1. Introduction
Over the past several years, rapid development in Finsler geometry has been observed. A significant class of Finsler metrics, referred to as
-metrics, has gained attention for its special characteristics. These metrics are expressed in the form
, where
denotes a Riemannian metric,
represents a 1-form on
M, and
is a smooth positive function defined on a specific open interval and
. In particular, the Finsler metric
is called a Kropina metric when
. Kropina metrics were initially innovated by Berwald in relation to a two-dimensional Finsler space with rectilinear extremal, and later studied by Kropina [
1]. Recently, geometers discovered important geometric properties of Kropina metrics, which have diverse and significant applications [
2,
3,
4,
5,
6,
7,
8].
Consider an object moving in a metric space, such as Euclidean space, driven by an interval force and an external force field. The shortest time problem aims to determine a curve from one point to another in the space, along which it takes the least time for the object to travel. This is called the Zermelo navigation problem [
9]. Later, Randers spaces were viewed from a new perspective by Shen [
10]. He identified these metrics with the solution of Zermelo’s navigation problem on some Riemannian spaces and described a Randers metric by a new Riemannian metric
h and a vector field
W with
. Based on this, in 2004, Bao, Robles, and Shen [
11] established the necessary and sufficient conditions for a Randers metric to be of constant flag curvature. Additionally, they obtained the classification of a Randers metric with constant flag curvature using the navigation method. Kropina metrics can be easily treated as the limit of the navigation problem for Randers metrics as
[
2,
4,
12,
13]. Zhang and Shen [
4] obtained the expression of Ricci curvature for Kropina metrics and certified that a non-Riemannian Kropina metric, which has a constant Killing 1-form
, is an Einstein metric in the same way that
is an Einstein metric. More generally, Xia [
5] classified Kropina metrics with weakly isotropic flag curvature via the navigation data. Based on these results, Yoshikawa and Sabau [
6] obtained the classification theorem for Kropina metrics with constant flag curvature via the navigation data. Cheng, Li, and Yin [
7] characterized conformal vector fields on Kropina manifolds by the navigation data, and fully ascertained conformal vector fields on Kropina manifolds which have weakly isotropic flag curvature.
Ricci curvature in Finsler geometry is the natural generalization of that in Riemannian geometry. Nevertheless, there is no unified definition of Ricci curvature tensor in Finsler geometry. Therefore, several different versions of definitions of scalar curvature can be found in Finsler geometry. Here, we take the notion of scalar curvature introduced by Akbar-Zadeh [
14]. The scalar curvature
R of a Finsler metric
F is defined as
where
. Tayebi [
15] studied general fourth-root manifolds with isotropic scalar curvature. Also, he characterized Bryant metrics with isotropic scalar curvature. Meanwhile, Chen and Xia [
16] explored an
-metric that is conformally flat and of weakly isotropic scalar curvature. They proved that its scalar curvature must vanish. Cheng and Gong [
17] obtained that a Randers metric with weakly isotropic scalar curvature must have isotropic
S-curvature. They claimed that a conformally flat Randers metric, which is of weakly isotropic scalar curvature, must be either Minkowskian or Riemannian. Recently, in Zhu and Song’s manuscript [
18], they proved that a Kropina metric is of weakly isotropic scalar curvature if and only if it is an Einstein metric. Further, they gave a negative answer to the Yamabe problem on Kropina metrics with isotropic
S-curvature.
Moreover, Li and Shen [
19] introduced a new notion of Ricci curvature tensor
Sevim, Shen, and Ulgen [
20] discussed several Ricci curvature tensors and their relationship with the Ricci curvature to provide a better understanding of non-Riemannian quantities. Liu, Zhang, and Zhao [
8] obtained expressions of Ricci curvature tensor
and scalar curvature
of Kropina metrics. And they characterized Kropina metrics with isotropic scalar curvature
.
In this paper, we mainly focus on the scalar curvature introduced by Akbar-Zadeh [
14], study Kropina metrics with isotropic scalar curvature via the navigation data, and obtain the following results.
Theorem 1. Let be a Kropina metric on M. Then F is of isotropic scalar curvature if and only if F is an Einstein metric.
Although our conclusion is the special case of Zhu and Song’s [
18], the methods of the proof are different. Here, we use the navigation method, which can simplify the proof process, instead of divisibility analysis.
