Conformally related Douglas metrics in dimension two are Randers
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Archiv der Mathematik
Conformally related Douglas metrics in dimension two are Randers Vladimir S. Matveev
and Samaneh Saberali
Abstract. We show that two-dimensional conformally related Douglas metrics are Randers. Mathematics Subject Classification. 53B40, 58E10. Keywords. Finsler metric, Geodesics, Affine connection.
1. Introduction. A Finsler metric F is called Douglas if there exists an affine connection Γ = (Γijk ) such that each geodesic of F , after some re-parameterisation, is a geodesic of Γ. We assume without loss of generality that Γ is torsion free. Such Finsler metrics were considered by Douglas in [7,8] and were named Douglas metrics (or metrics of Douglas type) in [2]. Though results of our paper are local, let us note that a partition of unity argument shows that the existence of such a connection locally, in a neighborhood of any point, implies its existence globally. Prominent examples of Douglas metrics are Riemannian metrics (with Γ being the Levi–Civita connection), Berwald metrics (in this case, as Γ, we can take the associated connection), and locally projectively flat metrics (in this case, in the local coordinates such that the geodesics are straight lines, one can take the flat connection Γ ≡ 0). In the present paper, we study the following question: can two conformally related Finsler metrics F and eσ(x) F both be Douglas? We do not require that the connection Γ is the same for both metrics, in fact, by [3], two conformally equivalent metrics can not have the same (unparameterized) geodesics unless the conformal coefficient is constant. Of course two conformally related Riemannian metrics are both Douglas. Another trivial example is as follows: let F be Douglas and σ be a constant. Then eσ F is also Douglas. Let us give a less trivial example:
V.S. Matveev and S. Saberali
Arch. Math.
Example 1.1. = α + β, where Consider the Randers metric F α(x, y) = gij y i y j for a Riemannian metric g and β is a 1-form. Assume in addition that β is closed, locally it is equivalent to the condition that β = df for a function f on the manifold. Then the metric F is Douglas since adding the closed 1-form β does not change the geodesics, so the geodesics of F are (up to a re-parameterisation) geodesics of the Levi–Civita connection of g. Next, for any function γ of one variable, the conformally related metric F˜ = eγ(f (x)) F = eγ(f (x)) α + eγ(f (x)) β is also Douglas. Indeed, since the 1- form eγ(f (x)) β is closed, the geodesics of F˜ are geodesics of eγ(f (x)) α, i.e., geodesics of the Levi–Civita connection of the Riemannian metric e2γ(f (x)) g. It is easy to see (see e.g. [4, Theorem 3.1]) that in the class of Randers metrics the above example is the only possible example (of conformally related Douglas metrics with nonconstant conformal coefficient). Indeed, Douglas metrics are geodesically-reversible, in the sense that for any geodesic, its certain orientation-reversing unparameterisation is also a geodesic. Now, it is known (see e.g. [10, Theorem 1]) that for a geodesically reversible Randers met
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