Some geometric correspondences for homothetic navigation
aa r X i v : . [ m a t h . DG ] J a n SOME GEOMETRIC CORRESPONDENCESFOR HOMOTHETIC NAVIGATION
MING XU, VLADIMIR MATVEEV, KE YAN, SHAOXIANG ZHANG
Abstract.
In this paper, we provide conceptional explanations for the geodesic andJacobi field correspondences for homothetic navigation, and then let them guide usto the shortcuts to some well known flag curvature and S-curvature formulas. Theyalso help us directly see the local correspondence between isoparametric functionsor isoparametric hypersurfaces, which generalizes the classification works of Q. Heand her coworkers for isoparametric hypersurfaces in Randers space forms and Funkspaces.
Mathematics Subject Classification (2010) : 53B40, 53C42, 53C60.
Key words : flag curvature, geodesic, homothetic vector field, isoparametric func-tion, Jacobi field, Zermelo navigation, Introduction
Zermero navigation (or navigation for simplicity) is an important technique whichhelps us produce new Finsler metrics and study their geometric properties. The sim-plest non-Riemannian Finsler metrics, Randers metrics, can be produced by navigationfrom Riemannian metrics (see Subsection 5.4.2 in [21]). If we use a homothetic vectorfield in the navigation datum to produce the new metric, we simply call this procedurea homothetic navigation . Killing navigation , which uses a Killing vector field in thenavigation datum, provides an important subclass of homothetic navigation. Homo-thetic navigation and Killing navigation are crucial for classifying Randers metrics ofconstant flag or Ricci curvature [3, 4] and studying closed geodesics [16, 25] in Finslergeometry.Comparing the geometry before and after a homothetic navigation, we see manysimilar features and beautiful correspondences. For example, X. Mo and L. Huangproved their flag curvature formula for homothetic navigation [17].
Theorem 1.1.
Let ˜ F be the Finsler metric on M defined by navigation from the datum ( F, V ) , in which V is a homothetic vector field with dilation c , then we have the equalitybetween flag curvatures, K ˜ F ( x, ˜ y, ˜ P ) = K F ( x, y, P ) − c . (1.1) Here x is any point with F ( x, − V ( x )) < , y is any nonzero vector in T x M , the tangentplane P is spanned by y and u ∈ T x M satisfying h u, y i Fy = 0 , and the tangent plane ˜ P is spanned by u and ˜ y = y + F ( x, y ) V ( x ) . The notions of homothetic vector field and its dilation are according to the conventionof Subsection 5.4.2 in [21]. See Section 3 for equivalent definitions for them. Theequality (1.1) when c = 0, i.e., the flag curvature formula for Killing navigation, wasfirstly found by P. Foulon [8]. There are many interesting applications [10, 24] for hisformula. Recently, Q. He and her coworkers classified locally all isoparametric hypersurfacesin a Randers space form (
M, F ), with respect to the Busemann-Hausdorff (or B.H. forsimplicity) volume form dµ F BH [11]. Their classification result can be summarized asthe following theorem. Theorem 1.2.
Let ˜ F be a Randers metric defined by navigation from the datum ( F, V ) ,in which F is a Riemannian metric with constant curvature and V is a homotheticor Killing vector field for the metric F . Then locally around any x ∈ M where F ( x , − V ( x )) < , any hypersurface is isoparametric for ( F, dµ F BH ) if and only ifit is isoparametric for ( ˜ F , dµ ˜ F BH ) . According to the work [4] of D. Bao, C. Robles and Z. Shen, any Randers space form,i.e., Randers manifold with constant flag curvature, can be produced by homothetic orKilling navigation from a Riemannian space form. In Riemannian geometry, any localisoparametric hypersurface in a complete space form M can be extended to a global onein the universal cover of M . When M is an Euclidean space or a hyperbolic space, itsglobal isoparametric hypersurfaces are classified by E. Cartan [5]. When M is a unitsphere, the classification was recently completed by Q. Chi [6] (see also the surveys[19, 22] and the references therein).Many proofs in the literature on Zermero navigation, for example, those for The-orem 1.1 in [17] and Theorem 1.2 in [11], have involved some sophistical notions orcomplicated calculations. But we believe that there must exist more straightforwardexplanations and easier proofs.In a recent paper [9], P. Foulon and the second author showed a simple proof forP. Foulon’s flag curvature formula for Killing navigation. Their method inspired us tostudy the case of homothetic navigation, and see how some geometrical properties canbe naturally fitted into a system of correspondences. Firstly, we have a conceptionalexplanation for the geodesic correspondence, reproving Theorem 4.1 which firstly ap-peared in [14]. Then as a corollary, we get the correspondence for orthogonal Jacobifields (see Theorem 4.2). Since flag curvature can be described by Jacobi fields (seeLemma 5.1), we can use the above correspondences to propose an alternative proof forTheorem 1.1, with a crystal theme and minimized core calculation (see Lemma 5.2). Byalmost the same argument, we can even prove Theorem 1.1 when F is pseudo-Finsler(i.e., Theorem 1.3 in [15]; see Remark 5.4).As a byproduct, similar thought and Lemma 5.2 help us prove Theorem 1.3.
Let ˜ F be the Finsler metric defined by navigation from the datum ( F, V ) , in which V is a homothetic vector field with dilation c . Then for the metrics F and ˜ F , and their B.H. volume forms dµ F BH and dµ ˜ F BH respectively, we have the followingequality between the S-curvatures S F ( x, y ) and S ˜ F ( x, ˜ y ) , S ˜ F ( x, ˜ y ) = S F ( x, y ) + ( n + 1) c, in which x ∈ M satisfies F ( x, − V ( x )) < , y is any F -unit vector in T x M and ˜ y = y + V ( x ) . Theorem 1.3 seems known in folklore. Its special case when F is Riemannian isincluded in Theorem 5.10 in [21].As an application of Theorem 1.1, We discuss OME GEOMETRIC CORRESPONDENCES FOR HOMOTHETIC NAVIGATION 3
Question 1.4.
