Two Kinds of Weakly Berwald Special (α, β) -metrics of ... · Riemannian geometry, which is first...

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

ISSN: 2231-5373 http://www.ijmttjournal.org 23

Two Kinds of Weakly Berwald Special (α, β) -

metrics of Scalar flag curvature Thippeswamy K.R

#1, Narasimhamurthy S.K

#2

1.2Departme 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/


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




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 curvature with flag

curvature Suppose that the -curvature

is isotropic, where

is a scalar function on . Then there is a

scalar function on such that




(2.13)

This shows that -curvature has important influence

on the geometric structures of Finsler metrics. For a

Finsler metric , the Landsberg curvature

and the mean Landsberg

curvature are defined respectively by

A Finsler metric is called weak Landsberg metric if

the mean Landsberg curvature vanishes, i.e.,

. For an metric

, Li and Shen[8] obtained the following formula

of the mean Landsberg curvature

Besides, they also obtained

The horizontal covariant derivatives and

of with respect to and respectively

are given by

Where ,

.

Further we have,

=

If a Finsler metric is of constant flag curvatuType

equation here.re (see[13]), then

.

So, if an metric , is

of constant flag curvature , then

Contracting the above equation by yields the

following equation

(2.16)

3. Characterization of Weakly-Berwald

metrics of Scalar flag curvature

In -dimensional Finsler manifold we

characterize the two important class of Weakly-

Berwald metrics of Scalar flag curvature. So

first we have to prove the following lemma.

Lemma 3.1. Let be an -dimensional

Finsler manifold Suppose that

metrics and

(where are

of non-Randers type. Then

Proof; we just give the proof for

for

(where

are constants) is also similar. So we

omit it. By a direct computation, we have

where

–




Assume that . Then A=0. Multiplying

with yields

Hence we have,

–

Clearly, observe that

is not divisible

by Since we’ve by (3.1), which is a

contradiction with . So . By

using this lemma 3.1, now we can prove the

following

Theorem 3.3. Let be an n-

dimensional Finsler manifold . Assume

that metrics and

(where are

constants) are of scalar flag curvature

Then is weak Berwald metric if and

only if is Berwald metric and . In this

case, must be locally Minkowskian.

Proof: By the above lemma and (2.9) we

know that metrics and

(where

are constants) can’t represents the

Riemannian metrics, where a constant

and .

The sufficiency is obvious. We just prove the

necessity. First, we assume that the metric

is weak Berwald. By lemma(3.1), we know

that with

( 3 . 2 )

By theorem (2.2) and (2.13), must be of

isotropic flag curvature Further ,

by schur lemma[13], must be of constant flag

curvature. From (3.2), we can simplify (2.8),

(2.14) and (2.15) as follows

In addition, from (2.9) we obtain

–

Thus (2.16) can be expressed as follows

By lemma(3.1), we’ve

Note that . We have

– (3.3)

In this case,

Then (3.3) becomes

Multiplying this equation with

yields

(3.4)

Note that, the left of (3.4) is divisible by . Hence we can obtain that the flag curvature because

and is not divisible by Substituting

into (3.3), we’ve i.e., is closed. By (3.2), we know that is parallel with respect to .

Then is a Berwald metric, where a

constant. Hence must be locally Minkowskian.




Case II: (where are constants). In this case,

Then (3.3) become

where

Obviously, we have and By

and clearly note that is not divisible by . Then

we obtain . Hence is closed. By (3.2),

we know that is parallel with respect to . Then

is a Berwald metric. From (3.3), we find that

Hence is locally Minkowskian.

References

[1] S. Bacso, X. Cheng and Z. Shen, Curvature Properties of

metrics, Adv. Stud. Pure Math. Soc., Japan (2007).

[2] S.S. Chern and Z. Shen, Riemann-Finsler geometry, World - Scientific Publishing Co., Singapore (2005).

[3] X. Cheng , X. Mo.,and Z. Shen, On the flag curvature of Finsler metrics of scalar curvature, J. London Math. Soc., 68

(2003), 762-780.

[4] X. Cheng and Z. Shen, A Class of Finsler metrics with isotropic S-curvature, Israel J. Math., 169 (2009), 317-340.

[5] X. Cheng, On metrics of Scalar flag curvature with constant S-curvature, J. Acta Math. Sinica, 26(2010), 1701-1709.

[6] X. Cheng, H. Wang and M. Wang, metrics with relatively isotropic mean Landsberg curvature,, J. Publ. Math.

Debrecen , 72 (2008), 475-485.

[7] X. Chun Huan and X. Cheng, On a Class of weak Landsberg

metrics, J. Mathematical Research and Exposition., 29 (2009), 227-236.

[8] Li Benling and Z. Shen, On a Class of weak Landsberg metrics, J. Science China Series., 50 (2004), 573-589.

[9] C. Ningwei, On the S-curvature of metrics, J. Acta Math. Sci., 26 (2006), 1047-1056.

[10] Z. Shen, Finsler metrics with and , Canadian J. Math., 55 (2003), 112-132.

[11] Z. Shen, Differential Geometry of Spray and Finsler spaces,

Kluwer Acad. Dordrecht, (2001). [12] Z. Shen Lectures on Finsler Geometry, World - Scientific

Publishing Co., Singapore (2001).

[13] Z. Shen, Landsberg curvature, S-curvature and Riemann curvature, In A Sample of Finsler Geometry, MSRI Series,

Cambridge University Press, Cambridge (2004). [14] Z. Shen, Volume Comparison and its applications in

Riemann-Finsler Geometry, J. Adv. in Math., 128 (1997),

306-328. [15] Z. Shen and Yildirim G.C, On a Class of Projectively flat

metrics wtih constant flag curvature, Cana- dian J. Math., 60

(2008), 443-456. [16] Z. Shen, Two-dimensional Finsler metrics wtih constant flag

curvature, J. Manuscripta Mathamatica.,109 (2002), 349-366.

[17] A. Tayebi, Tayebeh Tabatabaeifar, and Esmaeil Peyghan , On The Second Approximate Matsumato Metric,, Bull.

Korean Math. Soc. 51 (2014), No. 1, pp. 115-128.

[18] A. Tayebi and M. Rafie Rad, S-curvature of isotropic Berwald metrics,, Sci. China Ser. A 51 (2008), no. 12, 2198-

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