The geometry of a Randers rotational surface with an arbitrary direction wind

In the present paper we study the global behaviour of geodesics on a Randers metric, defined on a topological cylinder, obtained as the solution of the Zermelo's navigation problem. Our wind is not necessarily a Killing field. In special we concentrate our study on the geodesics equation on the base manifold, the conjugate and cut loci.


Introduction
A Finsler structure on a surface M can be regarded as a smooth 3-manifold Σ ⊂ T M for which the canonical projection π : Σ → M is a surjective submersion and having the property that for each x ∈ M, the π-fiber Σ x = π −1 (x) is a strictly convex curve including the origin O x ∈ T x M. Here we denote by T M the tangent bundle of M. This is actually equivalent to saying that such a geometrical structure (M, F ) is a surface M endowed with a Minkowski norm in each tangent space T x M that varies smoothly with the base point x ∈ M all over the manifold. Obviously Σ is the unit sphere bundle {(x, y) ∈ T M : F (x, y) = 1}, also called the indicatrix bundle. Even though the these notions are defined for arbitrary dimension, we restrict to surfaces hereafter ( [BCS]).
On the other hand, such a Finsler structure defines a 2-parameter family of oriented paths on M, one in every oriented direction through every point. This is a special case of the notion of path geometry. We recall that, roughly speaking, a path geometry on a surface M is a 2-parameter family of curves on M with the property that through each point x ∈ M and in each tangent direction at x there passes a unique curve in the family. The fundamental example to keep in mind is the family of lines in the Euclidean plane.
To be more precise, a path geometry on a surface M is a foliation P of the projective tangent bundle PT M by contact curves, each of which is transverse to the fibers of the canonical projection π : PT M → M. Observe that even though PT M is independent of any norm F , actually there is a Riemannian isometry between PT M and Σ, fact that allows us to identify them in the Finslerian case( [B]).
The 3-manifold PT M is naturally endowed with a contact structure. Indeed, observe that for a curve be a smooth, immersed curve γ : (a, b) → M, let us denote byγ : (a, b) → PT M its canonical lift to the projective tangent bundle PT M. Then, the fact that the canonical projection is a submersion implies that, for each line L ∈ PT M, the linear map π * ,L : T L PT M → T x M, is surjective, where π(L) = x ∈ M. Therefore E L := π −1 * ,L (L) ⊂ T L PT M is a 2-plane in T L PT M that defines a contact distribution and therefore a contact structure on PT M. A curve on PT M is called contact curve if it is tangent to the contact distribution E. Nevertheless, the canonical liftγ to PT M of a curve γ on M is a contact curve.
A local path geometry on M is a foliation P of an open subset U ⊂ PT M by contact curves, each of which is transverse to the fibers of π : PT M → M.
The pair (P, Q) is called sometimes a generalized path geometry (see [Br]). The (forward) integral length of a regular piecewise C ∞ -curve γ : [a, b] → M on a Finsler surface (M, F ) is given by F (γ(t),γ(t))dt, whereγ = dγ dt is the tangent vector along the curve γ| [t i−1 ,t i ] . A regular piecewise C ∞ -curve γ on a Finsler manifold is called a forward geodesic if (L γ ) ′ (0) = 0 for all piecewise C ∞ -variations of γ that keep its ends fixed. In terms of Chern connection a constant speed geodesic is characterized by the condition D˙γγ = 0. Observe that the canonical lift of a geodesic γ to PT M gives the geodesics foliation P described above.
Using the integral length of a curve, one can define the Finslerian distance between two points on M. For any two points p, q on M, let us denote by Ω p,q the set of all piecewise C ∞ -curves γ : [a, b] → M such that γ(a) = p and γ(b) = q. Then the map gives the Finslerian distance on M. It can be easily seen that d is in general a quasidistance, i.e., it has the properties d(p, q) ≥ 0, with equality if and only if p = q, and d(p, q) ≤ d(p, r) + d(r, q), with equality if and only if r lies on a minimal geodesic segment joining from p to q (triangle inequality). A Finsler manifold (M, F ) is called forward geodesically complete if and only if any short geodesic γ : [a, b) → M can be extended to a long geodesic γ : [a, ∞) → M. The equivalence between forward completeness as metric space and geodesically completeness is given by the Finslerian version of Hopf-Rinow Theorem (see for eg. [BCS], p. 168). Same is true for backward geodesics. In the Finsler case, unlikely the Riemannian counterpart, forward completeness is not equivalent to backward one, except the case when M is compact.