When can the locally symmetric property of the Finsler metric F bepreserved after a homothetic navigation? Theorem 2 in [9] answers Question 1.4 for Killing navigation, which always preservesthe locally symmetric property. Here we answer Question 1.4 for non-Killing homotheticnavigation, which only preserves the locally symmetric property for flat metrics (seeTheorem 6.1).Finally, we study the correspondence between normalized isoparametric functions (orisoparametric hypersurfaces) before and after a homothetic navigation. The notion ofnormalized isoparametric function implies that its gradient vector field generates unitspeed geodesics, for which we already have the correspondence by Theorem 4.1. Com-paring the B.H. volume forms and applying fundamental properties of Lie derivative,we can easily prove a relation between the Laplacians (see Lemma 7.4). Now the localcorrespondence between normalized isoparametric functions is obvious (see Theorem7.6), and we can generalize Theorem 1.2 to the following
Theorem 1.5.
Let V be a homothetic or Killing vector field on the Finsler mani-fold ( M, F ) , and ˜ F the metric defined by navigation from the datum ( F, V ) . Thenlocally around any point x with F ( x , − V ( x )) < , a hypersurface is isoparametricfor ( F, dµ F BH ) if and only if it is isoparametric for ( ˜ F , dµ ˜ F BH ) . Our approach is more direct than that in [11, 12, 13], which studied the subman-ifold geometry in the Finsler context. Besides classifying isoparametric hypersurfacesin Randers space forms, Theorem 7.6 and Theorem 1.5 also help us understand theintrinsic relation between the classification works in [12] and [13], for isoparametrichypersurfaces in Minkowski spaces and Funk spaces respectively. Furthermore, theyprovide abundant examples of the isoparametric hypersurfaces in Finsler geometry (seethe remark at the end of the paper).For simplicity, we will mainly discuss non-Killing homothetic navigation in this paper.With very minor changes, all the statements for lemmas and theorems, and all thearguments can be transplanted to the easier case of Killing navigation.In Section 2, we summarize some necessary knowledge on Finsler geometry. InSection 3, we introduce the notions of homothetic vector field and navigation process.In Section 4, we discuss geodesic or Jacobi field correspondences with conceptionalproofs. In Section 5, we prove Theorem 1.1 and Theorem 1.3. In Section 6, we applyTheorem 1.1 to answer Question 1.4. In Section 7, we study the local correspondencebetween isoparametric functions and prove Theorem 1.5.2.
Preliminaries
In this section, we summarize some basic knowledge on Finsler geometry. See [2, 20,21] for more details. Throughout this paper, we assume M to be a smooth manifoldwhich real dimension is n > Finsler metric on M is a continuous function F : T M → [0 , + ∞ ) which satisfiesthe following conditions for any standard local coordinates x = ( x i ) ∈ M and y = y i ∂ x i ∈ T x M :(1) The restriction of F to T M \ λ ≥ F ( x, λy ) = λF ( x, y ).(3) When y = 0, the Hessian matrix ( g Fij ( x, y )) = ( [ F ( x, y )] y i y j ) is positive defi-nite. MING XU, VLADIMIR MATVEEV, KE YAN, SHAOXIANG ZHANG
We will call (
M, F ) a
Finsler manifold . The restriction of F to each tangent space T x M is called a Minkowski norm .The Hessian matrix ( g Fij ( x, y )) defines an inner product on T x M , i.e., h u, v i Fy = g Fij ( x, y ) u i v j = 12 [ F ( x, y + su + tv )] st | s = t =0 , (2.2)which depends on the choice of the nonzero base tangent vector y . Sometimes wesimply denote it as g Fy and call it the fundamental tensor . The fundamental tensor g Fy is independent of the choice of y in each tangent space if and only if ( M, F ) isRiemannian.Arc length and distance can be similarly defined on the Finsler manifold (
M, F ). Ageodesic γ ( t ) with t ∈ ( a, b ) is a smooth nonconstant curve which satisfies the locallyminimizing principle, i.e., for any t ∈ ( a, b ), we can find t and t with a < t < t
0. Here thecovariant derivative D F ˙ γ ( t ) is an ordinary differential operator acting on the space ofsmooth vector fields along γ ( t ). See Section 5.3 in [20] for its explicit expression. Wewill need the following property of covariant derivative. Lemma 2.1.
For any smooth vector fields U ( t ) and V ( t ) along the geodesic γ ( t ) (i.e., U ( t ) , V ( t ) ∈ T γ ( t ) M for all t , same below) on the Finsler manifold ( M, F ) , we have ddt h U ( t ) , V ( t ) i F ˙ γ ( t ) = h D F ˙ γ ( t ) U ( t ) , V ( t ) i F ˙ γ ( t ) + h U ( t ) , D F ˙ γ ( t ) V ( t ) i F ˙ γ ( t ) . (2.3)To be self contained, we sketch a short proof of Lemma 2.1 here. We can extend˙ γ ( t ) to a smooth vector field Y in a neighborhood U of γ , such that each integrationcurve of Y is a constant speed geodesic. The fundamental tensors g FY defines a smoothRiemannian metric on U . The covariant derivative along γ ( t ) for the Levi-Civita con-nection of g FY coincides with D F ˙ γ ( t ) (see Lemma 6.2.1 in [20]). So we only need to observe(2.3) in Riemannian geometry, which is a well known fact. Flag curvature is a natural generalization of sectional curvature in Riemannian ge-ometry. For any x ∈ M , y ∈ T x M , and tangent plane P ⊂ T x M containing y , the flagcurvature K F ( x, y, P ) is defined by K F ( x, y, P ) = h u, R Fy u i Fy h y, y i Fy h u, u i Fy − ( h y, u i Fy ) , where u is any vector in P such that P = span { y, u } . Here the linear operator R Fy : T x M → T x M is the Riemann curvature (see [2, 20] for its explicit formula).We call a smooth vector field J ( t ) along the unit speed geodesic γ ( t ) a Jacobi field if it satisfies the Jacobi equation D F ˙ γ ( t ) D F ˙ γ ( t ) J ( t ) + R F ˙ γ J ( t ) = 0 . For example, the variation of a smooth family of constant speed geodesics providesa Jacobi field along each geodesic in this family. Conversely, any Jacobi field can belocally realized in this way (see Lemma 4.3 for the special case we will use later).