The flag curvature in Finsler geometry generalizes the sectional curvature in Riemannian geometry. If the flag curvature
, then a Finsler metric
F is said to be of constant flag curvature. Xia [
5] classified Kropina metrics with constant flag curvature in three dimensions as follows.
Lemma 1 ([
5])
. (Three-dimensional rigidity.) Let F be a Kropina metric in three dimensions. Then, F is an Einstein metric if and only if it has non-negative constant flag curvature. In virtue of Lemma 1 and Theorem 1, we directly obtain the following result.
Theorem 2. (Three-dimensional rigidity). Let F be a Kropina metric in three dimensions. Then, F is of isotropic scalar curvature if and only if it has a non-negative constant flag curvature.
2. Preliminaries
For the sake of simplicity, we always set the dimension of Kropina metrics as in the following unless otherwise specified.
Let
M be an
-dimensional smooth manifold. A Finsler structure of
M is a function
with the following properties:
(1) Regularity: F is smooth on the entire slit tangent bundle ;
(2) Positive homogeneity: ;
(3) Strong convexity:
is positive-definite at every point of
. A smooth manifold
M endowed with a Finsler structure
F is called a Finsler manifold, which is denoted by
.
Let
be a Finsler manifold. The geodesics of a Finsler metric
F on
M are classified by the following ODEs:
where
. We call the local functions
geodesic coefficients (or spray coefficients).
For
and
, Riemann curvature
is defined by
The Ricci curvature of F is the trace of the Riemann curvature, i.e., .
The Hessian of the Ricci curvature also gives rise to a Ricci curvature tensor
Then, one can define the notion of scalar curvature
If
, where
is a scalar function and
is a 1-form on
M, then it is said that
F has weak isotropic scalar curvature. In particular, when
, i.e.,
it is said that
F is of isotropic scalar curvature.
Let
be a Kropina metric. Its fundamental tensor
is given by [
4]
where
. Moreover,
where
,
and
.
Let
denote the covariant derivative of
with respect to
. Set
The Ricci curvature of Kropina metrics is presented as follows.
Lemma 2 ([
4])
. Let F be a Kropina metric on M. Then, the Ricci curvature of F is given bywhere is the Ricci curvature of α, and A Kropina metric can also be characterized by a Riemannian metric
and a 1-form
with
. Between
and a pair
, there exists a one-to-one correspondence with
where
and
(see [
5]). We call this pair
the navigation data of a Kropina metric.
Note 1 ([
4])
. Let F be an -metric. Then, if and only if . Note 2 ([
4])
. For a 1-form on M, we say that is a conformal 1-form with respect to if it satisfies , where is a function on M. If , we say that is a Killing 1-form. Furthermore, is said to be a constant Killing 1-form if it is a Killing 1-form and has constant length, which is equivalent to . Note 3 ([
4])
. Let be a Kropina metric given by (2) with the navigation data . Thus, is a conformal 1-form with respect to if and only if is a Killing 1-form with respect to . 3. Scalar Curvature of Kropina Metrics
In this section, we shall present an expression of the scalar curvature of Kropina metrics. Then, we will obtain the necessary condition for the Kropina metric with isotropic scalar curvature.
Firstly, using the notion of Ricci curvature tensor and Lemma 2, we can derive the expression of Ricci curvature tensor for Kropina metrics.
Proposition 1. Let be a Kropina metric on M. Then, the Ricci curvature tensor of F is given bywhere denotes the Ricci curvature tensor of α. Proof. From Lemma 2, one can obtain the proposition by a direct computation. □
Using to contract the Ricci curvature tensor, we can directly obtain the expression of the scalar curvature R for Kropina metrics as follows.
Proposition 2. Let be a Kropina metric on M. Then, the scalar curvature of F is given bywhere denotes the scalar curvature of α. This is also calculated by Zhu and Song [
18], and there is a detailed calculation process.
Lemma 3. Let be a Kropina metric on M. Assume F is of isotropic scalar curvature. Then, β is a conformal 1-form with respect to α.
Proof. Assume the Kropina metric
F has isotropic scalar curvature. Then,
holds for some scalar function
. Plugging (
3) into it, we have
where
By (
4),
can be divided by
. Hence,
holds for some scalar function
. It means that
β is a conformal 1-form with respect to
α. This completes the proof of Lemma 3. □
4. Isotropic Scalar Curvature via the Navigation Data
In this section, we study Kropina metrics by the navigation data. We obtain an equivalent characterization for a Kropina metric with isotropic scalar curvature.
