On Projectively Flat Finsler Warped Product Metrics of Constant Flag Curvature

In this paper, we study locally projectively flat Finsler metrics of constant flag curvature. We find equations that characterize these metrics by warped product. Using the obtained equations, we manufacture new locally projectively flat Finsler warped product metrics of vanishing flag curvature. These metrics contain the metric introduced by Berwald and the spherically symmetric metric given by Mo-Zhu.


Introduction
In Finsler geometry, the flag curvature is analog of sectional curvature in Riemannian geometry. Furthermore, Finsler metrics of constant flag curvature are the natural extension of Riemannian metrics of constant sectional curvature. Beltrami's theorem tells us that a Riemannian metric is of constant sectional curvature if and only if it is locally projectively flat. However, the situation is much more complicated for Finsler metrics. In fact, there are lots of projectively flat Finsler metrics which are not of constant flag curvature [11]. Conversely, there are infinitely many non-locally projectively flat Finsler metrics with constant flag curvature [1]. An interesting problem then is to X. Mo  study locally projectively flat Finsler metrics of constant flag curvature. Recall that a Finsler metric F on a manifold M is said to be locally projectively flat if at any point there is a local coordinate system in which the geodesics are straight lines as point sets. Projectively flat Finsler metrics on a convex domain in R n are regular solutions to Hilbert's fourth problem [5]. In this paper, we will study locally projectively flat Finsler warped product metrics of constant flag curvature.
Finsler metrics in the form F =αφ(r , s) are called warped product metrics wherȇ α is a Riemannian metric (for definition, see Sect. 2). Finsler warped product metrics are the natural extension of Riemannian warped product metrics [3]. In Riemannian geometry, these metrics have mainly been used in the efforts to construct new examples of Riemannian manifolds with prescribed conditions on the curvatures.
Very recently, Chen, Shen, and Zhao have obtained the characterization of Einstein Finsler warped product metrics F =αφ(r , s) by introducing function (see the first equation of (2.4) below) [4]. In this paper, we show that φ and are mutually determined for a Douglas warped product metric, in particular, for a locally projectively flat Finsler warped product metric (see Lemma 2.3 below). Furthermore, the functions φ(r , s) and (r , s) satisfy the same second-order PDE for a locally projectively flat Finsler warped product metric of constant flag curvature. This is indeed an amazing phenomenon. Precisely we have the following: is locally projectively flat with constant flag curvature if and only ifα has constant sectional curvature κ and where f (r ) and g(r ) are differentiable functions which satisfy After noting this interesting fact, we produce infinitely many locally projectively flat Finsler warped product metrics of vanishing flag curvature in Sect. 6. We have the following: where c 0 is a constant and h is any differentiable function satisfying |h| > 1, h r = 0 and c 0 > 0. Then on I ×M the following Finsler warped product metric is locally projectively flat with zero flag curvature, whereα has constant sectional curvature κ = 1.
We have the following two interesting special cases: is the warped product form of the Berwald's metric [2,4]. F Ber is projectively flat with vanishing flag curvature.
is the warped product form of the Mo-Zhu's metric [10,13]. F M Z is locally projectively flat with vanishing flag curvature, but it is not projectively flat. In Sect. 5, we also construct a lot of locally projectively flat Finsler warped product metrics where (K , κ) = (0, 0) (see Proposition 5.1 below). In fact we will show the following result: Any locally projectively flat Finsler warped product metric of zero flag curvature must be of Berwald type or square type (see Sect. 5 below).
For related results of locally projectively flat Finsler metrics of constant flag curvature, we refer the reader to [7,12,15].

Preliminaries
Let M be a manifold and let Let F be a Finsler metric on an n-dimensional manifold M. For a non-zero vector y ∈ T x M, F induces an inner product g y on T x M by Here (x i , y i ) denotes the standard local coordinate system in T M, i.e., y i 's are determined by y = y i ∂ ∂ x i | x . For a two-dimensional plane P ⊂ T x M and a non-zero vector y ∈ T x M, the flag curvature K(y, P) is defined by K(y, P) := g y (u, R y (u)) g y (y, y)g y (u, u) − g y (y, u) 2 where P = y ∧ u and R y is the Riemannian curvature of F [9,15]. A Finsler metric F on a manifold M is said to be of scalar flag curvature if the flag curvature K(y, P) = K(x, y) is a scalar function on the slit tangent bundle T M\{0}. In particular, F is said to be of constant flag curvature if K(y, P) = constant. In general, the flag curvature is a function K(y, P) of tangent planes P ∈ T x M and directions y ∈ P.
Let I be an interval of R andM be an (n − 1)-dimensional manifold equipped with a Riemannian metricα. Finsler metrics on the product manifold M := I ×M, given in the form 1 +v, and φ is a suitable function defined on a domain of R 2 are called Finsler warped product metrics [4].
Let B n (r ) denote the Euclidean ball of radius r and let F be a Finsler metric on B n (r ). F is said to be spherically symmetric if it satisfies F(Ax, Ay) = F(x, y) for all x ∈ B n (r ), y ∈ T x B n (r ), and A ∈ O(n). A Finsler metric F on B n (r ) is spherically symmetric if and only if there is a function φ : [0, r ) × R → R such that F(x, y) = |y|φ |x|, x,y |y| where (x, y) ∈ T B n (r ) \ {0} [6].