OME GEOMETRIC CORRESPONDENCES FOR HOMOTHETIC NAVIGATION 5
We call the Jacobi field J ( t ) orthogonal , if J ( t ) is contained in the g F ˙ γ ( t ) -orthogonalcomplement of ˙ γ ( t ), i.e., h J ( t ) , ˙ γ ( t ) i F ˙ γ ( t ) = 0, for each value of t .Busemann-Hausedorff (B.H. in short) volume form on the Finsler manifold ( M, F )can be locally presented as dµ F BH = σ F dx · · · dx n . Here σ F = Vol( S n (1))Vol( { y = ( y i ) | F ( x, y i ∂ x i ) ≤ } ) , in which Vol( · ) denotes the volume with respect to the standard measure in an Euclideanspace.For all standard local coordinates, τ F ( x, y ) = ln q det( g Fij ( x, y )) σ F globally defines a smooth function on T M \
0, called the distortion function . The
S-curvature S F ( x, y ) is defined as the derivative of τ F ( x, y ) in the direction of the geodesicspray, or equivalently, the derivative of ddt τ F ( γ ( t ) , ˙ γ ( t )) | t =0 , in which γ ( t ) is the constantspeed geodesic γ ( t ) on ( M, F ), satisfying γ (0) = x and ˙ γ (0) = y .3. Homothetic vector field and Zermero navigation
Let V be a smooth vector field on the Finsler manifold ( M, F ). Around each x ∈ M , V generates a family (a one-parameter local subgroup) of local diffeomorphisms Ψ t .We call V a homothetic vector field on ( M, F ) if(Ψ ∗ t F )( x, y ) = F (Ψ t ( x ) , (Ψ t ) ∗ ( y )) = e − ct F ( x, y ) , (3.4)for each x ∈ M , y ∈ T x M and t ∈ R , whenever Ψ t ( x ) is well defined. The constant c in (3.4) is called the dilation of V . Notice that (3.4) indicates Ψ t are local homothetictranslations. By (3.4) and (2.2), it is easy to see that, whenever y ∈ T x M is nonzeroand Ψ t ( x ) is well defined, we have h (Ψ t ) ∗ u, (Ψ t ) ∗ v i F (Ψ t ) ∗ y = e − ct h u, v i Fy , ∀ u, v ∈ T x M. (3.5)The homothetic vector field V is a Killing vector field if its dilation c is zero.Since the local homothetic or isometric translations Ψ t maps constant speed geodesicsto constant speed geodesics, the restriction of the homothetic or Killing vector field V to any constant speed geodesic γ ( t ) is a Jacobi field. So h V ( γ ( t )) , ˙ γ ( t ) i F ˙ γ ( t ) is a linearfunction. More precise information is given by the following lemma. Lemma 3.1.
Let V be a homothetic vector field with dilation c . Then its restrictionto a unit speed geodesic γ ( t ) satisfies h V ( γ ( t )) , ˙ γ ( t ) i F ˙ γ ( t ) ≡ c − ct, (3.6) in which c is some real constant. Proof.
The lemma is obvious when V is constantly zero.When V is not constantly zero, we first prove this lemma locally where V is nottangent to γ ( t ). We can find local coordinates x = ( x , x ′ ) = ( x i ) ∈ M and y = y i ∂ x i ,such that γ ( t ) = (0 , t, , . . . ,
0) and V coincides with ∂ x . Since V is a homotheticvector field with dilation c , the metric F can be presented as F ( x, y ) = e − cx F ( x ′ , y ). MING XU, VLADIMIR MATVEEV, KE YAN, SHAOXIANG ZHANG
From the assumption that γ ( t ) is a unit speed geodesic, i.e., D F ˙ γ ( t ) ˙ γ ( t ) = 0, we canget ∂∂x [ F ( x, ∂ x )] y = [ F ( x, ∂ x )] x y = [ F ( x, ∂ x )] x = − c, when x = x = · · · = x n = 0. Solving this differential equation with respect to thevariable x , we see h V ( γ ( t )) , ˙ γ ( t ) i F ˙ γ ( t ) = [ F ( x, ∂ x )] y | (0 ,t, ,..., is a linear function of t which slope is − c .By continuity, Lemma 3.1 is valid everywhere for all unit speed geodesics.By similar argument with local coordinates, it is also easy to see Lemma 3.2.
Let V be a homothetic vector field with dilation c on the Finsler manifold ( M n , F ) , and Ψ t the local homothetic translation generated by V . Then we have theequality for the B.H. volume forms (Ψ t ) ∗ dµ F BH = dµ Ψ ∗ t F BH = e − cnt dµ F BH , whenever Ψ t is well defined. Assume that V is a smooth vector field satisfying F ( x, − V ( x )) < U ⊂ M . Then the equality F ( x, y ) = ˜ F ( x, ˜ y ) defines a new Finsler metric on U ,in which ˜ y = y + F ( x, y ) V ( x ) for any x ∈ U and y ∈ T x M (see Section 5.4 in [21]). Wewill call ˜ F the metric defined by navigation from the datum ( F, V ).A relation between the fundamental tensors of F and ˜ F is revealed by the followinglemma (see Lemma 4.4 in [23] or the equality (5) in [9]). Lemma 3.3.
Let ˜ F be the Finsler metric on M defined by navigation from the datum ( F, V ) , then for any x ∈ U and nonzero vector y ∈ T x M , we have h u, v i ˜ F ˜ y = 11 + h V ( x ) , y i Fy h u, v i Fy , for any u and v in the g Fy -orthogonal complement of y in T x M . In each tangent space T x M for x ∈ U , the indicatrix S ˜ Fx M = { y ∈ T x M | ˜ F ( y ) = 1 } is a parallel shifting of the indicatrx S Fx M by the vector V ( x ), so it is easy to prove(see Proposition 5.3 in [21]) Lemma 3.4.
Let ˜ F be the Finsler metric on M defined by navigation from the datum ( F, V ) . Then dµ F BH = dµ ˜ F BH inside U . Geodesic and Jacobi field correspondences
Unless otherwise specified, we keep the following setup for the rest of the paper.Let V is a homothetic vector field on the Finsler manifold ( M, F ) with dilation c = 0.We will fix a point x ∈ M with F ( x , − V ( x )) < U of x where the metric ˜ F can be defined in U bynavigation from the datum ( F, V ). The parameter t for a unit speed geodesic passing x ∈ U when t = 0, or for the local homothetic translations Ψ t generated by V , isunderstood to be sufficiently close to zero.In [14], L. Huang and X. Mo proved the following correspondence between unit speedgeodesics before and after a homothetic navigation. OME GEOMETRIC CORRESPONDENCES FOR HOMOTHETIC NAVIGATION 7
Theorem 4.1.
