On Curvatures of Homogeneous Finsler Space with Special (

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IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 2 Ver. IV (Mar. - Apr. 2017), PP 47-53 www.iosrjournals.org

On Curvatures of Homogeneous Finsler Space with Special (đ?œś, đ?œˇ)-Metric Chandru K. and Narasimhamurthy S.K. Department of Mathematics, Kuvempu University, Jnana Sahyadri, Shankaraghatta-577 451, Shimoga, Karnataka, INDIA.

Abstract: In this paper, we give an explicit formula of the S-curvature of homogeneous Finsler space with special (đ?›ź, đ?›˝)-metric and proved that a homogeneous Finsler space with almost isotropic S-curvature must have vanishing S-curvature. We also derived an explicit formula of the mean-Berwald curvature đ??¸đ?‘–đ?‘— of homogeneous Finsler space. Keywords: Finsler space, (đ?›ź, đ?›˝)-metric, Homogeneous Finsler space, S-curvature, Mean-Berwald curvature.

I.

Introduction

Let (đ?‘€, đ??š) be a Finsler space, where đ?‘€ be the connected smooth manifold and đ??š be the Finsler metric. (đ?›ź, đ?›˝)-metrics are the class of Finsler metrics, for example Randers metric, Kropina metric, Matsumoto metric, đ?›˝ etc. (đ?›ź, đ?›˝)-metrics are of the form đ??š = đ?œ™(đ?‘ ), where đ?‘ = , here đ?›ź is Riemannian metric and đ?›˝ is the one form. đ?›ź One of the central problems in Finsler geometry is to study curvature properties of special class of Finsler spaces such as homogeneous-Finsler spaces with (đ?›ź, đ?›˝)-metrics. A Finsler space (đ?‘€, đ??š) becomes homogeneous if the group of isometries đ??ź(đ?‘€, đ??š) of (đ?‘€, đ??š) acts transitively on đ?‘€. As described in [7], for homogeneous Finsler space, đ?‘€ can be written as a coset space đ??ş/đ??ť with (đ?›ź, đ?›˝)đ?›˝ metric of the form đ??š = đ?œ™(đ?‘ ), where đ?‘ = , with đ?›ź a đ??ş invariant Riemannian metric on đ??ş/đ??ť and đ?›˝ a đ??ş-invariant đ?›ź vector field on đ??ş/đ??ť. Therefore the Lie algebra g of đ??ş expressed as composition of đ?‘• and đ?‘š. đ?‘” = đ?‘• + đ?‘š (direct sum of subspaces) (1.1) such that đ??´đ?‘‘ đ?‘• đ?‘š ⊂ đ?‘š, đ?‘• ∈ đ??ť, and we can identify đ?‘€ with the tangent space of (đ??ş/đ??ť) at the origin đ?‘œ = đ??ť. Further, the one form đ?›˝ corresponds to a vector field đ?‘‹ on đ??ş/đ??ť which is generated by đ??´đ?‘‘(đ??ť) −invariant vector in đ?‘š with length < 1. The goal of this paper is to derive an explicit formula for the S-curvature of homogeneous Finsler space with special (đ?›ź, đ?›˝)-metric đ??š =

�2 �

+ đ?›˝.The notion of S-curvature for a Finsler space introduced by Z. Shen in [14]. It

is a quantity to measure the rate of change of the volume form of a Finsler space along the geodesics. Recently many geometers studied curvature properties of homogeneous Finsler space [7, 8]. In 2007, S. Deng and Z. Hou studied that a homogeneous Finsler spaces with non-positive flag curvature [8]. In 2010, S. Deng obtained the explicit formula of S-curvature of homogeneous Randers space with almost isotropic S-curvature must have vanishing S-curvature [7]. As for reference, there is an explicit formula for S-curvature in a local standard coordinate system by Z. Shen [14]. However, for a homogeneous spaces there should have a formula which does not use local coordinates. In this article we are studied curvature properties of homogeneous Finsler space with special (đ?›ź, đ?›˝)-metric đ??š =

�2 �

+ đ?›˝ and find the formula for S-curvature and mean-Berwald curvature.

In the following, we shall use Einstein summation convention unless otherwise stated.

II.

Preliminaries

Finsler space is a smooth manifold possessing a Finsler metric. A standard definition of a Finsler space is defined by: Definition 2.1. A Finsler space is a triple đ??š đ?‘› = (đ?‘€, đ??ˇ, đ??š), where đ?‘€ is an đ?‘›-dimensional manifold, đ??ˇ is an open subset of a tangent bundle đ?‘‡đ?‘€ and đ??š is a Finsler metric defined as a function đ??š âˆś đ?‘‡đ?‘€ → [0,1), with the following properties: 1. Regular: đ??š is đ??ś 1 on the entire tangent bundle đ?‘‡đ?‘€\{0}. 2. Positive homogeneous: đ??š(đ?‘Ľ, đ?œ†đ?‘Ś) = đ?œ†đ??š(đ?‘Ľ, đ?‘Ś). 3. Strong convexity: The đ?‘› Ă— đ?‘› Hessian matrix đ?‘”đ?‘–đ?‘— = đ??š 2 đ?‘Ś đ?‘– đ?‘Ś đ?‘— is positive definite at every point on đ?‘‡đ?‘€\{0}, where đ?‘‡đ?‘€\{0} denotes the tangent vector đ?‘Ś is non zero in the tangent bundle đ?‘‡đ?‘€. DOI: 10.9790/5728-1302044753

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