1. Introduction
The concept of projective change between two Finsler spaces has been studied by many geometers [1] - [6]. An interesting result concerned with the theory of projective change was given by Rapscak [7]. He proved necessary and sufficient conditions for projective change. S. Bacso and M. Matsumoto [8] discussed the projective change between Finsler spaces with -metric. H. S. Park and Y. Lee have studied on projective changes between a Finsler space with - metric and the associated Riemannian metric.
In Riemannian geometry, two Riemannian metrics and on a manifold M are projectively related if and only if their spray coefficients have the relation , where is a scalar function on M and . In Finsler geometry, two Finsler metrics F and on a manifold M are called projectively related if , where and are the geodesic coefficients of F and , respectively and is a scalar function on the slit tangent bundle .
In [9], we introduced the generalized -metric
(1.1)
where is a Riemannian metric, is a 1-form.
We know from [4], that two Finsler metrics F and are projectively related if and only if their spray coefficients have the following relation:
(1.2)
where is a scalar function on and homogeneous of degree one in y.
Also, from [1] we know that a Finsler metric is called a projectively flat metric if it is projectively related to a Minkowskian metric. From [4], we know that the Randers metric is projectively flat if and only if is projectively flat and is closed.
The purpose of the present paper is to continue the study on the generalized -metric and to investigate the locally projective flatness. Also, the projective change between between generalized -metric and Randers metric , where and are
two Riemannian metrics, and are 1-forms. Further, we characterized such projective change. Precisely, we have the following
Theorem 1.1. Let and , be two -
metrics, where and are two Riemannian metrics; and are 1- forms. Then F is projectively related to , if and only if the following equations, holds
where and are the coefficients of the covariant derivative of with respect to ; is a scalar function and is a 1-form on M.
Corollary 1.1. Let and , be two -
metrics, where and are two Riemannian metrics; and are 1- forms. Then F is projectively flat if the following relation holds:
(1.3)
where and are the coefficients of the covariant derivative of with respect to ; is a scalar function and is a 1-form on M.
Theorem 1.2. Let the -metric an
n-dimensional manifold M, with is a Riemannian metric; is a 1-form. Then F is locally projectively flat if and only if
(1.4)
Finally, we have shown that the generalized -metric satisfy the sign property.
2. Preliminaries
Definition 2.1. [1] Let
(2.1)
where are the spray coefficients of F. The tensor is called the Douglas tensor. If Douglas tensor vanishes then Finsler metric is called Douglas metric.
Some interesting results concerning Douglas metrics are recently obtained in [10] & [11].
The function is a positive function on an open interval and it satisfies the following condition:
(2.2)
Also, F is a Finsler metric if and only if for any .
In general, the -metrics are defined as follows:
Definition 2.2. [1] For a given Riemannian metric and one form , satisfying for , then: , , is called -metric.
The covariant derivative of with respect to is . Also, in [1], the following notations are given:
(2.3)
It is clear that if and only if is closed. Also, we can take:
If we consider the fundamental tensor of Randers space , then we have the following formulae
The geodesic coefficients of F and the geodesic coefficients of , are related as follows (see [1]):
(2.4)
where
(2.5)
In [2] and [4], the condition for an -metric to be locally projectively flat is presented as follows:
Lemma 2.1. A Finsler space is locally projectively flat if and only if
(2.6)
In [12], we have the following condition for an -metric to be a Douglas metric
(2.7)
where and .
Theorem 2.3. [12] Let be an -metric on an open
subset , where and one form . Let . Suppose that the following conditions holds
a) is not parallel with respect to ;
b) F is not of Randers type;
c) everywhere or on U. Then F is a Douglas metric on U if and only if the function satisfies the following ODE
(2.8)
and the covariant derivative of with respect to satisfies the following equation
(2.9)
where is a scalar function on U and are constants with .
Remark: The above equation holds good in dimension .
3. Main Results
By the Theorem 2.1, we compute the coefficients for ,
taking into account that , where , using Equation (2.9), we get
(3.1)
Next, we obtain
(3.2)
Make use of (2.5) for , we get
(3.3)
Plugging (3.3) in (2.4), we get
(3.4)
where is given in (3.2).
Now, we can formulate the first result:
Remark. The -metric is a Douglas metric with respect to Theorem 2.1, if and only if (3.1) is of the form
for some scalar function , where represents the coefficients of the covariant derivative with respect to . In this case is closed.
If is closed, then and .
Replace (3.2) in (3.4), we get:
(3.5)
We consider a scalar function on , i.e.,
(3.6)
From (3.5) and (3.6), we get
(3.7)
Since RHS of above equation is in quadratic form, thus there must be a 1-form , such that
Then, we get
(3.8)
Using (3.1) and (3.8) and also the above remark, we can conclude the following result
Theorem 3.4. Let and , be two -
metrics, where and are two Riemannian metrics; and are 1- forms. Then F is projectively related to , if and only if the following equations, holds
where and are the coefficients of the covariant derivative of with respect to ; is a scalar function and is a 1-form on M.
The proof is obtained using (3.1) and (3.8). Also, we can now formulate the following corollary:
Corollary 3.2. Let and , be two -
metrics, where and are two Riemannian metrics; and are 1- forms. Then F is projectively flat if the following relation holds:
(3.9)
where and are the coefficients of the covariant derivative of with respect to ; is a scalar function and is a 1-form on M.
Theorem 3.5. Let the -metric an
n-dimensional manifold M, with is a Riemannian metric; is a 1-form. Then F is locally projectively flat if and only if
(3.10)
Proof: We apply lemma 1.1, using
First, we compute
(3.11)
Then, we obtain
(3.12)
From (3.11), replacing k and i and substituting , we get
(3.13)
Finally, substituting (3.12) and (3.13) in (2.6), we obtain
(3.14)
Thus
This completes the proof of necessity. The converse part follow easily.
Theorem 3.6. Let the -metric given by (1.1), be locally projectively flat. Assume that is locally projectively flat. Then
(3.15)
where
Since is locally projectively flat and from (2.6), we get
(3.16)
From (3.10) and (3.16), we get
(3.17)
Use definitions of P and Q and dividing with in (3.17), we get
Hence the proof.
From [13], we have the following:
Definition 3.3. We say that an -metric on a manifold M, satisfy the sign property, if the function
has a fix sign on a symmetric interval . Here, with s is denoted .
Let us consider the metric (1.1), , with .
In this case, we have:
We conclude that, for , has a fix sign.
Thus metric (1.1) satisfy the sign property.
4. Conclusion
In this paper, we have obtained some important results concerning the projective change and locally projective flatness of the generalized -metric
(
, and
are constants). Further, we have
shown that the generalized -metric satisfy the sign property.
Acknowledgements
The authors express their sincere thanks to the reviewer for his valuable comments that greatly improved the manuscript.
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