Boundary rigidity for Randers metrics

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY-NC 4.0 https://creativecommons.org/licenses/by-nc/4.0/ Boundary rigidity for Randers metrics © 2021 Annales Fennici Mathematici Published version Mönkkönen, Keijo Mönkkönen, K. (2022). Boundary rigidity for Randers metrics. Annales Fennici Mathematici, 47(1), 89-102. https://doi.org/10.54330/afm.112492 2022 Annales Fennici Mathematici Volumen 47, 2022, 89–102 Boundary rigidity for Randers metrics Keijo Mönkkönen Abstract. If a non-reversible Finsler norm is the sum of a reversible Finsler norm and a closed 1-form, then one can uniquely recover the 1-form up to potential fields from the boundary distance data. We also show a boundary rigidity result for Randers metrics where the reversible Finsler norm is induced by a Riemannian metric which is boundary rigid. Our theorems generalize Riemannian boundary rigidity results to some non-reversible Finsler manifolds. We provide an application to seismology where the seismic wave propagates in a moving medium. Reunajäykkyys Randers-metriikoille Tiivistelmä. Jos ei-reversiibeli Finsler-normi on summa reversiibelistä Finsler-normista ja suljetusta 1-muodosta, niin 1-muodon voi määrätä reunaetäisyysdatasta potentiaalikenttiä vaille yksikäsitteisesti. Osoitamme myös reunajäykkyystuloksen Randers-metriikoille, missä reversiibeli Finsler-normi on reunajäykän Riemannin metriikan indusoima. Lauseemme yleistävät Riemannilaisia reunajäykkyystuloksia joillekin ei-reversiibeleille Finsler-monistoille. Tarjoamme sovelluksen seismologiaan, missä seisminen aalto etenee liikkuvassa väliaineessa. 1. Introduction In this article we study a certain type of inverse problem for a special class of Finsler norms. The inverse problem we consider is known as the boundary rigidity problem: does the boundary distance data determine the Finsler norm uniquely up to the natural gauge in question? Here we present the problem and our results in a general level; more detailed information can be found in Sections 1.1, 1.2 and 2. Let Mbe a smooth manifold with boundary ∂M. A Finsler norm Fon Mis a non-negative function on the tangent bundle F:TM →[0,∞)such that for each x∈Mthe map y7→ F(x, y)defines a positively homogeneous norm in TxM. In general, Finsler norms are homogeneous only in positive scalings and they induce a distance function on Mwhich is not necessarily symmetric in contrast to the Riemannian distance function. Let βbe a smooth 1-form on Mand Fra reversible Finsler norm, i.e. Fr(x, −y) = Fr(x, y)for all x∈Mand y∈TxM. If the norm of βwith respect to Fris small enough, we can define the non-reversible Finsler norm F=Fr+β. The Finsler norm Fis non-reversible in the sense that F(x, −y) = F(x, y)for all x∈Mand y∈TxMif and only if β= 0. We can thus think that βis an anisotropic perturbation to the reversible Finsler norm Fr. We further assume that βis closed (dβ= 0) which implies that Fand Frhave the same geodesics as point sets and that Fhas reversible geodesics. Suppose we know the boundary distance data of F=Fr+β, i.e. we know the lengths of all geodesics of Fconnecting two points on the boundary ∂M. The question is: can we say something about βand Frfrom this information? We prove https://doi.org/10.54330/afm.112492 2020 Mathematics Subject Classification: Primary 53C24, 53A35, 86A22. Key words: Inverse problems, boundary rigidity, travel time tomography. c 2022 The Finnish Mathematical Society 90 Keijo Mönkkönen that if Mis simply connected, then one can uniquely recover the 1-form β(up to potential fields) and the boundary distance data of Frfrom the boundary distance data of F(see theorem 1.3 for a precise statement). Riemannian metrics form a special class of reversible Finsler norms. Suppose that Fris induced by a Riemannian metric gand write Fr=Fg. If kβkg<1, then F=Fg+βdefines a non-reversible Finsler norm called Randers metric. We say that the Riemannian manifold (M, g)is boundary rigid, if the boundary distance data determines the metric guniquely up to boundary preserving diffeomorphism. We prove that if Mis simply connected and (M, g)is boundary rigid, then (M, F)is also boundary rigid in the sense that one can uniquely recover the 1-form βup to potential fields and the Riemannian metric gup to boundary preserving diffeomorphism from the boundary distance data of F. See theorem 1.5 for a precise