For any unit speed geodesic γ ( t ) for the metric F with γ (0) = x ∈U , ˜ γ ( t ) = Ψ t ( γ ( e ct − c )) is a unit speed geodesic for the metric ˜ F with ˜ γ (0) = x .Conversely, any unit speed geodesic ˜ γ ( t ) for the metric ˜ F with ˜ c (0) = x can be presentedin this way. Following a similar thought as in [9], we propose a conceptional proof of it.
Proof.
Firstly we assume γ ( t ) is a unit speed geodesic on ( M, F ) and prove ˜ γ ( t ) is aunit speed geodesic for the metric ˜ F .Direct calculation shows F ( γ ( e ct − c ) , ddt γ ( e ct − c )) = F ( γ ( e ct − c ) , e ct ˙ γ ( e ct − c )) = e ct . By (3.4), we have F (Ψ t ( γ ( e ct − c )) , (Ψ t ) ∗ ( ddt γ ( e ct − c ))) = e − ct F ( γ ( e ct − c ) , ddt γ ( e ct − c ))= e − ct F ( γ ( e ct − c ) , e ct ˙ γ ( e ct − c )) = F ( γ ( e ct − c ) , ˙ γ ( e ct − c ) = 1 , and by the notion of navigation,˙˜ γ ( t ) = (Ψ t ) ∗ ( ddt γ ( e ct − c )) + V (˜ γ ( t ))is a ˜ F -unit tangent vector. To summarize, ˜ γ ( t ) is a ˜ F -unit speed curve.Assume conversely that ˜ γ ( t ) is not a geodesic, i.e., the local minimizing principle isnot valid somewhere on ˜ γ ( t ), then we can find a pair of real numbers t ′ and t ′′ , with t ′ < t ′′ sufficiently close to each other, and satisfying the following:(1) the segment of γ ( t ) with t ∈ [ e ct ′ − c , e ct ′′ − c ] is the unique minimizing geodesicbetween its end points for the metric F .(2) ˜ γ ( t ) with t ∈ [ t ′ , t ′′ ] is not minimizing for the metric ˜ F .Because of (2), we can find another ˜ F -unit speed smooth curve ˜ γ ( t ) such that itcoincides with ˜ γ ( t ) when t / ∈ ( t ′ , t ′′ ). From ˜ γ ( t ), we can trace back to find an F -unitspeed smooth curve γ ( t ), such that ˜ γ ( t ) = Ψ t ( γ ( e ct − c )). The curve γ ( t ) is differentfrom γ ( t ). But both coincide when t / ∈ ( e ct ′ − c , e ct ′′ − c ) and have the same F -length c ( e ct ′′ − e ct ′ ) for the segment t ∈ [ e ct ′ − c , e ct ′′ − c ]. This is a contradiction to (1). So˜ γ ( t ) must be a geodesic for the metric ˜ F .This proves the first statement in Theorem 4.1.To prove the other statement in Theorem 4.1, we observe that at ˜ γ (0) = γ (0) = x ,˙˜ γ (0) = ˙ γ (0) + V ( x ) can exhaust all ˜ F -unit tangent vectors. All other arguments aresimilar.Using Theorem 4.1, we can prove a correpondence between the orthogonal Jacobifields for the metrics F and ˜ F respectively. Theorem 4.2.
For any orthogonal Jacobi field J ( t ) along the unit speed geodesic γ ( t ) for the metric F , ˜ J ( t ) = (Ψ t ) ∗ ( J ( e ct − c )) (4.7) defines an orthogonal Jacobi field along the unit speed geodesic ˜ γ ( t ) = Ψ t ( γ ( e ct − c )) forthe metric ˜ F . Conversely, any orthogonal Jacobi field ˜ J ( t ) along ˜ γ ( t ) for the metric ˜ F MING XU, VLADIMIR MATVEEV, KE YAN, SHAOXIANG ZHANG can be presented by (4.7) for some orthogonal Jacobi field J ( t ) along γ ( t ) for the metric F . Proof.
Firstly, we assume J ( t ) is an orthogonal Jacobi field along the unit speedgeodesic γ ( t ) for the metric F and prove ˜ J ( t ) is an orthogonal Jacobi field along ˜ γ ( t )for the metric ˜ F .The orthogonal property of J ( t ) can be equivalently described as the claim that J ( e ct − c ) is tangent to the indicatrix of F in T γ ( e ct − c ) M at ˙ γ ( e ct − c ). Since Ψ t is alocal homothetic translation, ˜ J ( t ) = (Ψ t ) ∗ ( J ( e ct − c )) is also tangent to the indicatrixof F in T ˜ γ ( t ) M at (Ψ t ) ∗ ( ddt γ ( e ct − c )). Since the indicatrix of ˜ F is a parallel shiftingof that of F by the value of V , ˜ J ( t ) is tangent to the indicatrix of ˜ F in T ˜ γ ( t ) M atthe ˜ F -unit vector ˙˜ γ ( t ) = (Ψ t ) ∗ ( ddt γ ( e ct − c )) + V (˜ γ ( t )) as well. To summarize, we haveproved the orthogonal property for ˜ J ( t ). Then we will prove ˜ J ( t ) is a Jacobi field forthe metric ˜ F .By Lemma 4.3 below, J ( t ) can be realized as J ( t ) = ∂∂s γ ( s, t ) | s =0 for a smoothvariation γ ( s, t ) of γ ( t ) = γ (0 , t ), such that for each s , γ ( s, t ) is a unit speed geodesicfor the metric F . Then we have ˜ J ( t ) = ∂∂s ˜ γ ( s, t ) | s =0 , where ˜ γ ( s, t ) = Ψ t ( γ ( s, e ct − c )).By Theorem 4.1, ˜ γ ( s, t ) is a smooth variation of ˜ γ ( t ) = ˜ γ (0 , t ), such that for each s ,˜ γ ( s, t ) is a unit speed geodesic for the metric ˜ F . So ˜ J ( t ) is a Jacobi field along ˜ γ ( t ) forthe metric ˜ F .This argument proves the first statement in Theorem 4.2. The proof for the secondstatement is similar. Lemma 4.3.
