine the Finsler spaces due to deformation of special Finsler (α, β) metrics. Consequently, we obtain the nonholonomic frame with the 2 i j i help of Riemannian metric α = aij (x)y y , one form metric β = bi (x)y and Douglas metric 2 L(α, β) = α + βα such as forms I. L(α, β)
=
II. L(α, β)
=
αβ, β2 β3 α+ β = αβ + . α α
The first metric of the above deformation is obtained by the product of Riemannian metric and one form and second one is the product of Douglas metric and 1-form metric.
Key Words: Finsler space, (α, β)-metrics, Riemannian metric, one form metric, Douglas metric, GL-metric, nonholonomic Finsler frame.
AMS(2010): 53C60, 53B40. §1. Introduction P.R. Holland [1], [2] studies about the nonholonomic frame on space time which was based on the consideration of a charged particle moving in an external electromagnetic field in the year 1982. In the year 1987, R.S. Ingarden [3] was the first person, to point out that the Lorentz force law can be written in above case as geodesic equation on a Finsler space called Randers space. Further in 1995, R.G. Beil [5], [6] have studied a gauge transformation viewed as a nonholonomic frame on the tangent bundle of a four dimensional base manifold. The geometry that follows from these considerations gives a unified approach to gravitation and gauge symmetries. In the present paper we have used the common Finsler idea to study the existence of a nonholonomic frame on the vertical sub bundle VTM of the tangent bundle of a base manifold M. In this case we have considered the fundamental tensor field might be the deformation of two different special Finsler spaces from the (α, β)- metrics. First we consider a nonholonomic frame for a Finsler space with (α, β)- metrics such as first product of Riemannian metric and 1form metric and second is the product of Douglas metric and 1-form metric. Further we obtain 1 Received
August 2, 2017, Accepted February 20, 2018.