statement. Our proofs are mainly based on the following two facts. First, if two Finsler norms differ only by a closed 1-form, then they are projectively equivalent (they have the same geodesics modulo orientation preserving reparametrizations). Second, since F=Fr+β, we can express the length of any curve γwith respect to Frin terms of the symmetric part of the length functional LF(γ). Similarly, the integral ´γβcan be expressed in terms of the antisymmetric part of LF(γ). This allows us to reduce the boundary rigidity problem of Fto the boundary rigidity problem of Fr. Boundary rigidity has been studied earlier mainly on Riemannian manifolds. Boundary rigidity is known for example for simple subspaces of Euclidean space [34], simple subspaces of symmetric spaces of constant negative curvature [10], conformal simple metrics which agree on the boundary [26, 55] and for certain two-dimensional manifolds including compact simple surfaces [25, 43, 45, 50]. It is also conjectured that compact simple manifolds of any dimension are boundary rigid [43]. Our results generalize the boundary rigidity results to certain Randers metrics whenever the boundary rigidity of the unperturbed Riemannian manifold is known (see theorem 1.5). For a more comprehensive treatment of the boundary rigidity problem in Riemannian geometry, see the review [55]. Closest to our main theorems are rigidity results for magnetic geodesics on Riemannian manifolds. In [27] the authors prove boundary rigidity in the presence of a magnetic field (see also [7] for a generalization). Magnetic geodesics can be seen as geodesics of a Randers metric under additional assumptions for the vector potential which induces the magnetic field (the magnetic field has to be “weak") [36, 56]. There is also a correspondence between Randers metrics and stationary Lorentzian metrics [16, 17, 40] (see [57] for a boundary rigidity result on stationary Lorentzian manifolds). We note that projectively flat Finsler norms (geodesics of the Finsler norm are segments of straight lines) on compact convex domains in R2are completely determined by their boundary distance data [2, 3, 41]. In fact, this holds for a more general class of projective metrics in the plane [41]. Some geometric results similar to the boundary rigidity are known on Finsler manifolds. It was shown in [30] that the collection of boundary distance maps, which measure distances from the interior to the boundary, determines the topological and differential structures of the Finsler manifold. Further, it was shown in [31] that the broken scattering relation (lengths of all geodesics with endpoints on the boundary and reflecting once in the interior) determines the isometry class of reversible Finsler manifolds admitting a strictly convex foliation. The boundary rigidity problem is known in seismology as the travel time tomography problem where one tries to recover the speed of sound inside the Earth by Boundary rigidity for Randers metrics 91 measuring travel times of seismic waves on the surface. The ray paths of the seismic waves correspond to geodesics and the travel times correspond to lengths of the geodesics. The travel time tomography problem was solved in the beginning of 20th century for spherically symmetric metrics g=c−2(r)ewhere eis the Euclidean metric and c=c(r)is a radial sound speed satisfying the Herglotz condition (see equation (1)) [35, 58]. Our results apply to the situation where the seismic wave propagates in a moving medium: one can uniquely recover both the sound speed and the velocity of the medium up to potential fields from travel time measurements (see Theorem 1.5 and Section 1.2). The linearization of the boundary rigidity or travel time tomography problem leads to tensor tomography where one wants to characterize the kernel of the geodesic ray transform on symmetric 2-tensor fields [52]. For results in this direction and a general overview of tensor tomography, see the reviews [37, 47]. 