Abstract
The purpose of this paper is to examine the condition for a Finsler space with a generalized -metric to be projectively flat. In addition, we establish that the Finsler space with generalized -metric is a flat-parallel Minkowski space and derive the condition under which the -change for the aforementioned metric is projective. We also explored the projective nature of -change for various significant Finsler metrics derived from the generalized -metric.
1. Introduction
Let be an n-dimensional Finsler space for the n-dimensional differentiable manifold with a basic function . In 1972, Matsumoto [1] introduced the concept of a -metric, . If is a positively homogeneous function of and of degree one, where is a Riemannian metric and is a one form on , then a Finsler metric is called as -metric . The Randers, Kropina, and Matsumoto metrics are the three most interesting instances of -metrics.
A Finsler space is called projectively flat, or with rectilinear geodesic, if the space is covered by coordinate neighborhoods in which the geodesics can be represented by linear equations of the coordinates. Such a coordinate system is called rectilinear. The condition for a Finsler space to be projectively flat was studied by Berwald [2] in tensorial form and completed by Matsumoto [3]. Hashiguchi and Ichijyo’s paper [4] gives interesting results on the projective flatness of Randers spaces.
We have two important projective invariant tensors: the Weyl tensor and the Douglas tensor . A Finsler space with both of these tensors vanishes as a projectively flat space that can be projectively mapped to a locally Minkowski space. Randers spaces, Kropina spaces, and generalized Kropina spaces with are examples of Finsler spaces with -metric . Matsumoto [5] demonstrated the criteria for the above spaces to be projectively flat. The concept of projectively flat Finsler space with the -metric has been studied by many authors [6–10].
Furthermore, Matsumoto [11, 12] defined a -change and flat-parallel Minkowski space with -metrics in 1988 and 1991, respectively. He dealt flat parallelness of Randers metric, Kropina metric, and their generalized form in his paper [13]. The flat parallelness of the Matsumoto metric was studied by Aikou et al. [14].
In the present paper, we studied the generalized -metric [15]. The purpose of this research work is to study the condition for a Finsler space with the generalized -metric to be projectively flat using Matsumoto’s results. Furthermore, we proved that Finsler space with generalized -metric is a flat-parallel Minkowski space and obtained the condition for a projective change . We also investigated how -change is projective for various significant Finsler metrics derived from the generalized -metric.
We used terminology and notations from Matsumoto’s monograph [16] throughout this paper.
2. Preliminaries
Let represent an n-dimensional Finsler space and represent the corresponding Riemannian space, where . Let be the Christoffel symbols with respect to and the covariant differentiation with respect to be indicated by (;). From the differential 1-form , we define
Then, we consider the Berwald connection of the Finsler space with the -metric . As is well known, we have
Then, the previous paper [12, 17] gives the equation to find the difference :
We consider a locally Minkowski space , that is, admits a covering by coordinate neighborhoods in each of which the fundamental function is a function of alone. We denote by a Riemannian curvature tensor with respect to .
Definition 1 (see [13]). A locally Minkowski space with -metric is called flat parallel, if is locally flat and is parallel with respect to .
Theorem 2 (see [12]). A is a locally Minkowski if and only if are functions of alone, and of the Riemannian is written aswhere denotes the terms obtained from the preceding terms by interchanging indices and and is Riemannian .
In this paper, we refer to the contraction of with 0 subscript, and the partial differentiation by and of with and subscripts.
According to Theorem 1 of [5], a Finsler space with an -metric is projectively flat if and only if the space is covered by coordinate neighborhoods on which satisfieswhere is given by
By the homogeneity of , we know that ; therefore, (6) can be rewritten as
If , then we can eliminate in (5) and it can be written in the following form:
Thus, we have [18].
Theorem 3. If , then a Finsler space with an -metric is projectively flat if and only if (8) is satisfied.
Remark 4. According to [19], if contains as a factor, then and the dimension is equal to two. Throughout this paper, we assume that the dimension is more than two and , i.e., .
3. Projectively Flat Finsler Space with Generalized -Metric
Let be a Finsler space with the generalized -metric given by
In this section, we shall find the conditions for with the generalized metric (9) to be projectively flat. The partial derivative with respect to and of (9) are given by
If , then we have which leads to a contradiction. Thus, we can apply Theorem 3.