For any orthogonal Jacobi field J ( t ) along the unit speed geodesic γ ( t ) for the metric F satisfying γ (0) = x , we can find a smooth map γ ( s, t ) with s ∈ ( − ǫ, ǫ ) such that for each fixed s , γ ( s, t ) is a unit speed geodesic for the metric F , γ (0 , t ) = γ ( t ) and ∂∂s γ (0 , t ) = J ( t ) . Proof.
Let t and t with t < < t be real numbers which are sufficiently closeto 0, and γ i ( s ) smooth curves with γ i (0) = γ ( t i ) and ˙ γ i (0) = J ( t i ), for i = 1 and 2respectively. For each fixed s sufficiently close to 0, there exists a unique unit speedgeodesic γ ( s, t ) from γ ( s ) to γ ( s ), which can be suitably extended and parametrizedsuch that γ ( s, t ) = γ ( s ).We only need to prove that J ( t ) coincides with the Jacobi field ¯ J ( t ) = ∂∂s γ (0 , t ). Foreach fixed s , we denote l ( s ) = d F ( γ ( s ) , γ ( s )) the distance from γ ( s ) to γ ( s ). By theorthogonal property of J ( t ), i.e., h J ( t ) , ˙ γ ( t ) i F ˙ γ ( t ) = h J ( t ) , ˙ γ ( t ) i F ˙ γ ( t ) = 0 , the first variation indicates dds l (0) = 0. So we have ¯ J ( t ) = ˙ γ (0) = J ( t ). Meanwhilewe also have ¯ J ( t ) = ˙ γ (0) = J ( t ). When t and t are sufficiently close to 0, γ ( t )is not a conjugate point of γ ( t ) along the geodesic γ ( t ), i.e., the values at t = t and t uniquely determine the Jacobi field. So we must have J ( t ) ≡ ¯ J ( t ), which ends theproof of this lemma.For any tangent vector u in the g F ˙ γ (0) -orthogonal complement of ˙ γ (0) at γ (0) = x ,we denote J Fγ ; u the set of all orthogonal Jacobi fields J ( t ) along γ ( t ) for the metric F ,satisfying J (0) = u . Then Theorem 4.2 immediately implies the following OME GEOMETRIC CORRESPONDENCES FOR HOMOTHETIC NAVIGATION 9
Corollary 4.4.
The correspondence from J ( t ) to ˜ J ( t ) in Theorem 4.2 is one-to-onebetween J Fγ ; u and J ˜ F ˜ γ ; u . Proofs of Theorem 1.1 and Theorem 1.3
To prove Theorem 1.1, we need the following description of flag curvature by Jacobifields.
Lemma 5.1.
Let γ ( t ) be a unit speed geodesic for the metric F with γ (0) = x ∈ M and ˙ γ (0) = y ∈ T x M . Suppose the tangent plane P ⊂ T x M is spanned by y and thenonzero vector u satisfying h u, y i Fy = 0 . Then we have K F ( x, y, P ) = ( h u, u i Fy ) − / max J ( t ) ∈J Fγ ; u {− d dt | t =0 [( h J ( t ) , J ( t ) i F ˙ γ ( t ) ) / ] } , (5.8) Proof.
Let J ( t ) be any orthogonal Jacobi field along the unit speed geodesic γ ( t ) forthe metric F , satisfying J (0) = u . Using Lemma 2.1, we can get d dt [( h J ( t ) , J ( t ) i F ˙ γ ( t ) ) / ] = ddt h D F ˙ γ ( t ) J ( t ) , J ( t ) i F ˙ γ ( t ) ( h J ( t ) , J ( t ) i F ˙ γ ( t ) ) / ! = h D F ˙ γ ( t ) J ( t ) , D F ˙ γ ( t ) J ( t ) i F ˙ γ ( t ) h J ( t ) , J ( t ) i F ˙ γ ( t ) − ( h D F ˙ γ ( t ) J ( t ) , J ( t ) i F ˙ γ ( t ) ) ( h J ( t ) , J ( t ) i F ˙ γ ( t ) ) / + h J ( t ) , D F ˙ γ ( t ) D F ˙ γ ( t ) J ( t ) i F ˙ γ ( t ) ( h J ( t ) , J ( t ) i F ˙ γ ( t ) ) / ≥ h J ( t ) , D F ˙ γ ( t ) D F ˙ γ ( t ) J ( t ) i F ˙ γ ( t ) ( h J ( t ) , J ( t ) i F ˙ γ ( t ) ) / = − h J ( t ) , R F ˙ γ ( t ) J ( t ) i F ˙ γ ( t ) ( h J ( t ) , J ( t ) i F ˙ γ ( t ) ) / , in which we have used Cauchy inequality. So at t = 0, we have K F ( x, y, P ) = h R F ˙ γ (0) J (0) , J (0) i F ˙ γ (0) h J (0) , J (0) i F ˙ γ (0) ≥ − ( h u, u i Fy ) − / d dt [( h J ( t ) , J ( t ) i F ˙ γ ( t ) ) / ] | t =0 . (5.9)This calculation proves K F ( x, y, P ) ≥ ( h v, v i Fy ) − / max J ( t ) ∈J Fγ ; u {− d dt ( h J ( t ) , J ( t ) i F ˙ γ ( t ) | t =0 ) / } . (5.10)Notice that there exists a unique Jacobi field J ( t ) along c ( t ) such that J (0) = u and D F ˙ γ ( t ) J ( t ) | t =0 = 0. This Jacobi field is orthogonal, i.e., J ( t ) ∈ J Fγ ; u , because h J ( t ) , ˙ γ ( t ) i F ˙ γ ( t ) is a linear function of t , and when t = 0, we have h J (0) , ˙ γ (0) i F ˙ γ (0) = h u, y i Fy = 0 and by Lemma 2.1, ddt | t =0 h J ( t ) , ˙ γ ( t ) i F ˙ γ ( t ) = h D F ˙ γ ( t ) J ( t ) | t =0 , ˙ γ (0) i Fy + h J (0) , D F ˙ γ ( t ) ˙ γ ( t ) | t =0 i Fy = 0 . From previous calculation, it is easy to see that the equality and maximum in (5.10)is achieved simultaneously by this J ( t ). This ends the proof of Lemma 5.1.The most crucial calculation for a homothetic navigation is contained in the followinglemma. Lemma 5.2.