Any geodesic γ emanating from a point p in a compact Finsler manifold loses the global minimising property at a point q on γ. Such a point q is called a cut point of p along γ. The cut locus of a point p is the set of all cut points along geodesics emanating from p. This kind of points often appears as an obstacle when we try to prove some global theorems in differential geometry being in the same time vital in analysis, where appear as a singular points set. In fact, the cut locus of a point p in a complete Finsler manifold equals the closure of the set of all non-differentiable points of the distance function from the point p. The structure of the cut locus plays an important role in optimal control problems in space and quantum dynamics allowing to obtain global optimal results in orbital transfer and for Lindblad equations in quantum control.
The notion of cut locus was introduced and studied for the first time by H. Poincare in 1905 for the Riemannian case. In the case of a two dimensional analytical sphere, S. B. Myers has proved in 1935 that the cut locus of a point is a finite tree in both Riemannian and Finslerian cases. In the case of an analytic Riemannian manifold, M. Buchner has shown the triangulability of the cut locus of a point p, and has determined its local structure for the low dimensional case in 1977 and 1978, respectively. The cut locus of a point can have a very complicated structure. For example, H. Gluck and D. Singer have constructed a C ∞ Riemannian manifold that has a point whose cut locus is not triangulable (see [SST] for an exposition). There are C k -Riemannian or Finsler metrics on spheres with a preferential point whose cut locus is a fractal ( [IS]).
In the present paper we will study the local and global behaviour of the geodesics of a Finsler metric of revolution on topological cylinders. In special, we will determine the structure of the cut locus on the cylinder for such metrics and compare it with the Riemannian case.
Will focus on the Finsler metrics of Randers type obtained as solutions of the Zermelo's navigation problem for the navigation data (M, h), where h is the canonical Riemannian metric on the topological cylinder h = dr 2 + m 2 (r)dθ 2 , and W = A(r) ∂ ∂r + B ∂ ∂θ is a vector field on M. Observe that our wind is more general than a Killing vector field, hence our theory presented here is a generalization of the classical study of geodesics and cut locus for Randers metrics obtained as solutions of the Zermelo's navigation problem with Killing vector fields studied in [HCS] and [HKS]. Nevertheless, by taking the wind W in this way we obtain a quite general Randers metric on M which is a Finsler metric of revolution and whose geodesics and cut locus can be computed explicitly.
Our paper is organized as follows. In the Section 2, we recall basics of Finsler geometryusing the Randers metrics that we will actually use in order to obtain explicit information on the geodesics behaviour and cut locus structure. We introduce an extension of the Zermelo's navigation problem for Killing winds to a more general case W = V + W , where only W is Killing. We show that the geodesics, conjugate locus and cut locus can be determined in this case as well.
In the section 3 we describe the theory of general Finsler surfaces of revolution. In the case this Finsler metric is a Riemannian one, we obtain the theory of geodesics and cut locus known already ( [C1], [C2]).
In the Section 4 we consider some examples that ilustrate the theory depicted until here. In particular, in subsection 4.1 we consider the general wind W = A(r) ∂ ∂r + B ∂ ∂θ which obviously is not Killing with respect to h, where A = A(r) is a bounded function and B is a constant and determine its geometry here. Essentially, we are reducing the geodesics theory of the Finsler metric F , obtained from the Zermelo's navigation problem for (M, h) and W , to the theory of a Riemannian metric (M, α). Moreover, in the particular case W = A ∂ ∂r + B ∂ ∂θ in Section 4.2, where A, B are constants, the geodesic theory of F can be directly obtained from the geometry of the Riemannian metric (M, h). A similar study can be done for the case W = A(r) ∂ ∂r . We leave a detailed study of these Randers metrics to a forthcoming research.

Finsler metrics. The Randers case
Finsler structures are one of the most natural generalization of Riemannian metrics. Let us recall here that a Finsler structure on a real smooth n-dimensional manifold M is a function F : where O is the zero section, has the homogeneity property F (x, λy) = λF (x, y), for all λ > 0 and all y ∈ T x M and also has the strong convexity property that the Hessian matrix is positive definite at any point (x, y) ∈ T M.