Substituting (10) into (8), we have
The above equation is rewritten as a polynomial of sixth degree in as follows:where
Since and are rational and is irrational in , we have
The term which does not contain in (14) is . Thus, there exists a homogeneous polynomial of degree six in such that
Since , we must have a function satisfying
Contracting (17) by , we havei.e., . Furthermore, contracting this equation by , we have . Substituting this equation into (18), we have . Thus, from (17), we getwhich gives . Contracting this equation by , we have by virtue of . Thus, we get . Hence, from (19), we have , provided that .
On the other hand, from (15), we have 1-form such that
Upon replacing , , and (20) into (11), we haveby virtue of . Then, (21) is written in the form , where
Since and are rational and is irrational in , we have and .
First, it follows from thati.e.,which shows that the associated Riemannian space is projectively flat.
Then, from and (23), we have
Contracting (25) by , we have , from which , provided , i.e., . From and , we have .
On the other hand, it is easily verified that (11) is a consequence of (23) and . Consequently, we have the following.
Theorem 5. A Finsler space with the generalized -metric (9) is projectively flat if and only if the associated Riemannian space is projectively flat and .
4. Flat-Parallel Minkowski Space with Generalized -Metric
In [20], Kim and Choi defined a procedure to show that -metrics are flat-parallel Minkowski space.
Substituting and in (3), we havewhere the index zero means, as usual, contraction by . It is remarked that for a locally Minkowski space, and are polynomials in of degree 2 and 1, respectively. If (26) gives necessarily, then we have and , and (4) shows that Consequently, the Finsler space with -metrics defines a flat-parallel Minkowski space.
In this section, we shall apply the above procedure to the generalized -metric (9).
Substituting (10) into (26), we have
Since is irrational in , (27) leads us to
Thus, from (28), we have . Hence, we conclude the following.
Theorem 6. A Finsler space with the generalized -metric (9) is a flat-parallel Minkowski space.
5. -Change with Generalized -Metric
Let and be two Finsler spaces on the same underlying manifold . If any geodesic of is a geodesic of and vice versa, then is called projective to and change of the metric is called projective. It is well known that is projective if and only if there exists a positively homogeneous function of degree 1 in satisfying .
On the other hand, we shall introduce -change [13] as follows.
Definition 7. Let be an -metric. The change of the metric is called -change.
If we denote by the associated Riemannian space with a Finsler space with -metric, then the -change is the change from to . There is a theorem between projective change and -change as follows:
Theorem 8 (see [13]). A -change is projective, if and only if we havewhere , and .
Now, we consider a change . Then, from Theorem 8, we can obtain the condition for a change to be projective.
Substituting (10) into (29), we have
Since is an irrational polynomial of , (30) leads us to . Substituting this into (30), by virtue of , we have . Further, contracting this by and using and , we have . Conversely, if , then it satisfies (30). Thus, we have the following.
Theorem 9. A change is projective if and only if we have .
From the above theorem and (30), we have discussed three cases as follows: (i) , (ii) , and (iii) . Case (i): Let in (30), then we have . Since , we have . From , we have . Conversely, if , it satisfies . Thus, we have the following.
Corollary 10 (see [13]). A Randers change is projective if and only if we have .
Case (ii): Let in (30), then we have Since is irrational in , we have . Substituting in (31), we have . Further, contracting this by and using and , we have . Conversely, if , it satisfies (31). Thus, we have the following.
Corollary 11. A Berwald change is projective, if and only if we have .
Case (iii):
Let in (30), then we have
Since is irrational in , we have . Substituting in (32), we have . Further, contracting this by and using and , we have . Conversely, if , it satisfies (32). Thus, we have the following.
Corollary 12. A square root metric change is projective, if and only if we have .
On the other hand, in Theorem 5, we dealt with the condition that a Finsler space with the generalized -metric be projectively flat. By combining Theorems 5 and 9, we can give a more geometrical meaning, similar to Matsumoto’s Theorem ([5], Theorem 2). Thus, we have the following.
Corollary 13. A Finsler space with the generalized -metric (9) is projectively flat if and only if a change is projective and the associated Riemannian space with metric is projectively flat.
6. Conclusion
The infinitesimal transformations in Finsler geometry, such as conformal, projective, semiprojective, and -changes, play an important role not only in differential geometry but also in application to other branches of science, especially in the process of geometrization of physical theories. In this paper, we have obtained results concerning the projective flatness and flat parallelness of the generalized -metric (9). Further, we have shown that the -change of the aforementioned metric (9) is projective. Also, we have discussed how -change is projective for some important Finsler metrics arising from the generalized -metric (9).
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.