Weakly Douglas Finsler metrics
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Weakly Douglas Finsler metrics M. Atashafrouz1 · B. Najafi1 · A. Tayebi2
© Akadémiai Kiadó, Budapest, Hungary 2020
Abstract In this paper, we define a weaker notion of Douglas metrics, namely weakly Douglas metrics. A Finsler metric satisfies a projectively invariant equation D i jkl = T jkl y i for some tensor T jkl is called a weakly Douglas metric. We show that every Randers manifold of dimension n ≥ 3 is a weakly Douglas metric if and only if it is a Douglas metric. Then, we prove that every Kropina manifold is a weakly Douglas metric if and only if it is a Douglas metric. It turns out that every Kropina surface is a Douglas surface. Keywords Generalized Douglas–Weyl metric · Douglas metric · Berwald metric Mathematics Subject Classification 53B40 · 53C60
1 Introduction Let (M, F) be a Finsler manifold. In local coordinates, a curve c = c(t) is called a geodesic if and only if its coordinates (ci (t)) satisfy the ODE c¨i + 2G i (c) ˙ = 0, where the local functions G i = G i (x, y) are called the spray coefficients. Define B y : Tx M ⊗ Tx M ⊗ Tx M → Tx M by B y (u, v, w) := B i jkl (y)u j v k wl ∂∂x i |x , where B i jkl :=
∂ 3Gi . ∂ y j ∂ y k ∂ yl
The quantity B is called Berwald curvature. Indeed, L. Berwald first discovered that the third order derivatives of spray coefficients give rise to an invariant for Finsler metrics. F is called a Berwald metric if B = 0. As an extension of Berwald metric, Bácsó–Matsumoto introduced the notion of Douglas metrics which are projective invariants in Finsler geometry [3]. For non-zero vector y ∈
B
B. Najafi [email protected] A. Tayebi [email protected]
1
Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
2
Department of Mathematics, University of Qom, Qom, Iran
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M. Atashafrouz et al.
Tx M0 , define D y : Tx M ⊗ Tx M ⊗ Tx M → Tx M by D y (u, v, w) := D i jkl (y)u i v j w k ∂∂x i |x , where ∂3 2 ∂G m i i i G − (1.1) y . D jkl := ∂ y j ∂ y k ∂ yl n + 1 ∂ ym D is called the Douglas curvature. F is called a Douglas metric if D = 0 [4]. There is a weak notion of Douglas metrics. A Finsler metric F on a manifold M is called weakly Douglas metric if its Douglas tensor satisfies the equation D i jkl = S jkl y i ,
(1.2)
where S jkl is a Finslerian tensor on M. Indeed, S jkl = F −2 yi D i jkl . Every Douglas metric is a weakly Douglas metric. Here, we give some non-trivial examples of weakly Douglas metrics. Example 1.1 It is known that the Berwald curvature of every Finslerian surface (M, F) is given by B i jkl = F −1 C jkl y i + λ(h i j h kl + h ik h jl + h il h jk ),
(1.3)
where h i j := F Fy i y j is the angular metric, Ci jk := 1/4[F 2 ] y i y j y k is the Cartan torsion of F and λ = λ(x, y) is a homogeneous functions on T M of degrees − 1 with respect to y. It is easy to see that (1.3) implies that F satisfies (1.2) with S jkl = (F −1 − 2λ)C jkl − h jk (λ.l + λF −1 Fl ).
(1.4)
Thus, every Finslerian surface is a weakly Douglas metric. Remark 1.2 It is easy to see that by contr
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