1.1. The main results. Before stating our main results let us briefly introduce some notation; more details can be found in Section 2. The proofs of the main theorems can be found in Section 3. We denote by Man n-dimensional smooth manifold with boundary ∂M where n≥2. We let Fbe a Finsler norm and Frrefers to a reversible Finsler norm, i.e. Fr(x, −y) = Fr(x, y)for all x∈Mand y∈TxM. Riemannian metrics are a special case of Finsler norms: if gis a Riemannian metric, then it induces a reversible Finsler norm Fgas Fg(x, y) = pgij(x)yiyj. We denote by βa smooth closed 1-form (dβ= 0) and kβkF∗= supx∈MF∗(x, β(x)) is the dual norm of βwith respect to the co-Finsler norm F∗in T∗M. We say that the Finsler norm Fis admissible, if for every two boundary points x, x0∈∂M there is unique geodesic γof Fwith finite length going from xto x0. If Fis admissible, then we define the (not necessarily symmetric) map dF(·,·): ∂M ×∂M → [0,∞)by setting dF(x, x0) = LF(γ)where LF(γ)denotes the length of the curve γ with respect to F. We call the map dF(·,·)the boundary distance data of F. Finally, we say that the Riemannian manifolds (M, g1)and (M, g2)are boundary rigid, if dg1(x, x0) = dg2(x, x0)for all x, x0∈∂M implies that g2= Ψ∗g1where Ψ: M→M is a diffeomorphism such that Ψ|∂M = Id. In other words, g1and g2are isometric as Riemannian metrics. We recall that a diffeomorphism Ψ: (M, F2)→(M, F1)is an isometry between Finsler manifolds if Ψ∗F1=F2, or equivalently Ψpreserves the Finslerian distance [6]. We make the following observations before giving our first theorem. Remark 1.1. We note that Finsler norms are very flexible with respect to the boundary distance data, i.e. they are not usually boundary rigid in the same sense as Riemannian metrics. Let Ψ: M→Mbe a diffeomorphism which is identity on the boundary. If F1is an admissible Finsler norm and φis a scalar field which is constant on the boundary and its differential dφhas sufficiently small norm with respect to Ψ∗F1, then F1and F2= Ψ∗F1+ dφgive the same boundary distance data (Finslerian isometries preserve geodesics [6] and addition of dφonly changes parametrizations of geodesics [21]). Especially, if F1is reversible, then {Ψ∗F1+ dφ: Ψ|∂M = Id and φ|∂M =constant}provides a large family of Finsler norms which give the same boundary distances but are not isometric to F1(since Ψ∗F1+ dφis non-reversible whenever φis not constant). See also [12, 22, 23, 38] for results and constructions of non-isometric Finsler norms giving the same boundary distances. Remark 1.2. Finsler norms F1and F2which satisfy F2= Ψ∗F1+ dφfor some scalar field φand diffeomorphism Ψare sometimes called almost isometric Finsler 92 Keijo Mönkkönen norms and the map Ψ: (M, F2)→(M, F1)is called almost isometry [14, 28, 36, 39]. We show in theorem 1.5 that under certain assumptions the boundary distance data determines Randers metrics up to an almost isometry (see also remark 1.6). Almost isometries have many good properties: they for example are projective transformations which preserve (minimizing) geodesics up to reparametrization [39]. Almost isometries can also be defined on general quasi-metric spaces (X, d). It follows that if Ψ: (X1, d1)→(X2, d2)is an almost isometry between quasi-metric spaces, then Ψis an isometry between the metric spaces (X1,e d1)→(X2,e d2)where e di(p, q) = 1 2(di(p, q) + di(q, p)) is the symmetrized metric [14, 39]. Especially, in the case of metric spaces almost isometries are isometries. Our first theorem says that one can uniquely recover (up to potential fields) the perturbation βand the boundary distance data of Frfrom the boundary distance data of F=Fr+β. Theorem 1.3. Let Mbe a compact and simply connected smooth manifold with boundary. For i∈ {1,2}let Fi=Fr,i +βibe admissible Finsler norms where Fr,i is an admissible and reversible Finsler norm and βiis a smooth closed 1-form such that kβikF∗ r,i <1. Then the following are equivalent: (i) dF1(x, x0) = dF2(x, x0)for all x, x0∈∂M. (ii) There is unique scalar field φvanishing on the boundary such that β2= β1+ dφ, and dFr,1(x, x0) = dFr,2(x, x0)for all x, x0∈∂M. Remark 1.4. Since βiis closed and Mis simply connected, it follows that βi= dφifor some scalar field φi. Thus Fr,i and Fi=Fr,i +βi=Fr,i + dφiare almost isometric (but not isometric) Finsler norms (see remark 1.2). Trivially one can define φ=φ2−φ1so that dφ=β2−β1. The assumption dF1(x, x0) = dF2(x, x0)for all x, x0∈∂M is then used to show that φis constant on the boundary (and one can choose this constant to be zero). Let us clarify some of our assumptions in theorem 1.3. We need the assumption kβikF∗ r,i <1to guarantee that the sum Fr,i +βidefines a Finsler norm. Reversibility of Fr,i is needed so that any curve has the same length with respect to Fr,i as any of its reversed reparametrizations. The condition