Concircular π-Vector Fields and Special Finsler Spaces

Nabil L Youssef,Amr Soleiman

doi:10.4236/apm.2013.32040

Abstract

The aim of the present paper is to investigate intrinsically the notion of a concircular π-vector field in Finsler geometry. This generalizes the concept of a concircular vector field in Riemannian geometry and the concept of concurrent vector field in Finsler geometry. Some properties of concircular π-vector fields are obtained. Different types of recurrence are discussed. The effect of the existence of a concircular π-vector field on some important special Finsler spaces is investigated. Almost all results obtained in this work are formulated in a coordinate-free form.

Highlights

The concept of a concurrent vector field in Riemannian geometry had been introduced and investigated by K
In [9], we investigated intrinsically concurrent vector fields in Finsler geometry
The notion of a concircular vector field in Riemannian geometry has been studied by Adat and Miyazawa [1]

Summary

Introduction

The concept of a concurrent vector field in Riemannian geometry had been introduced and investigated by K. Concurrent vector fields in Finsler geometry had been studied locally by S. In [9], we investigated intrinsically concurrent vector fields in Finsler geometry. We introduce and investigate intrinsically the notion of a concircular πvector field in Finsler geometry, which generalizes the concept of a concircular vector field in Riemannian geometry and the concept of a concurrent vector field in Finsler geometry. Global formulation of different aspects of Finsler geometry may help better understand these aspects without being trapped into the complications of indices. This is one of the motivations of the present work, where almost all results obtained are formulated in a coordinate-free form

Notation and Preliminaries

Concircular π-vector fields on a Finsler manifold

Special Finsler spaces admitting concircular π-vector fields