An ubiquitous family of Finsler structures: the Randers metrics
Initially introduced in the context of general relativity, Randers metrics are the most ubiquitous family of Finsler structures. A Randers metric on a surface M is obtained by a rigid translation of an ellipse in each tangent plane T x M such that the origin of T x M remains inside it.
Formally, on a Riemannian manifold (M, α), a Randers metric is a Finsler structure (M, F ) whose fundamental function F : T M → [0, ∞) can be written as where α(x, y) = a ij (x)y i y j and β(x, y) = b i (x)y i , such that the Riemannian norm of β is less than 1, i.e. b 2 := a ij b i b j < 1.
It is known that Randers metrics are solutions of the Zermelo's navigation problem [Z] which we recall here. The solution was given by Zermelo in the case the open sea is an Euclidean space, by [Sh] in the Riemannian case and studied in detailed in [BRS].
Indeed, for a time-independent wind W ∈ T M, on a Riemannian manifold (M, h), the paths minimizing travel-time are exactly the geodesics of the Randers metric where W = W i (x) ∂ ∂x i , y 2 h = h(y, y), λ = 1 − |W | 2 h , and W 0 = h(W, y). Requiring W h < 1 we obtain a positive definite Finslerian norm. In components, a ij = 1 [R] for a general discussion). The Randers metric obtained above is called the solution of the Zermelo's navigation problem for the navigation data (M, h) and W .
Remark 2.1 Obviously, at any x ∈ M, the condition F (y) = 1 is equivalent to y − W h = 1 fact that assures that, indeed, the indicatrix of (M, F ) in T x M differs from the unit sphere of h by a translation along W (x) (see Figure 1).
More generally, the Zermelo's navigation problem can be considered where the open sea is a given Finsler manifold (see [Sh]).
We have Proposition 2.2 Let (M, F ) be a Finsler manifold and W a vector field on M such that F (−W ) < 1. Then the solution of the Zermelo's navigation problem with navigation data F, W is th Finsler metric F obtained by solving the equation Indeed, if we consider the Zermelo's navigation problem where the open sea is the Finsler manifold (M, F ) and the wind W , by rigid translation of the indicatrix Σ F we obtain the closed, smooth, strongly convex indicatrix Σ F , where F is solution of the equation F y F − W = 1 which is clearly equivalent to (2.2) due to positively of F and homogeneity of F .
To get a genuine Finsler metric F , We need for the origin O x ∈ T x M to belong to the interior of Σ Remark 2.3 Consider the Zermelo's navigation problem for (M, F ) and wind W , where F is a (positive-defined) Finsler metric. If we solve the equation let F we obtain the solution of this Zermelo's navigation problem.
In order that F is Finsler we need to check: Since indicatrix of F is the rigid translation by W of the indicatrix of F , and indicatrix of F is strongly convex, it follows indicatrix of F is also strongly convex.
Hence, we need to find the condition for (ii) only. Denote the unit balls of F and F , respectively. The Zermelo's navigation problem shows Hence Then, the solution of the Zermelo's navigation problem with navigation data (M, F 1 ) and W is also a Randers metric F = α + β, where Proof. (Proof of Proposition 2.4) Let us consider the equation and squaring this formula, we get the equation substituting (2.5), (2.6) in (2.4) gives the 2nd degree equation The discriminant of (2.7) is Let us observe that F 1 (−W ) < 1 implies η > 0. Indeed The solution of (2.7) is given by On the other hand, since This implies a ij is positive defined.

A two steps Zermelo's navigation
We have discussed in the previous section the Zermelo's navigation when the open sea is a Riemannian manifold and when it is a Finsler manifold, respectively. In order to obtain a more general version of the navigation, we combine these two approaches. We have Theorem 2.5 Let (M, h) be a Riemannian manifold and V , W two vector fields on M.
Let us consider the Zermelo's navigation problem on M with the following data (I) Riemannian metric (M, h) with wind V + W and assume condition V + W h < 1; (II) Finsler metric (M, F 1 ) with wind W and assume W satisfies condition F 1 (−W ) < 1, where F 1 = α + β is the solution of the Zermelo' s navigation problem for the navigation data (M, h) with wind V , such that V h < 1.
Then, the above Zermelo's navigation problems (I) and (II) have the same solution F = α + β.
