1. Introduction
The curvature properties of metrics play very important roles in Riemannian and Finsler geometry. Riemannian curvature and Ricci curvature are the most important Riemannian geometric quantities in Finsler geometry. In 1988, the concept of Ricci curvature was first proposed by Akbar-Zadeh, and its tensor form can be naturally obtained [
1]. In recent years, many scholars have conducted a great deal of research on them. Cheng-Shen-Tian proved that the polynomial
-metric is an Einstein metric if and only if it is Ricci-flat [
2]. Zhang-Shen gave the expression of Ricci curvature of Kropina metric. Furthermore, they proved that a non-Riemannian Kropina metric with a constant Killing form
is an Einstein metric if and only if
is also an Einstein metric [
3]. By using navigation date
, they proved that
n (≥2)-dimensional Kropina metric is an Einstein metric if and only if Riemann metric
h is an Einstein metric and
W is a Killing vector field with respect to
h. Xia gave the expression for the Riemannian curvature of Kropina metrics and proved that a Kropina metric is an Einstein metric if and only if it has non-negative constant flag curvature [
4]. Cheng-Ma-Shen studied and characterized projective Ricci-flat Kropina metrics and obtained its equivalent characterization Equation [
5].
Unlike the notion of Riemannian curvature, there is no unified definition of scalar curvature in Finsler geometry, although several geometers have offered several versions of the definition of the Ricci curvature tensor [
1,
6,
7,
8]. In 2015, Li–Shen introduced a new definition of the Ricci curvature tensor [
6]. This tensor is symmetric. They proved that a Finsler metric
F has isotropic Ricci curvature tensor if and only if it has isotropic Ricci curvature and
-curvature tensor satisfies
, where
. It was further proven that for Randers metrics, they are isotropic Ricci curvature tensors if and only if they are of isotropic Ricci curvature.
In Finsler geometry, there are several versions of the definition of scalar curvature. We used Akbar-Zadeh’s definition [
1] of the scalar curvature, based on Li–Shen’s definition of the (symmetric) Ricci curvature tensor [
6]. For a Finsler metric
F on an
n-dimensional manifold
M, the scalar curvature
R of
F is defined as
. Tayebi studied general fourth-root metrics [
9]. They characterized general fourth-root metrics with isotropic scalar curvature and also for general fourth-root metrics with isotropic scalar curvature under conformal variation. Finally, they characterized Bryant metric with isotropic scalar curvature. Chen–Xia studied a conformally flat
-metric with weakly isotropic scalar curvature [
10]. They proved that if conformally flat polynomial
-metrics have weakly isotropic scalar curvature
R, then
R vanishes.
In this paper, we obtain a characterization of Kropina metrics with isotropic scalar curvature and have the following results.
Theorem 1. Let F be a Kropina metric on an n (≥3)-dimensional manifold M. Then, F is of isotropic scalar curvature if and only ifwhere , g, h, , are expressed by (15), (23), (26), (27), respectively. In this case, scalar curvature is 2. Preliminaries
Let
M be an
n-dimensional
manifold. A Finsler structure of
M is a function
with the following properties:
- (1)
Regularity: F is on the slit tangent bundle ;
- (2)
Positive homogeneity: , ;
- (3)
Strong convexity:
is positive-definite at every point of
.
Let
be an
n-dimensional Finsler manifold. Suppose that
. The geodesics of a Finsler metric
on
M are classified by the following ODEs:
where
. The local functions
are called geodesic coefficients (or spray coefficients). Then, the
S curvature with respect to a volume form
is defined by
For
and
, Riemann curvature
is defined by
The trace of Riemann curvature is called Ricci curvature of F, i.e., .
Riemann curvature tensor is defined by
Let
, and
is called a Ricci curvature tensor. The scalar curvature
R of
F is defined by
Let
be a scalar function on
M,
be a 1-form on
M. If
then it is said that
F is of weak isotropic scalar curvature. Especially when
, i.e.,
, it is said that
F is of isotropic scalar curvature.
Let
F be a Finsler metric on
M. If
, where
is a Riemannian metric,
is a 1-form; then,
F is a Kropina metric. Its fundamental tensor
is given by [
4]
where
. Moreover,
where
,
.
Let
denote the covariant derivative of
with respect to
. Set
The Ricci curvature of Kropina metrics is given by the following.
Lemma 1 ([
3]).
Let F be a Kropina metric on M. Then, the Ricci curvature of F is given bywhere is the Ricci curvature of α, and Lemma 2 ([
4]).
Let F be a Kropina metric on n-dimensional M. Then, the followings are equivalent:- (i)
F has an isotropic S curvature, i.e., ;
- (ii)
;
- (iii)
;
- (iv)
β is a conformal form with respect to α,
where and are functions on M.
3. Ricci Curvature Tensor and Scalar Curvature Tensor of Kropina Metrics
By the definition of Ricci curvature tensor and Lemma 1, we obtain the Ricci curvature tensor of Kropina metrics.
