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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
1.2Departme 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]:
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
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 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
International Journal of Mathematics Trends and Technology (IJMTT) – Volume 44 Number 3 April 2017
ISSN: 2231-5373 http://www.ijmttjournal.org Page 126
(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 follow- ing 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
–
International Journal of Mathematics Trends and Technology (IJMTT) – Volume 44 Number 3 April 2017
ISSN: 2231-5373 http://www.ijmttjournal.org Page 127
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.
International Journal of Mathematics Trends and Technology (IJMTT) – Volume 44 Number 3 April 2017
ISSN: 2231-5373 http://www.ijmttjournal.org Page 128
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.
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