Proof. (Proof of Theorem 2.5) Let us consider case (I), i.e. the sea is the Riemannian metric (M, h) with the wind W := V + W such that V + W h < 1. The associated Randers metric through the Zermelo's navigation problem is given by α + β, where Next, we will consider the case (II) which we regard as a two steps Zermelo type navigation: Step 1. Consider the Zermelo's navigation with data (M, h) and wind V , Step 2. Consider the Zermelo's navigation with data (M, F 1 = α + β) obtained at step 1, and wind W such that We will show that a ij = a ij and b It follows that In a similar manner, It can be also seen that hence a ij = a ij and the identity of formulas (2.8) and (2.10) is proved. In order to finish the proof we show that the conditions h < 1 and F (−W ) < 1 are actually equivalent. Geometrically speaking, the 2-steps Zermelo's navigation is the rigid translation of Σ h by V followed by the rigid translation of Σ F 1 by W . This is obviously equivalent to the rigid translation of Σ h by W = V + W .
The geometrical meaning of (i) is that the origin O x ∈ T x M is in the interior of the translated indicatrix Σ F (see Figure 2. On the other hand, the relation in (ii) shows that the origin O x is in the interior of the translated indicatrix Σ h by V and Σ F 1 by W .
This equivalence can also be checked analytically.
For initial data (M, h) and V , we obtain by Zermelo's navigation the Randers metric Consider another vector field W and compute Conversely, if V +W 2 h < 1, by reversing the computation above, we obtain F The 2-steps Zermelo's navigation problem discussed above, can be generalized to ksteps Zermelo's navigation.
Remark 2.6 Let (M, F ) be a Finsler space and let W 0 , W 1 , . . . , W k−1 be k linearly independent vector fields on M. We consider the following k-step Zermelo's navigation problem.

Solution of F
Then F k is the Finsler metric obtained as solution of the Zermelo's navigation problem with data F 0 , W := W 0 + · · · + W k−1 with condition F 0 (− W ) < 1.

Geodesics, conjugate and cut loci
Proposition 2.7 Let (M, h) be a Riemannian manifold, V a vector field such that V h < 1, and let F = α + β be the solution of the Zermelo's navigation with data (M, h) and V .
Remark 2.8 The equation (2.12) can be written in coordinates In the 2-dimensional case, we get the 1st order PDE It can easily be seen that in the case of a surface of revolution h = dr 2 + m 2 (r)dθ 2 the wind V = A(r) ∂ ∂r is a solution of (2.12) and of (2.13).
Theorem 2.9 Let (M, h) be a simply connected Riemannian manifold and V = V i ∂ ∂x i a vector field on M such that V h < 1, and let F = α + β be the Randers metric obtained as the solution of the Zermelo's navigation problem with this data.
If V satisfies the differential relation then the followings hold good.
1. There exists a smooth function f : M → R such that β = df .
2. The Randers metric F is projectively equivalent to α, i.e. the geodesics of (M, F ) coincide with the geodesics of the Riemannian metric α as non-parametrized curve.
3. The Finslerian length of any C ∞ piecewise curve γ : [a, b] → M on M joining the points p and q is given by where L α (γ) is the Riemannian length with respect to α of γ.
4. The geodesic γ is minimizing with respect to α if and only if it is minimizing with respect to F . 5. For any two points p and q we have where d α (p, q) is the Riemannian distance between p and q with respect to α of γ.
6. For an F -unit speed geodesic γ, if we put p := γ(0) and q := γ(t 0 ), then q is conjugate to p along γ with respect to F if and only if q is conjugate to p along γ with respect to α.
7. The cut locus of p with respect to F coincide with the cut locus of p with respect to α.
On the other hand, since M is simply connected manifold, any closed 1-form is exact, hence in this case (2.14) is equivalent to β = df .
2. Follows immediately from the classical result in Finsler geometry that a Randers metric α +β is projectively equivalent to its Riemannian part α if and only if dβ = 0 (see for instance [BCS], p.298).
3. The length of the curve γ[a, b] → M, given by x i = x i (t) is defined as 4. It follows from 3.
5. It follows immediately from 2 and 3 (see [SSS] for a detailed discussion on this type of distance).
Let us consider a Jacobi field Y (t) along γ such that and construct the geodesic variation γ : Since the variation vector field ∂γ ∂u u=0 is Jacobi field it follows that all geodesics γ u (t) in the variation are α-geodesics for any u ∈ (−ε, ε).