that βiis closed is used in three places. First, it is equivalent to that Fiand Fr,i have the same geodesics up to orientation preserving reparametrizations (Fiand Fr,i are projectively equivalent, see lemma 2.2). Second, closedness of βiis also equivalent to that Fihas reversible geodesics (Fiis projectively reversible, see lemma 2.1). Third, dβi= 0 implies that βi is exact since Mis assumed to be simply connected. All these properties are in a crucial role in our proofs. The existence of unique geodesics connecting boundary points is used in the proof as well and for this reason we assume that the Finsler norms are admissible. We note that since Fiand Fr,i are projectively equivalent, the admissibility of Fr,i implies the admissibility of Fi, and vice versa. We also note that dφis closed so the conclusion β2=β1+dφis compatible with the assumptions on βi. The conclusion that βidiffer only by a potential is similar to the solenoidal injectivity result for the geodesic ray transform of 1-forms [4, 47]. As an application of theorem 1.3 we have the following boundary rigidity result for Randers metrics (see [27, Theorem 6.4] for a similar result). Boundary rigidity for Randers metrics 93 Theorem 1.5. Let Mbe a compact and simply connected smooth manifold with boundary. For i∈ {1,2}let Fi=Fgi+βibe admissible Finsler norms where giis an admissible Riemannian metric and βiis a smooth closed 1-form such that kβikgi<1. Assume that (M, gi)are boundary rigid. Then the following are equivalent: (a) dF1(x, x0) = dF2(x, x0)for all x, x0∈∂M. (b) There is unique scalar field φvanishing on the boundary and a diffeomorphism Ψwhich is identity on the boundary such that β2=β1+ dφand g2= Ψ∗g1. (c) There is unique scalar field φvanishing on the boundary and a diffeomorphism Ψwhich is identity on the boundary such that β2= Ψ∗β1+ dφand g2= Ψ∗g1. Remark 1.6. Theorem 1.5 part (c) implies that F2= Ψ∗F1+dφ, i.e. the Randers metrics F1and F2are almost isometric (see remark 1.2). Hence we obtain a boundary rigidity result for Randers metrics in the special case when the 1-form βis closed and the Riemannian metric gis boundary rigid. This generalizes earlier boundary rigidity results to non-reversible (and hence non-Riemannian) Finsler norms. Note that the diffeomorphism Ψ: (M, F2)→(M, F1)in part (c) is an almost isometry but not an isometry since this would require that Ψ∗β1=β2[9]. Also note that if β1= 0 and β26= 0, then F1and F2can not be isometric since F1is reversible and F2is non-reversible. The assumptions of theorem 1.5 are the same as in theorem 1.3 except that we also assume the boundary rigidity of (M, g). We can simultaneously recover the metric g and the 1-form βfrom the boundary distance data dF(·,·)since the reversibility of Fgimplies that the data for βiand gi“decouple”: for any curve γone can obtain ´γβfrom the antisymmetric part and Lg(γ)from the symmetric part of the length functional LF(γ). We note that in theorems 1.3 and 1.5 we only use the lengths of geodesics connecting boundary points as data. Admissible Finsler norms as we have defined are closely related to simple Finsler norms and simple Riemannian metrics. A Riemannian metric gon a smooth manifold Mwith boundary is simple if it is non-trapping (geodesics have finite length), geodesics have no conjugate points and the boundary ∂M is strictly convex with respect to g(the second fundamental form on ∂M is positive definite). See [48, Section 3.7] for many equivalent definitions of simple Riemannian metrics. The concept of a simple Finsler norm can be defined analogously [13, 38]. The simplicity of the Finsler norm or Riemannian metric implies that there exists unique minimizing geodesic between any two points of the manifold [13, 38, 48]. More generally, if the manifold admits a convex function which has a minimum point, then there is a finite number of geodesics between any two non-conjugate points [15, 32] (see also [49]). We remark that one can take (M, gi)to be a compact simple surface in theorem 1.5 since simple Riemannian metrics are admissible and in two dimensions they are boundary rigid [50]. If g1and g2are simple metrics which are conformal and agree on the boundary, then they are boundary rigid in any dimension n≥2[26, 45, 55]. Theorem 1.5 has an application to Randers metrics arising in seismology (see Section 1.2 for more details). Let M=B(0, R)be a closed ball of radius R > 0and g=c−2(r)ewhere eis the Euclidean metric and c=c(r)is a radial sound speed satisfying the Herglotz condition (1) d drr c(r)>0, r ∈[0, R]. 