Let v be a tangent vector at γ ( e ct − c ) in the g F ˙ γ ( e ct − c ) -orthogonal com-plement of ˙ γ ( e ct − c ) . Then (Ψ t ) ∗ ( v ) is a tangent vector at Ψ t ( γ ( e ct − c )) in the g ˜ F ˙˜ γ ( t ) -orthogonal complement of ˙˜ γ ( t ) , which satisfies h (Ψ t ) ∗ ( v ) , (Ψ t ) ∗ ( v ) i ˜ F ˙˜ γ ( t ) = 1 c + 1 · e − ct h v, v i F ˙ γ ( e ct − c ) , (5.11) in which c is the constant in Lemma 3.1. Proof.
Firstly, the argument in the proof of Theorem 4.1 covers the first statement ofLemma 5.2, i.e., h (Ψ t ) ∗ ( v ) , ˙˜ γ ( t ) i ˜ F ˙˜ γ ( t ) = 0. So we only need to verify (5.11).Denote y ( t ) = ddt γ ( e ct − c ) and ¯ y ( t ) = ˙˜ γ ( t ) − V (˜ γ ( t )) = (Ψ t ) ∗ ( y ( t )) . By Lemma 3.3, h (Ψ t ) ∗ ( v ) , (Ψ t ) ∗ ( v ) i ˜ F ¯ y ( t ) = 11 + h V (˜ γ ( t )) , ¯ y ( t ) i F ¯ y ( t ) h (Ψ t ) ∗ ( v ) , (Ψ t ) ∗ ( v ) i F ¯ y ( t ) . (5.12)By (3.5), we get h (Ψ t ) ∗ ( v ) , (Ψ t ) ∗ ( v ) i F ¯ y ( t ) = h (Ψ t ) ∗ ( v ) , (Ψ t ) ∗ ( v ) i F (Ψ t ) ∗ ( y ( t )) = e − ct h v, v i Fy ( t ) , (5.13)and h V (˜ γ ( t )) , ¯ y ( t ) i F ¯ y ( t ) = h (Ψ t ) ∗ ( V ( γ ( e ct − c ))) , (Ψ t ) ∗ ( y ( t )) i F (Ψ t ) ∗ ( y ( t )) = e − ct h V ( e ct − c ) , y ( t ) i F ˙ γ ( e ct − c ) = e − ct h V ( e ct − c ) , ˙ γ ( e ct − c ) i F ˙ γ ( e ct − c ) . (5.14)By Lemma 3.1, h V ( e ct − c ) , ˙ γ ( e ct − c ) i F ˙ γ ( e ct − c ) = c − c · e ct − c = ( c + 1) − e ct , (5.15)in which c is the constant in Lemma 3.1.Summarizing (5.12)-(5.15), we get h (Ψ t ) ∗ v, (Ψ t ) ∗ v i ˜ F ˙˜ γ ( t ) = 11 + e − ct (( c + 1) − e ct ) · e − ct h v, v i F ˙ γ ( e ct − c ) = 1 c + 1 · e − ct h v, v i F ˙ γ ( e ct − c ) . This ends the proof of Lemma 5.2.Now we summarize all the observations in these two sections to prove Theorem 1.1.
Proof of Theorem 1.1.
Firstly, we assume the dilation c is not zero. For any fixed x ∈ M , we only need to restrict our discussion locally in a suitable open neighborhoodof x . Let y be any F -unit tangent vector in T x M , then there exists a unique unitspeed γ ( t ) for the metric F , such that γ (0) = x and ˙ γ (0) = y . By Theorem 4.1,˜ γ ( t ) = Ψ t ( γ ( e ct − c )) is a unit speed geodesic for the metric ˜ F , satisfying ˜ γ (0) = x and˙˜ γ (0) = ˜ y = y + V ( x ). OME GEOMETRIC CORRESPONDENCES FOR HOMOTHETIC NAVIGATION 11
For any orthogonal Jacobi field J ( t ) in J Fγ ; u , we denote f J ( t ) = ( h J ( t ) , J ( t ) i F ˙ γ ( t ) ) / .By Lemma 5.1, we have K F ( x, y, P ) = ( h u, u i Fy ) − / max J ( t ) ∈J Fγ ; u {− f J ′′ (0) } . (5.16)By Theorem 4.2, ˜ J ( t ) = (Ψ t ) ∗ ( J ( e ct − c )) ∈ J ˜ F ˜ γ ; u , i.e., it is an orthogonal Jacobi fieldalong ˜ γ ( t ) for the metric ˜ F satisfying ˜ J (0) = u . We denote ˜ f J ( t ) = h ˜ J ( t ) , ˜ J ( t ) i ˜ F ˙ γ ( t ) .Then by Corollary 4.4 and Lemma 5.1, K ˜ F ( x, ˜ y, ˜ P ) = ( h u, u i ˜ F ˜ y ) − / max J ( t ) ∈J Fγ ; u {− ˜ f J ′′ (0) } . (5.17)By Lemma 5.2, for the same J ( t ) ∈ J Fγ ; u , we have˜ f J ( t ) ≡ r
11 + c e − ct f J ( e ct − c ) , (5.18)where c is the constant in Lemma 3.1. Evaluate (5.18) at t = 0, we can determine theconstant r
11 + c = ˜ f J (0) f J (0) = ( h u, u i ˜ F ˜ y ) / ( h u, u i Fy ) / . It is easy to calculate that˜ f ′′ J (0) = ( h u, u i ˜ F ˜ y ) / ( h u, u i Fy ) / · d dt ( e − ct f J ( e ct − c )) | t =0 = ( h u, u i ˜ F ˜ y ) / ( h u, u i Fy ) / · ( c f J (0) + f J ′′ (0)) . (5.19)Finally, summarizing (5.16), (5.17) and (5.19), we get K ˜ F ( x, ˜ y, ˜ P ) = ( h u, u i ˜ F ˜ y ) − / max J ( t ) ∈J Fγ ; u {− ˜ f J ′′ (0) } = ( h u, u i Fy ) − / max J ( t ) ∈J Fγ ; u {− ( c f J (0) + f J ′′ (0)) } = ( h u, u i Fy ) − / max J ( t ) ∈J Fγ ; u {− f J ′′ (0) } − c = K F ( x, y, P ) − c . This ends the proof of Theorem 1.1 when c = 0. With only some minor modifications,this argument can also prove the case c = 0, which ends the proof of Theorem 1.1. Remark 5.3. In [9] , Theorem 1.1 with c = 0 is proved by a slightly different approach. The authors of [9] used another description for the flag curvature, i.e., K F ( x, y, P ) = 12 h u, u i Fy max J ( t ) ∈J Fγ ; u {− d dt | t =0 ( f J ( t )) } , (5.20)to prove Theorem 1.1 for Killing navigation. The maximum in (5.20) can only beachieved when D F ˙ γ ( t ) J ( t ) | t =0 = 0. Implied by Lemma 5.2, D ˜ F ˙˜ γ ( t ) ˜ J ( t ) | t =0 = 0 when c = 0. So their proof can not be directly generalized to homothetic navigation. Remark 5.4.