Similarly with the case of base manifold, every curve in the variation can be reparametrized as an F -geodesic. In other words, for each u ∈ (−ε, ε) it exists a parameter changing t = t(s, u), ∂t ∂s > 0 such that γ(t, u) = γ(t(s, u), u).
If we evaluate this relation for u = 0 we get i.e. the Jacobi field Y ( a) is linear combination of the tangent vector ∂γ ∂t (a) and Y (a). Let us assume q = γ( a) is conjugate point to p along the F -geodesic γ, i.e. Y ( a) = 0. It results dγ dt (a) cannot be linear independent, hence Y (a) = 0, i.e. q = γ(a) is conjugate to p along the α-geodesic γ.
Conversely, if q = γ(a) is conjugate to p along the α-geodesic γ then (2.17) can be written as Y (a) = Y (s(a)) − dγ ds (s(a)) ds dt dt du and the conclusion follows from the same linearly independence argument as above.

Observe that
Indeed, if Cut α (p) = ∅ all α-geodesics from p are globally minimizing. Assume q ∈ Cut F (p) and we can consider q end point of Cut F (p), i.e. q must be F -conjugate to p along the geodesic σ(s) from p to q. This implies the corresponding point on σ(t) is conjugate to p, this is a contradiction.
Converse argument is identical.
Let us assume Cut α (p) and Cut F (q) are not empty sets.
If q ∈ Cut α (p) then we have two cases: (i) q is an end point of Cut α (p), i.e. it is conjugate to p along a minimizing geodesic γ from p to q. Therefore q is closes conjugate to p along the F -geodesic γ which is the reparametrization of γ (see 6).
(ii) q is an interior point of Cut α (p). Since the set of points in Cut α (p) founded at the intersection of exactly minimizing two geodesics of same length is dense in the closed set Cut α (p) it is enough to consider this kind of cut points. In the case q ∈ Cut α (p) such that there are 2 α-geodesics γ 1 , γ 2 of same length from p to q = γ 1 (a) = γ 2 (a), then from (4) it is clear that the point q = γ 1 ( a) = γ 2 ( a) has the same property with respect to F . Hence Cut α (p) ⊂ Cut F (p). This inverse conclusion follows from the same argument as above by changing roles of α with F .
Remark 2.10 See [INS] for a more general case.
We recall the following well-known result for later use.
Then the Legendre dual of F is Hamiltonian function F * = α * + β * where α * 2 = h ij (x)p i p j and β * = V i (x)p i . Here (x, p) are the canonical coordinates of the cotangent bundle T * M.
The following result is similar to the Riemannian counterpart and we give it here with proof.
We recall that a smooth vector field X on a Finsler manifold (M, F ) is called Killing field if every local one-parameter transformation group {ϕ t } of M generated by X consists of local isometries. It is clear from our construction above that W is Killing field on the surface of revolution (M, F ). We also have Proposition 2.12 Let (M, F ) be a Finsler manifold (any dimension) with local coordinates (x i , y i ) ∈ T M and X = X i (x) ∂ ∂x i a vector field on M. The following formulas are equivalent (i) X is Killing field for (M, F ); (iv) X i|j + X j|i + 2C p ij X p|q y q = 0, where " | " is the h-covariant derivative with respect to the Chern connection.
Lemma 2.13 With the notation in Lemma 2.11, the vector field ∂y i is the canonical lift of W to T M. In local coordinates this is equivalent to (2.18) Since the left hand side is 0-homogeneous in the y-variable, this relation is actually equivalent to the contracted relation by y i y j , i.e. (2.18) is equivalent to where we use C ijk y i = 0. We get the equivalent relation ∂g ij ∂x p W p y i y j + 2g pj ∂W p ∂x i y i y j = 0. (2.19) Observe that g ij g * jk = δ k i is equivalent to ∂g ij ∂x p g * ik = −g ij ∂g * ik ∂x p , hence (2.19) reads ∂g ij ∂x p W p g * ik p k g * jl p l + 2g pj ∂W p ∂x i g * ik p k g * jl p l = 0 and from here −g ij ∂g * ik ∂x p W p p k g * jl p l + 2g pj ∂W p ∂x i g * ik p k g * jl p l = 0. We finally obtain On the other hand, we compute which is the same with (2.20). Here we have used the 0-homogeneity of g * ij (x, p) with respect to p. We also observe that for any functions f, g : T * M → R we have {f 2 , g} = 2f {f, g}. Therefore, the following are equivalent Proposition 2.14 ( [FM]) Let (M, F ) be a Finsler manifold and W = W i (x) ∂ ∂x i a Killing filed on (M, F ) with F (−W ) < 1. If we denote by F the solution of the Zermelo's navigation problem with data (F, W ), then the following are true 1. The F -unit speed geodesics P(t) can be written as where ϕ t is the 1-parameter flow of W and ρ is an F -unit speed geodesic.