Lemma 2.1 [4] A spherically symmetric metric is a Finsler warped product metric.
Proof In fact, whereα + is the standard Euclidean metric on the unit sphere S n−1 , where v 1 = dr(y). [4,9]. It follows that det From (2.5) and (2.6) one obtains It implies that By using (2.7), we have It follows that Plugging these into (2.8) yields Then (2.3) holds.  (2.10) or F is of Berwald type where is given in the first equation of (2.4).
which completes the proof of Lemma 2.3.

Locally Projectively Flat Finsler Metrics
In this section, we are going to find equations that characterize a locally projectively flat Finsler warped product metric According to Theorem 1.1 in [10], F is of scalar flag curvature if and only ifα has constant sectional curvature κ and where and are given in (2.4).
A direct calculation gives the following formula: Combining this with (3.1), we obtain the following: We mention that two characterizing equations of Finsler warped product metrics of scalar curvature were established first by Chen-Shen-Zhao [4]. Later, Liu-Mo-Zhang found that one equation is sufficient, using the Weyl curvature [10]. Very recently, Liu-Mo have constructed infinitely many non-spherically symmetric warped product Finsler metrics of scalar flag curvature by refining Chen-Shen-Zhao and Liu-Mo-Zhang equations characterizing Finsler warped product metrics of scalar flag curvature [9].

Theorem 3.2 LetM be an
where f (r ) and g(r ) are differentiable functions which satisfy (1.3).
where f = f (r ) and g = g(r ) are differentiable functions. Plugging (3.5) into the second equation of (2.4) yields − s = sφ rs −φ r 2φ ss . It follows that (3.6) holds if and only if Combining (3.5) with (3.6) we have Furthermore, we assume that F is of scalar flag curvature. Combining (3.9) with Lemma 3.1, we conclude that (1.3) holds. According to Douglas' result, Finsler metric F(u, v) on M n with n ≥ 3 is locally projectively flat if and only if F has vanishing Douglas curvature and scalar flag curvature.
First suppose that F is locally projectively flat. Then (3.7) and (1.3) hold where κ is the constant sectional curvature of (M,α). Conversely, suppose thatα has constant sectional curvature κ such that (3.4) and (1.3) hold. Equation (3.4) implies (3.7). It follows that F has vanishing Douglas curvature. Furthermore, the constancy of flag curvature ofα, (3.9) and (1.3) tell us that F is of scalar flag curvature. Note that n ≥ 3. Hence, F is locally projectively flat.

Finsler Metrics of Constant Flag Curvature
First we refine Chen-Shen-Zhao and Liu-Mo-Zhang equations that characterize Finsler warped product metrics of constant flag curvature. where and are given in (2.4). In this case, the flag curvature K of F satisfies We mention that the Chen-Shen-Zhao paper was published in 2018, in which the authors obtain two PDEs depending on the constant K that characterize Finsler warped product metrics of constant flag curvature. Theorem 4.1 tells us that these PDEs can be replaced by ones independent of the constant K .
Proof of Theorem 4.1 According to Theorem 1.2 in [10], F is of constant flag curvature if and only ifα has constant sectional curvature κ and where ε and ν are given in Sect. 3, and where is given in (3.2). In this case, the flag curvature of F is given by It follows that the second equation of (4.4) holds if and only if (4.2) holds. By (3.5), (4.7) and (4.6), we obtain The following proposition will be used in Sect. 5.
where κ is the constant sectional curvature ofα.
Proof Differentiating (4.3) with respect to s and using (4.2) one obtains Together with (4.3), we have Thus we complete the proof of the proposition.  Our classification of locally projectively flat Finsler warped product metrics with vanishing flag curvature is given in two steps. First, we are going to study the case whenα satisfies κ = 0 by a deep analyzing. Then we shall study the case when κ = 0. In particular, we are going to manufacture locally projectively flat Finsler warped product metrics with K = 0 (see Proposition 5.1 and the proof of Theorem 1.2 below). Case 1. κ = 0 By using (5.1) we obtain

Projective Finsler Metrics of Zero Flag Curvature
where we have used (3.6). According to Corollary 3.2 in [8], F is of Berwald type.
If = 0 then where we have used (5.2). It follows that According to Lemma 2.3, we denote the corresponding functions with respect to ± by φ ± . By using (2.10), (5.4), and (5.7), we have where we have used the following formula where we have used (5.6). Plugging (5.9) into (5.8) and using (5.5) we obtain It follows that We obtain that F is of square type. By investigating φ + , we have the following:

Proposition 5.1 Let φ(r , s) be a function defined by
where λ is a constant and h is any differentiable function satisfying λh r < 0 and h > 0. Then on M = I ×M the following Finsler metric is locally projectively flat with zero flag curvature, whereα has zero sectional curvature.
Then the warped product Finsler metric given by is a locally projectively flat Finsler metric with zero flag curvature, where r and s satisfy (6.2). Finsler metric (6.3) is the Finsler warped product form of the metric found by Mo and Zhu recently [13]. Mo and Zhu showed that F is not projectively flat.
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