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  • International Journal of Mathematics Trends and Technology (IJMTT) – Volume 44 Number 3 April 2017

    ISSN: 2231-5373 http://www.ijmttjournal.org Page 123

    Two Kinds of Weakly Berwald Special (α, β) - metrics of Scalar flag curvature

    Thippeswamy K.R #1

    , Narasimhamurthy S.K #2

    1.2 Departme of Mathematics, Kuvempu University, Shankaraghatta - 577451, Shimoga, Karnataka, india.

    Abstract: In this paper, we study two important class

    of special (α, β)-metrics of scalar flag curvature in

    the form of and

    (where and

    are constants) are of scalar flag curvature. We prove

    that these metrics are weak Berwald if and only if

    they are Berwald and their flag curvature vanishes.

    Further, we show that the metrics are locally

    Minkowskian.

    Key Words: Finsler metric, metrics, S-

    curvature, Weak berwald metric, flag cur-vature,

    locally minkowski metric.

    1. Introduction

    The flag curvature in Finsler geometry is a

    natural extension of the sectional curvature in

    Riemannian geometry, which is first introduced by

    L. Berwald. For a Finsler manifold the flag

    curvature is a function of tangent planes

    and directions . is said to be of

    scalar flag curvature if the flag curvature

    is independent of flags

    associated with any fixed flagpole . One of the

    fundamental problems in Riemann-Finsler geometry

    is to study and characterize Finsler metrics of scalar

    flag curvature. It is known that every Berwald

    metric is a Landsberg metric and also, for any

    Berwald metric the -curvature vanishes [2]. In

    2003, Shen proved that Randers metrics with

    vanishing -curvature and are not

    Berwaldian [10]. He also proved that the Bishop-

    Gromov volume comparison holds for Finsler

    manifolds with vanishing -curvature.

    The concept of metrics and its curvature

    properties have been studied by various authors

    [1],[8],[4],[7]. X. Cheng, X. Mo and Z. Shen

    (2003) have obtained the results on the flag

    curvature of Finsler metrics of scalar curvature [3].

    Yoshikawa, Okubo and M. Matsumoto(2004)

    showed the conditions for some metrics to

    be weakly- Berwald. Z. Shen and Yildirim (2008)

    obtained the necessary and sufficient conditions for

    the metric to be projectively flat. They

    also obtained the necessary and sufficient conditions

    for the metric

    to be projectively flat

    Finsler metric of constant flag curvature and proved

    that, in this case, the flag curvature vanishes

    Recently, X. Cheng(2010) has worked on

    metrics of scalar flag curvature with

    constant S-curvature. The main purpose of the

    present paper is to study and characterize the two

    important class of weakly-Berwald metrics

    and

    (where and are constants) are of scalar flag

    curvature The terminologies and notations are

    referred to [5][2].

    2. Preliminaries

    Let be an n-dimensional manifold and

    denote the tangent bundle of .

    A Finsler metric on is a functions

    with the following properties:

    a)

    b)

    Minkowskian norm

    The pair is called Finsler manifold;

    Let be a Finsler manifold and

    (2.1)

    For a vector induces an inner

    product on as follows

    where and

    Further, the Cartan torsion C and the mean Cartan

    torsion are defined as follows[2]:

    http://www.ijmttjournal.org/

  • International Journal of Mathematics Trends and Technology (IJMTT) – Volume 44 Number 3 April 2017

    ISSN: 2231-5373 http://www.ijmttjournal.org Page 124

    (2.2)

    Where

    (2.3)

    (2.4)

    Where

    Let be a Riemannian metric and

    be a 1-form on an n-dimentional

    manifold . The norm

    ,

    Let be a C∞ positive function on an open

    interval satisfying the following conditions

    – –

    Then the functions is a Finsler metric

    if and only if . Such Finsler metrics are

    called metrics.

    Let denote the horizontal covariant

    derivative with respect to and , respectively.

    Let

    (2.5)

    . (2.6)

    For a positive function on

    and a number let

    Where

    The geodesic of a Finsler metric is

    characterized by the following system of second

    order ordinary differential equations:

    Where

    is called as geodesic coefficients of F. For an

    metric using a

    Maple program, we can get the following [5];

    (2.8)

    Where denote the spray coefficients of .

    We shall denote

    and .

    Furthermore, let

    By a direct computation, we can obtain a formula for

    the mean Cartan torsion of metrics as

    follows[6]:

    – . (2.9)

    According to Diecke’s theorem, a Finsler metric is

    Riemannian if and only if the mean Cartan torsion

    vanishes, . Clearly, an metric

    is Riemannian if and only if

    (see [6]).

    For a Finsler metric on a manifold ,

    the Riemann curvature is

    defined by

    .

    Let

    http://www.ijmttjournal.org/

  • International Journal of Mathematics Trends and Technology (IJMTT) – Volume 44 Number 3 April 2017

    ISSN: 2231-5373 http://www.ijmttjournal.org Page 125

    For a flag where , the

    flag curvature of is defined by

    , (2.10)

    where .

    We say that Finlser metric is of scalar flag

    curvature if the flag curvature is

    independent of the flag By the definition , is of

    scalar flag curvature if and only if in a

    standard local coordinate system,

    , (2.11)

    where ([2]).

    The Schur Lemma in Finsler geometry tell us that, in

    dimension , if is of isotropic flag curvature ,

    then it is of constant flag curvature , =

    constant.

    The Berwald curvature

    and mean Berwald curvature

    are defined respectively by

    A Finsler metric F is called weak Berwald metric if

    the mean Berwald curvature vanishes, i.e.,

    A Finsler metric is said to be of isotropic

    mean Berwald curvature if

    where is a scalar function on the manifold

    .

    Theorem 2.1. [9] For special metric

    and

    (where are constants) on an n-

    dimensional manifold . Then the following are

    equivalent:

    (a) is of isotropic -curvature,

    (b) is of isotropic mean Berwald curvature

    (c) is a killing 1-form with constant length with

    respect to , i.e., (d) -curvature vanishes , (e) is a weak Berwald metric, where is scalar function on the manifold . Note that, the discussion in [9] doesn’t involve

    whether or not is Berwald metric. By the

    definitions, Berwald metrics must be weak Berwald

    metrics but the converse is not true.

    For this observation, we further study the metrics

    and

    (where are constants).

    For a Finsler metric on and n-

    dimensional manifold , the Busemann-Hausdorff

    volume form is

    given by

    ,

    denotes the euclidean volume in . The -

    curvature is given by

    (2.12)

    Clearly, the mean Berwald curvature

    can be characterized by use of -curvature as

    follows:

    A Finsler metric F is said to be of isotropic S-

    curvature if where

    is a scalar function on the manifold . -

    curvature is closely related to the flag curvature.

    Shen, Mo and cheng proved the following important

    result.

    Theorem 2.2. [3] Let be an n-dimensional

    Finsler manifold of scalar flag curvatur