Proposition 1. Let F be a Kropina metric on an n-dimensional manifold M. Then, the Ricci curvature tensor of F is given bywhere denotes the Ricci curvature tensor of α. Contracting the Ricci curvature tensor with , we can obtain the expression of the scalar curvature R of Kropina metrics as following.
Proposition 2. Let F be a Kropina metric on an n-dimensional manifold M. Then, the scalar curvature of F is given bywhere denotes the scalar curvature of α. 4. The Proof of Main Theorem
In this section, we will prove Theorem 1.
Proof. “Necessity”. Assume Kropina metric
F is of isotropic scalar curvature, i.e.,
. Substituting (
2) into
yields
where
By (
3), we have that
divides
. Thus, there exists a scalar function
such that
, which is the second formula of (
1). Thus, we deduce that
where
,
,
.
Substituting the above equations into (
3) yields
where
By (
4), we have that
divides
, i.e., there exists a scalar function
such that
Differentiating the above equation with respect to
yields
Contracting this formula with
or
yields, respectively,
Combining the above two formulas, we obtain
and
Substituting (
5)–(
7) into (
4), we obtain
where
By (
8), we have that
divides
. Then, there exists a scalar function
such that
which is the first formula of (
1). Differentiating the above equation with respect to
or
, respectively, we obtain
Contracting (
10) with
yields
Contracting (
11) with
or
, respectively, we obtain
Comparing (
13) and (
14) yields
and
Substituting (
15) and (
16) into (
12) yields
Combining (
5), (
6), (
9), and (
15), we obtain the third formula of (
1).
Substituting (
9), (
15)–(
17) into (
8) yields
where
By (
18), we have that
divides
. Then, there exists a scalar function
such that
which is the fourth formula of (
1). Differentiating the above equation with respect to
yields
Contracting (
20) with
or
, respectively, we have
Comparing (
21) and (
22) yields
By (
7) and (
24), we obtain
Therefore, by (
16), we have
Substituting (
26) into (
17) yields
Substituting (
23) into (
18) yields
“Sufficiency”. It is obviously true.
This completes the proof of Theorem 1. □
5. Other Related Results
In this section, we consider in Theorem 1.
Corollary 1. Let F be a Kropina metric on an n-dimensional manifold M. Assume . Then F is of isotropic scalar curvature if and only if In this case, .
Proof. Sufficiency is obviously true. Next we prove necessity. Assume that
F is of isotropic scalar curvature, i.e.,
. By Theorem 1, obviously,
, (
5), (
19), (
26) and (
27) are true. When
, (
26) and (
27) can be simplified as
Substituting
, (
29) and (
30) into (
5), we obtain
where
. This is the first formula of (
28).
Substituting
and (
29) into (
19), we obtain
where
. This is the third formula of (
28).
By Theorem 1, in this case, . □
Corollary 2. Let F be a Kropina metric on an n-dimensional compact manifold M. Then F is of isotropic scalar curvature if and only if In this case, .
Proof. Sufficiency is obviously true. Next we prove necessity. Assume that
F is of isotropic scalar curvature, i.e.,
. By Theorem 1, (
24) and (
26) are true. Substituting (
26) into (
24), we obtain
Using the divergence theorem, when M is a compact manifold, . By Corollary 1, Corollary 2 is true. □
Based on Lemma 2 and Theorem 1, we obtain the following result.
Theorem 2. Let a Kropina metric F be of isotropic scalar curvature. Then, F is of isotropic S curvature if and only if S = 0.
Proof. Assume that F is of isotropic scalar curvature. By Theorem 1, we know that . By Lemma 2, the result is obviously true. □
Lemma 3 ([
6]).
For a Finsler metric or a spray on a manifold M, if and only if . Remark 1. Li–Shen defined with the S curvature in [11], where Based on Theorem 2, we know that for a Kropina metric with isotropic scalar curvature vanishes, i.e., . This means that . 6. Conclusions
In this paper, we study the Kropina metric with isotropic scalar curvature. Firstly, we obtain the expressions of Ricci curvature tensor and scalar curvature. Based on these, we characterize Kropina metrics with isotropic scalar curvature by tensor analysis in Theorem 1. In Corollary 2, we discuss the case of a compact manifold. Kropina metrics with isotropic scalar curvature deserve further study by the navigation method.
Author Contributions
Conceptualization, X.Z.; Validation, L.L.; Formal analysis, L.L. and X.Z.; Investigation, L.L.; Writing—original draft, L.L.; Writing—review & editing, X.Z. and L.Z.; Project administration, X.Z.; Funding acquisition, X.Z. All authors have read and agreed to the published version of the manuscript.
Funding
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11961061, 11461064, and 12071283).
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
No data were used to support this work.
Acknowledgments
The authors are very grateful to anonymous reviewers for careful reading of the submitted manuscript and also for providing several important comments and suggestions.
Conflicts of Interest
The authors declare no conflict of interest.
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