3. For any x ∈ M and any flag (y, V ) with flag pole y ∈ T x M and transverse edge V ∈ T x M, the flag curvatures K and K of F and F , respectively, are related by provided y + W and V are linearly independent.
In the 2-dimensional case, since any Finsler surface is of scalar flag curvature, we get Corollary 2.15 In the two-dimensional case, with the notation in Proposition 2.14, the Gauss curvature K and K of F and F are related by K(x, y) = K(x, y + W ), for any (x, y) ∈ T M.
Lemma 2.16 Let (M, F ) be a (forward) complete Finsler manifold, and let W be a Killing field with respect to F . Then W is a complete vector field on M, i.e. for any x ∈ M the flow ϕ x (t) is defined for any t.
Proof. (Proof of Lemma 2.16) Since W is Killing field, it is clear that its flow ϕ preserves the Finsler metric F and the field W . In other words, for any p ∈ M, the curve α : (a, b) → M, α(t) = ϕ x (t) has constant speed. Indeed, it is trivial to see that It means that the F -length of α is b − a, i.e. finite, hence by completeness it can be extended to a compact domain [a, b], and therefore α is defined on whole R. It results W is complete.
Theorem 2.17 Let (M, F ) be a Finsler manifold (not necessary Randers) and W = W i (x) ∂ ∂x i a Killing field for F , with F (−W ) < 1. If F is the solution of the Zermelo's navigation problem with data (M, F ) with the wind W then the followings hold good: and only if the corresponding point ρ(l) = ϕ(−l, P(l)) is the F -conjugate point to P(0) = ρ(0) along ρ. (iii) If ρ is a F -global minimizing geodesic from p = ρ(0) to a point q = ρ(l), then P(t) = ϕ(t, ρ(t)) is an F -global minimizing geodesic from p = P(0) to q = P(l), where l = d F (p, q).
(iv) If q ∈ cut F (p) is a F -cut point of p, then q = ϕ(l, q) ∈ cut F (p), i.e. it is a F -cut point of p, where l = d F (p, q).
Proof. (Proof of Theorem 2.17) (i) Since ϕ t (·) is a diffeomorphism on M (see Lemma 2.16), it is clear that its tangent map ϕ t * is a regular linear mapping (Jacobian of ϕ t is non-vanishing). Then Lemma 2.14 shows that J vanishes if and only if J vanishes, and the conclusion follow easily.
(ii) Let us denote by exp p : T p M → M and exp p : T p M → M the exponential maps of F and F , respectively. Then P(t) = ϕ(t, ρ(t)) implies (2.21) If (M, F ) is complete, Hopf-Rinow theorem for Finsler manifolds implies that for any p ∈ M, the exponential map exp p is defined on all of M. Taking into account Lemma 2.16, from (2.21) it follows exp p is defined on all of T p M, and again by Hopf-Rinow theorem we obtain that F is complete. The converse proof is similar.
For this, let us assume that, even though ρ is globally minimizing, the flow-corresponding geodesic P from p to q is not minimizing anymore. In other words, there must exist a shorter minimizing geodesic P s : [0, l 0 ] → M from p to q = P s (l 0 ) such that d F (p, q) = l 0 < l. (We use the subscript s for short).
We consider next, the F -geodesic ρ s : [0, l 0 ] → M obtained from P by flow deviation, i.e. ρ s (t) = ϕ(−t, P s (t)), and denote q 0 = ρ s (l 0 ) = ϕ(−l 0 , P(l 0 )). Then, triangle inequality in pq 0 q shows that where we denote by ξ the flow orbit from W through qm oriented from q 0 to q. In other wordsξ(t) = −W , and using the hypothesis F (−W ) < 1, it follows By comparing relations (2.21) with (2.22) it can be seen that this is a contradiction, hence P must be globally minimizing.