Our proof of Theorem 1.1 can be easily generalized to pseudo-Finslergeometry.
In [15], M. Javaloyes and H. Vit´orio proved Theorem 1.1 when F is pseudo-Finsler.Their proof applied the fanning curves approach [1].Our method can also be applied to prove their theorem (i.e., Theorem 1.3 in [15]).The geodesic correspondence is similar to the Finsler case (see Theorem 1.2 in [15]).From the view point of variation, the expected correspondence between orthogonalJacobi fields follows immediately. Lemma 5.2 with the key calculation can be provedby the same argument.When we use orthogonal Jacobi fields to describe flag curvature, we can not useLemma 5.1 directly. The reason is the following. When F is pseudo-Finsler, thefundamental tensor h· , ·i Fy may be indefinite, and then Cauchy inequality used in theproof of Lemma 5.1 fails. However, since the restriction of h· , ·i Fy to P = span { y, u } isnondegenerate and h u, y i Fy = 0, we must have h u, u i Fy = 0. By similar calculation as inthe proof of Lemma 5.1, we can show that K F ( x, y, P ) is the unique critical value ofthe functional L ( J ) = −|h u, u i Fy | / d dt | t =0 ( |h J ( t ) , J ( t ) i F ˙ γ ( t ) | / ) , ∀ J ( t ) ∈ J Fγ ; u , and the critical set is the affine subspace of all J ( t ) ∈ J Fγ ; u such that D F ˙ γ ( t ) J ( t ) | t =0 is a scalar multiple of u . So we can still use the calculation (5.19) to prove the flagcurvature equality (1.1).The geodesic correspondence for homothetic navigation and the key calculation inLemma 5.2 also help us prove Theorem 1.3. Proof of Theorem 1.3.
Let γ ( t ) be the unit speed geodesic om ( M, F ), satisfying γ (0) = x and ˙ γ (0) = y , and denote ˜ γ ( t ) = Ψ t ( γ ( e ct − c )). Firstly, we choose smoothvector fields e i ( t ), 1 ≤ i ≤ n , along the geodesic γ ( t ), such that e ( t ) = ˙ γ ( t ), and theyprovide a g F ˙ γ -orthonormal basis of T γ ( t ) M for each t . Secondly, we define the followingsmooth vector fields along ˜ γ ( t ),¯ e i ( t ) = (Ψ t ) ∗ ( e i ( e ct − c )) , for 1 ≤ i ≤ n, ˜ e ( t ) = ˙˜ γ ( t ) , and˜ e i ( t ) = ¯ e i ( t ) for 1 < i ≤ n. Then at each point ˜ γ ( t ), { ¯ e i ( t ) with 1 ≤ i ≤ n } and { ˜ e i ( t ) with 1 ≤ i ≤ n } are twobases for T ˜ γ ( t ) M . Denotevol( t ) = Vol( { ( y i ) | F ( γ ( t ) , y i e i ( t )) ≤ } ) , vol( t ) = Vol( { ( y i ) | F (˜ γ ( t ) , y i ¯ e i ( t )) ≤ } ) , and f vol( t ) = Vol( { ( y i ) | F (˜ γ ( t ) , y i ˜ e i ( t )) ≤ } ) , in which Vol is the standard measure in an Euclidean space. OME GEOMETRIC CORRESPONDENCES FOR HOMOTHETIC NAVIGATION 13
Using this setup, the distortions τ F ( γ ( t ) , ˙ γ ( t )) and τ ˜ F (˜ γ ( t ) , ˙˜ γ ( t )), for the metric F and ˜ F respectively, can be presented as τ F ( γ ( t ) , ˙ γ ( t )) = ln q det( h e i ( t ) , e j ( t ) i F ˙ γ ( t ) ) − ln vol( t ) + C , and (5.21) τ ˜ F (˜ γ ( t ) , ˙˜ γ ( t )) = ln r det( h ˜ e i ( t ) , ˜ e j ( t ) i ˜ F ˙˜ γ ( t ) ) + ln f vol( t ) + C , (5.22)in which C is some universal constant depending on n .By Lemma 5.2,det( h ˜ e i ( t ) , ˜ e j ( t ) i ˙˜ γ ( t ) ) = C · e − c ( n − t det( h e i ( e ct − c ) , e j ( e ct − c ) i F ˙ γ ( t ) ) , (5.23)in which C = 1 / (1 + c ) n − is some positive constant. By (5.14), (5.15), the homotheticproperty of V and multi-variable calculus, f vol( t ) = vol( t )1 + h V (Ψ t ( γ ( e ct − c ))) , ¯ e ( t ) i F ¯ e ( t ) = C ′ e ct vol( t )= C ′ e ct · e c ( n − t vol( e ct − c ) = C ′ e cnt vol( e ct − c ) , (5.24)in which C ′ is some positive constant.Summarizing (5.21)-(5.24), we get τ ˜ F (˜ γ ( t ) , ˙˜ γ ( t )) = τ F ( γ ( e ct − c ) , ˙ γ ( e ct − c )) + c ( n + 1) t, so S ˜ F ( x, ˜ y ) = ddt τ ˜ F (˜ γ ( t ) , ˙˜ γ ( t )) | t =0 = ( e ct − c ) ′ | t =0 · dds τ F ( γ ( s ) , ˙ γ ( s )) | s =0 + ( n + 1) c = S F ( x, y ) + ( n + 1) c. This ends the proof of Theorem 1.3 when c = 0. The case c = 0 can be provedsimilarly.6. Application to the study of locally symmetric property
In this section, we discuss an application of Theorem 1.1.Recall that a Finsler metric F is called locally symmetric ( in curvature sense ) if forany unit speed geodesic γ ( t ), we have D F ˙ γ ( t ) R F ˙ γ ( t ) ≡ x , we can find a local in-volutive isometry ρ x , such that x is an isolated fixed point of ρ x . Notice that locallysymmetric Finsler metric in metric sense must be Berwaldian [18], but there are manynon-Berwaldian Finsler spheres with constant curvature, which are locally symmetric.In [9], the second author and P. Foulon proved that the locally symmetric propertyis preserved by Killing navigation. However, for non-Killing homothetic navigation, thefollowing theorem indicates a very different phenomenon. Theorem 6.1.