Let
F and
be two Finsler metrics on
M. If
holds for some smooth function
on
M, then
F and
are said to be locally conformally related. And we call the smooth function
the conformal factor.
For conformally related Finsler metrics, Bácsó and Cheng [
21] gave some transformation conclusions. Here are some related results.
Proposition 3. Let be two Finsler metrics on M. Then
(1) , , , and ;
(2) F is an Einstein metric in the same way that is also an Einstein metric;
(3) F is of isotropic scalar curvature if and only if is of isotropic scalar curvature.
Lemma 4 ([
12]).
Let F be an -metric on M. Assume that α is an Einstein metric and β is a conformal 1-form with respect to α, i.e., , for some scalar functions and . Then the following hold:where and . To prove the main theorem, we shall express the Kropina metric
using the navigation data
as Formula (
2). Let
. Then,
.
Theorem 3. Let be a Kropina metric on M. Assume is the navigation data of the Kropina metric . Then, is of isotropic scalar curvature if and only if is an Einstein metric and is a constant Killing 1-form with respect to .
Proof. Necessity. Assume
has isotropic scalar curvature. Then,
holds for some scalar function
. By Proposition 3,
must be also of isotropic scalar curvature. Thus,
β is a conformal 1-form with respect to
α by Lemma 3. Furthermore, based on Note 1, Note 2, and Note 3,
is a constant Killing 1-form, namely,
and
. Substituting
,
into (
3), we obtain
where “‖” denotes the covariant derivative with respect to the Levi-Civita connection of
. Multiplying both sides of (
5) by
yields
The above equation shows that
can divide
. Thus, there exists a scalar function
such that
which means that
is an Einstein metric. Thus,
and
. Since
and
hold, we have
and
by Lemma 4. Substituting all of these into (
6) yields
It implies that .
Sufficiency. Suppose that
and
hold. Then we have
and
by Lemma 4. Plugging them into (
3) yields
, which means that
is of isotropic scalar curvature. It completes the proof of Theorem 3. □
The following lemma is necessary for the proof of the main theorem.
Lemma 5 ([
4]).
Let be a non-Riemannian Kropina metric with constant Killing 1-form β on M. Then, F is an Einstein metric if and only if α is also an Einstein metric. In this case, , where and are Einstein scalars of F and α, respectively. Moreover, F is Ricci-constant when . Now we are in the position to give a proof of Theorem 1.
Proof of Theorem 1. Let
be a Kropina metric on a manifold
M given by (
2) with the navigation data
. We claim that
is of isotropic scalar curvature if and only if
is an Einstein metric. Let
be the navigation data of the Kropina metric
. Suppose
is of isotropic scalar curvature. Then, we have that
is an Einstein metric and
is a constant Killing 1-form by Theorem 3. Hence,
is an Einstein metric, as stated in Lemma 5. Conversely, if a Finsler metric
is an Einstein metric, then
is of isotropic scalar curvature by the definition of scalar curvature.
Since , we conclude that F is of isotropic scalar curvature in the same way that F is an Einstein metric by Proposition 3. This completes the proof of Theorem 1. □
Let
be a Finsler manifold. Express the volume form of
F by
. For a non-zero vector
, the
S-curvature
is defined by
And the non-Riemannian quantity
-curvature on the tangent bundle
is defined by
where “;” and “.” denote the horizontal and vertical covariant derivatives with respect to the Chern connection, respectively. Further,
H-curvature can be expressed by
The notion of projective Ricci curvature
is given by
Lemma 6 ([
5]).
For a Kropina metric F on M, the following are equivalent: (1) F has isotropic S-curvature; (2) S-curvature vanishes; and (3) β is a conformal 1-form with respect to α. Proposition 4. Let F be a Kropina metric on M. If F is of isotropic scalar curvature, then (1) S-curvature vanishes; (2) χ-curvature vanishes; (3) H-curvature vanishes; and (4) .
Proof. Assume a Kropina metric is of isotropic scalar curvature. We have that β is a conformal 1-form with respect to α by Lemma 3. Thus, by Lemma 6. By definitions of χ-curvature, H-curvature, and projective Ricci curvature, we have , , and . This completes the proof. □