(iv) It follows from (iii) and the definition of cut locus.
Remark 2.18 Observe that statement (iii) and (iv) are not necessary and sufficient conditions, Indeed, from the proof of (iii) it is clear that for proving ρ global minimizer implies P global minimizer we have used condition F (−W ) < 1, which is equivalent to the fact that F -indicatrix includes the origin of T p M, a necessary condition for F to be positive defined (see Remark 2.3). Likewise, if we want to show that P global minimizer implies ρ global minimizer, we need F (W ) < 1, that is, the indicatrix Σ F translated by −W must also include the origin, i.e. the metric F 2 defined by F (y + F 2 W ) = F 2 , with the indicatrix Σ F 2 = Σ F − W is also a positive defined Finsler metric.
In conclusion if we assume F (−W ) < 1 and F (W ) < 1 then the statements (iii) and (iv) in Theorem 2.17 can be written with "if and only if". Observe that in local coordinates the conditions in Lemma 2.19 reads   where : is the covariant derivative with respect to the Levi-Civita connection of h.
Theorem 2.20 Let (M, h) be a simply connected Riemannian manifold and V = V i ∂ ∂x i , W = W i ∂ ∂x i vector fields on M such that (i) V satisfies the differential relation Then (i) The F -unit speed geodesics P(t) are given by where ϕ is the flow of W and σ(t) is an F 1 -unit speed geodesic. Equivalently, where γ(s) is an α-unit speed geodesic and s = s(t) is the parameter change t = s 0 F 1 ρ(τ ), dρ dτ dτ .
(ii) The point P(l) is conjugate to P(0) = p along the F − geodesic P(t) if and only if the corresponding point q = ρ(l) = ϕ(−l, P(l)) on the F -geodesic ρ is conjugate to p, or equivalently, q is conjugate to p along the α-geodesic from p to q.
Proof. (Proof of Theorem 2.20) All statements follows immediately by combining Theorem 2.9 with Theorem 2.17.
Remark 2.21 Informally, we may say that the cut locus of p with respect to F is the W -flow deformation of the cut locus of p with respect to F 1 , that is, the the W -flow deformation of the cut locus of p with respect to α, due to Theorem 2.9, 7.
3 Surfaces of revolution

Finsler surfaces of revolution
Let (M, F ) be a (forward) complete oriented Finsler surface, and W a vector field on M, whose one-parameter group of transformations {ϕ t : t ∈ I} consists of F -isometries, i.e.
This is equivalent with d F (ϕ t (q 1 ), ϕ t (q 2 )) = d F (q 1 , q 2 ), for any q 1 , q 2 ∈ M and any given t, where d F is the Finslerian distance on M. If ϕ t is not the identity map, then it is known that W must have at most two zeros on M.
We assume hereafter that W has no zeros, hence from Poincaré-Hopf theorem it follows that M is a surface homeomorphic to a plane, a cylinder or a torus. Furthermore, we assume that M is the topological cylinder S 1 × R.
By definition it follows that, at any x ∈ M \ {p}, W x is tangent to the curve ϕ x (t) at the point x = ϕ x (0). The set of points Orb W (x) := {ϕ t (x) : t ∈ R} is called the orbit of W through x, or a parallel circle and it can be seen that the period τ (x) := min{t > 0 : ϕ t (x) = x} is constant for a fixed x ∈ M.
Definition 3.1 A (forward) complete oriented Finsler surface (M, F ) homeomorphic to S 1 × R, with a vector field W that has no zero points, is called a Finsler cylinder of revolution, and ϕ t a rotation on M.
It is clear from our construction above that W is Killing field on the surface of revolution (M, F ).

The Riemannian case
The simplest case is when the Finsler norm F is actually a Riemannian one.
A Riemannian cylinder of revolution (M, h) is a complete Riemannian manifold M = S 1 × R = {(r, θ) : r ∈ R, θ ∈ [0, 2π)} with a warped product metric h = dr 2 + m 2 (r)dθ 2 . It follows that every profile curve {θ = θ 0 }, or meridian, is an h-geodesic, and that a parallel {r = r 0 } is geodesic if and only if m ′ (r 0 ) = 0, where θ 0 ∈ [0, 2π) and r 0 ∈ R are constants. It is clear that two meridians do not intersect on M and for a point p ∈ M, the meridian through p does not contain any cut points of p, that is, this meridian is a ray through p and hence d h (γ(0), γ(s)) = s, for all s ≥ 0.