94 Keijo Mönkkönen It follows that (M, g)is a non-trapping Riemannian manifold with strictly convex boundary [44, 55]. Let us further assume that ghas no conjugate points, i.e. g=c−2(r)eis a simple Riemannian metric. Then gis admissible and one can recover cand hence guniquely in theorem 1.5 (see [48, Remark 2.10] and [52, 55]). Especially, the diffeomorphism Ψbecomes identity in this case (Ψ = Id also for general conformal simple metrics which agree on the boundary). However, Ψcan be a nontrivial diffeomorphism for general spherically symmetric Riemannian metrics g (see [29, Appendix C]). We also note that there are sound speeds csatisfying the Herglotz condition (1) such that ghas conjugate points (and gis not admissible anymore, see [44, Section 3.3.2 and figure 6]). In Section 1.2 we give a physical interpretation for the 1-form βin Theorem 1.5 (βcorresponds to the flow field of a moving medium). 1.2. Application in seismology. Here we give an application of Theorems 1.3 and 1.5 to seismology where the seismic wave propagates in a moving medium. Assume that we have an object moving on a Riemannian manifold (M, g)with constant speed kUkg= 1. The speed is fixed, but the object can change the direction of the velocity vector Uarbitrarily. Let Wbe a vector field which can be interpreted as the additional velocity resulting from a time-independent external force field acting on the object. The net velocity is U+Wand we assume kWkg<1so that the object can move freely in any direction. Given any two points p, q ∈Mwe would like to know which path gives the least time when traveling from pto qtaking the drift Winto account. This is known as the Zermelo’s navigation problem (see [9, 20, 54]). It turns out that the unique solution is given by a geodesic of the Randers metric F=Fα+βwhere (see [20, Section 2.2]) αij =gij λ+Wi λ Wj λ, βi=−Wi λ, Wi=gijWj, λ = 1 − kWk2 g and we have left the dependence on x∈Mimplicit. Especially, if the parameter of a piecewise smooth curve γ: [0, T]→Mrepresents time, then (see [54, Lemma 3.1] and [21, Lemma 1.4.1]) (2) T=LF(γ). Let us interpret the object as a seismic wave (or ray) propagating in a moving medium. The manifold Mcorresponds to the Earth which can be modelled as a compact and simply connected smooth manifold with boundary (a ball). By the Fermat’s principle the path of the ray is a critical point of the travel time functional [5, 11, 19]. But since this functional equals to the length functional LF(γ)of the Randers metric F=Fα+βby equation (2), the ray paths of seismic waves correspond to geodesics of Fwhich is the unique solution to the Zermelo’s navigation problem. If our Riemannian metric is of the form g=c−2ewhere eis the Euclidean metric and c=c(x)is the sound speed, then kUkg= 1 is equivalent to kUke=cwhere k·keis the Euclidean norm of vectors. Thus Ucorresponds to the velocity of the propagating wave and the medium moves with velocity Wfor which kWke< c. The components of the Randers metric take the form αij =c−2δij 1−c−2kWk2 e +c−4WiWj (1 −c−2kWk2 e)2, βi=−c−2Wi 1−c−2kWk2 e . Note that here we have identified Wi=δijWj. Now if the 1-form βis closed, then Theorem 1.3 implies that one can uniquely recover βup to potential fields from travel Boundary rigidity for Randers metrics 95 time measurements of seismic waves (assuming admissibility of α). In addition, if the Riemannian manifold (M, α)is boundary rigid, then by Theorem 1.5 one can also uniquely recover the Riemannian metric αup to boundary preserving diffeomorphism from the travel time data. Let us do the following approximation. If we assume that kWke/c 1, then αij ≈c−2δij +Wi c2 Wj c2, βi≈ −Wi c2. When we only work to first order in kWke/c, the Riemannian metric αreduces to αij ≈c−2δij =gij and the ray paths of seismic waves correspond to geodesics of the Randers metric F= Fg+β. Similar linearization result is obtained in [33] for sound waves propagating in air under the influence of wind. We also note that the same result can be obtained from the linearization of travel time measurements [46]. If the sound speed c=c(r)is radial, csatisfies the Herglotz condition (1) and g=c−2(r)ehas no conjugate points, then Theorem 1.5 implies that in the first order approximation (with