ON A SPECIAL CONCIRCULAR LIE-RECURRENCE IN A FINSLER SPACE

V Pandey,P.N Pandey

doi:10.12732/ijpam.v92i3.4

Abstract

The concept of Lie-recurrence in a Finsler space was introduced by the second author [8] of the present paper in 1982.The Lie-recurrence in a Riemannian space was studied by K. L. Duggal [3] in 1992 but he used the term curvature inheriting symmetry in place of Lie-recurrence. K. L. Duggal also applied the theory to the study of fluid space time. Since then both the terms (Lie-recurrence and curvature inheriting symmetry) are in use. The present authors [11], Shivalika Saxena and P. N. Pandey [12], [13], C. K. Mishra and Gautam Lodhi [1] studied a Lie-recurrence (curvature inheriting symmetry) in a Finsler space and discussed the possibilities for contra and concurrent vector fields to generate Lie-recurrence. The present paper deals with a Lie-recurrence generated by a special concircular vector field and such Lie-recurrence is termed as a special concircular Lie-recurrence. We obtain certain results related to a special concircular Lie-recurrence in a general Finsler space as well as in birecurrent and bisymmetric Finsler spaces. AMS Subject Classification: 53B40

Highlights

Let Fn be an n-dimensional Finsler space equipped with a metric function F (xi, yi) satisfying the requisite conditions[2], the corresponding metric tensor g(xi, yi) and the Berwald connection G(xi, yi)
Let the components of the metric tensor g and coefficients of Berwald connection G be denoted by gij and Gijk respectively
Let us consider a Finsler space admitting the infinitesimal transformation (8) generated by a special concircular vector field vi(xj) characterized by

Summary

Introduction

Let Fn be an n-dimensional Finsler space equipped with a metric function F (xi, yi) satisfying the requisite conditions[2], the corresponding metric tensor g(xi, yi) and the Berwald connection G(xi, yi). Let the components of the metric tensor g and coefficients of Berwald connection G be denoted by gij and Gijk respectively. Berwald covariant derivative of a tensor field Tji with respect to xk is defined by. Hhi kj appearing in equation (5) are components of the Berwald curvature tensor. This tensor is skewsymmetric in last two lower indices and positively homogeneous of degree zero in yi. This tensor and the tensor Hkij appearing in equation (5) are connected by the following: a) Hjikhyj = Hki h, b) ∂ ̇j Hki h = Hjikh. The Lie-derivative of an arbitrary tensor field Tji with respect to above transformation [4] is given by

Special Concircular Lie-Recurrence

Special Concircular Lie-Recurrence in a Birecurrent Finsler Space

Special Concircular Lie-Recurrence in a Bisymmetric Finsler Space