The constant ν is called the Clairaut constant (see Figure 4). We recall the Theorem of cut locus on cylinder of revolution from [C1] Theorem 3.2 Let (M, h) is a cylinder of revolution with the warping function m : R → R is a positive valued even function, and the Gaussian curvature G h (r) = − m ′′ (r) m(r) is decreasing along the half meridian. If the Gaussian curvature of M is positive on r = 0, then the structure of the cut locus C q of a point θ(q) = 0 in M is given as follows: 1. The cut locus C q is the union of a subarc of the parallel r = −r(q) opposite to q and the meridian opposite to q if |r(q) < r 0 | := sup{r > 0|m ′ (r) < 0} and ϕ(m(r(q))) < π, i.e.
Remark 4.2 1. Observe that we actually perform a rigid translation of the Riemannian indicatrix Σ h by W , which is actually equivalent to translating Σ h by V followed by the translation of Σ F by W (see Remark 2.3).
2. Observe that the Randers metric given by (4.1) on the cylinder R × S 1 is rotational invariant, hence (M, α+ β) is a Finslerian surface of revolution. This type of Randers metircs are called Randers rotational metrics. Indeed, let us denote m F (r) := F ( ∂ ∂θ ). Observe that in the case A(r) is odd or even function, the function m F (r) is even function such that m F (0) > 0.
Theorem 2.20 implies Theorem 4.3 Let (M, h) be the topological cylinder R × S 1 with the Riemannian metric h = dr 2 + m 2 (r)dθ 2 and W = A(r) ∂ ∂r + B ∂ ∂θ , A 2 (r) + B 2 m 2 (r) < 1. If we denote by F = α + β the solution of Zermelo's navigation problem for (M, h) and W , then the followings are true.
(ii) The point q = P(l) is conjugate to P(0) = p along P if and only if q = (r(q), θ(q) − Bl) is conjugate to p with respect to α along the α-geodesic from p to q.
We have reduced the geometry of the Randers type metric (M, F ) to the geometry of the Riemannian manifold (M, α), obtained from (M, h) by (4.2).
Example 4.4 Let us observe that there are many cylinders (M, h) and winds W satisfying conditions in Theorem 4.3.
For instance, let us consider the topological cylinder R × S 1 with the Riemannian metric h = dr 2 + m 2 (r)dθ 2 defined using the warp function m(r) = e −r 2 .
Consider the smooth function A : R → − 1 If there exist a smooth function A : R → (−1, 1) and a constant B such that A 2 (r) + B 2 m 2 (r) < 1, G α (r) is decreasing along any half meridian and G α ≥ 0, then the α-cut locus and the F -cut locus of a point p = (r 0 , θ 0 ) is given in Theorem 3.2. Moreover, the F -cut locus of p is obtained by the deformation of the cut locus described in is an even bounded function such that m 2 < 1−A 2 B 2 , |A| < 1, B = 0. Proposition 4.1 and Theorem 4.3 can be easily rewritten for this case by putting A(r) = A = constant. We will not write them again here.
Instead, let us give some special properties specific to this case. A straightforward computation gives: Proposition 4.7 Let (M, h) be the Riemannian metric of the cylinder R × S 1 , and let W = A ∂ ∂r + B ∂ ∂θ , with A, B real constants such that m 2 < 1−A 2 B 2 , |A| < 1, B = 0. Then, the followings are true.
G α (r) = 1 λ 2 G h (r), where α is the Riemannian metric obtained in the solution of the Zermelo's navigation problem for (M, h) and V = A ∂ ∂r . (II) The geodesic flows S h and S α of (M, h) and (M, α), respectively, satisfy where ∆ = −2A 2 mm ′′ (y 2 ) 2 ∂ ∂y 1 is the difference vector field on T M endowed with the canonical coordinates (r, θ; y 1 , y 2 ). where A, B are constants, |A| < 1, B = 0, and wind W = A ∂ ∂r + B ∂ ∂θ then the followings hold good.