respect to kWke/c) one can uniquely recover the sound speed cand the velocity of the medium Wup to potential fields from travel time measurements. If the speed of sound cis constant, then the condition d(W/c2)=0 reduces to dW= 0, which in the case of a fluid flow means that Wis irrotational (or curl-free). Note that in the approximation we identify Wi=δijWj. In general, if cis not constant, then the condition d(W/c2) = 0 only means that the scaled flow field W/c2is irrotational. To summarize this section: our results (Theorems 1.3 and 1.5) apply to the propagation of seismic waves in a moving medium. Under certain assumptions one can recover the velocity of the medium from travel time measurements, and at the same time one reduces the travel time tomography problem in moving medium to the case where no flow field is involved. This allows one to recover the speed of sound as well in the first order approximation. 2. Finsler manifolds In this section we give a brief introduction to Finsler geometry. We only go through definitions and results which are needed in proving our main theorems. Basic theory of Finsler geometry can be found for example in [1, 8, 21, 53]. We use the Einstein summation convention, i.e. indices which appear both as a subscript and superscript are implicitly summed over. Let Mbe a smooth manifold. We denote by x∈Mthe base point on the manifold and by y∈TxMthe tangent vectors. A Finsler norm Fon Mis a nonnegative function on the tangent bundle F:TM →[0,∞)such that (F1) Fis smooth in TM \ {0}(smoothness outside zero section) (F2) F(x, y) = 0 if and only if y= 0 (positivity) (F3) F(x, λy) = λF(x, y)for all λ≥0(positive homogeneity of degree 1) (F4) 1 2 ∂2F2(x,y) ∂yi∂yjis positive definite whenever y6= 0 (convexity). The pair (M, F)is called a Finsler manifold. If Fis a Finsler norm, then one can define the reversed Finsler norm ←− Fby setting ←− F(x, y) = F(x, −y). It follows that ←− F also satisfies the properties (F1)–(F4). The conditions (F1)–(F4) imply that for every x∈Mthe map y7→ F(x, y) defines a positively homogeneous norm in TxM. If F(x, −y) = F(x, y)for all x∈M 96 Keijo Mönkkönen and y∈TxM, we say that the Finsler norm Fis reversible (or absolutely homogeneous). If Fis reversible, then the map y7→ F(x, y)defines a norm in TxM. Every Riemannian metric g=g(x)on Minduces a reversible Finsler norm Fgon Mby setting Fg(x, y) = qgij(x)yiyj. The condition 2 allows us to define the local metric gij =gij(x, y)as gij(x, y) = 1 2 ∂2F2(x, y) ∂yi∂yj. One can then define the Legendre transformation L:TM →T∗Musing the local metric gij (see for example [53, Chapter 3.1]). If F:TM →[0,∞)is a Finsler norm, then by using the Legendre transformation one obtains the dual norm (or co-Finsler norm) F∗:T∗M→[0,∞)satisfying the properties (F1)–(F4) in T∗M. The dual norm of a covector ω∈T∗ xMbecomes F∗(x, ω) = sup y∈TxM F(x,y)=1 ω(y). If F=Fgwhere g=g(x)is a Riemannian metric, then gij(x, y) = gij(x)is independent of yand the Legendre transformation Land its inverse correspond to the musical isomorphisms. In this article we study a class of non-reversible Finsler norms. Let F1be a Finsler norm on Mand βa smooth nonzero 1-form on M. Assume that the dual norm of βsatisfies kβkF∗ 1:= sup x∈M F∗ 1(x, β(x)) <1. Then F=F1+βdefines also a Finsler norm on M(see [53, Example 6.3.1] and [8, Chapter 11.1]). We study the special case F=Fr+βwhere Fris a reversible Finsler norm. It follows that Finsler norms of this kind are non-reversible since F(x, −y) = F(x, y)for all x∈Mand y∈TxMif and only if β≡0. If Fr=Fg where gis a Riemannian metric, then F=Fg+βis called a Randers metric (see [51] for the original definition of a Randers metric). Randers metrics are examples of Finsler norms which are not induced by any Riemannian metric (since Riemannian metrics are always reversible). The length of a piecewise smooth curve γ: [a, b]→Mis defined to be LF(γ) = ˆb a F(γ(t),˙γ(t)) dt. In general, LF(γ)is invariant only in orientation preserving reparametrizations. If in addition Fis reversible, then LF(γ)is also invariant in orientation reversing reparametrizations. When Fis induced by a Riemannian metric g, then we simply write Lg:= LFg. If Fis a Finsler norm such that F=F1+βwhere F1is a Finsler norm and βis a 1-form, then for any piecewise smooth curve γwe have LF(γ) = LF1(γ) + ˆγ β. Note that for the term coming from the 1-form βwe have ˆeγ β=±ˆγ β