Let ˜ F be the metric defined by navigation from the datum ( F, V ) , inwhich V is a homothetic vector field on M satisfying F ( x, − V ( x )) < in an open subset U and its dilation c is nonzero. Then the following two statements are equivalent: (1) The metric F has constant zero flag curvature in U . (2) Both F and ˜ F are locally symmetric in U . Proof.
Firstly, we prove the statement from (1) to (2). By Theorem 1.1, the flagcurvature of the metric ˜ F is constantly − c in U . As Finsler metrics of constant flagcurvature are locally symmetric, both F and ˜ F are locally symmetric in U , which provesthe statement from (1) to (2).Nextly, we prove the statement from (2) to (1). For any x ∈ U and F -unit vector y ∈ T x M , we denote K F max ( x, y ) = max { K F ( x, y, P ) |∀ tangent plane P with y ∈ P } , and K F min ( x, y ) = min { K F ( x, y, P ) |∀ tangent plane P with y ∈ P } . We claim K F max ( x, y ) ≡ K F min ( x, y ) ≡ λ = K F max ( x, y ) = 0 is as following. We choose a unit speed ge-odesic γ ( t ) in U for the metric F , such that γ (0) = x and ˙ γ (0) = y . The locallysymmetric property of F implies that K F max ( γ ( t ) , ˙ γ ( t )) is a constant function of t , i.e., K F max ( γ ( t ) , ˙ γ ( t )) ≡ K F max ( γ (0) , ˙ γ (0)) = λ . By Theorem 4.1, ˜ γ ( t ) = Ψ t ( γ ( e ct − c )) is aunit speed geodesic for the metric ˜ F , in which Ψ t ’s are the local homothetic transla-tions generated by V . Denote ¯ y ( t ) = (Ψ t ) ∗ ( ddt γ ( e ct − c )), then the homothetic propertyimplies K F max (˜ γ ( t ) , ¯ y ( t )) = e ct K F max ( γ ( t ) , ˙ γ ( t )) = λe ct . Applying Theorem 1.1 to all tangent planes P and ˜ P , containing ¯ y ( t ) and ˙˜ γ ( t ) respec-tively, we get K ˜ F max (˜ γ ( t ) , ˙˜ γ ( t )) = K F max (˜ γ ( t ) , ¯ y ( t )) − c = λe ct − c . (6.25)By the locally symmetric property of ˜ F , K ˜ F max (˜ γ ( t ) , ˙˜ γ ( t )) is a constant function of t .Since c = 0, we must have K F max ( x, y ) = λ = 0 for any x ∈ M and F -unit vector y ∈ T x M .Similarly, we can prove K F min ( x, y ) = 0 for any x ∈ M and F -unit vector y ∈ T x M .Then it is obvious to see K F ≡
0, and K ˜ F ≡ − c by Theorem 1.1. To summarize, thisargument proves the statement from (2) to (1).7. Homothetic navigation for isoparametric function
In this section, we keep all assumptions and the notations for the Finsler mani-fold (
M, F ) and the homothetic navigation as in Section 4. Further more, we con-sider some locally defined isoparametric functions in some open subset U of M where F ( x, − V ( x )) < f be a regular function in U . The notion of isoparametric property for f isdefined by the following conditions [12]:(1) The F -length function F ( ∇ F f ) for gradient vector field ∇ F f only depends onthe values of f , i.e., f is transnormal .(2) The Laplacian ∆ F f only depends on the values of f .Here ∇ F is the gradient operator for the metric F , defined by h∇ F f, W i F ∇ F f = df ( W )for any vector field W . The regularity of f ( x ) implies ∇ F f ( x ) is well defined and F ( · , ∇ F f ) is a positive smooth function in U . OME GEOMETRIC CORRESPONDENCES FOR HOMOTHETIC NAVIGATION 15
Denote div dµ the divergence operator with respect to the smooth volume form dµ , i.e.,div dµ W · dµ = L W dµ , in which W is any smooth vector field, and L is the Lie derivative.Then the (nonlinear) Laplacian ∆ F f can be presented as ∆ F f = div µ F BH ∇ F f .We can always replace an isoparametric or transnormal function f by ϕ ◦ f for somesuitable real smooth function ϕ , such that F ( ∇ F f ) ≡ , and f ( x ) = 0 for some fixed x ∈ U . (7.26)Notice that the isoparametric and transnormal properties are preserved, and the localfoliation of the level sets is unchanged. So we only need to consider f satisfying (7.26),which is simply called normalized around x .Now we consider the navigation with the datum ( F, V ), in which V is a homotheticvector field with dilation c = 0. We study its effect on the foliation M t = f − ( t ) locallydefined by a normalized transnormal function f ( x ) around each fixed x ∈ M . Weassume F ( x , − V ( x )) <
1, so that ˜ F is well defined around x . By Lemma 4.1 in [23],the integration curves of ∇ F f are unit speed geodesics on ( M, F ).We define a smooth map Ψ locally around x , such that Ψ | M e ct − c = Ψ t for eachvalue of t . Lemma 7.1. Ψ is an orientation preserving local diffeomorphism around M . Proof.
It is obvious that Ψ fixes each point of M . We only need to prove that forany x ∈ M , the tangent map Ψ ∗ : T x M → T x M is an orientation preserving linearisomorphism. Then Lemma 7.1 is obvious by this observation.The tangent map Ψ ∗ maps T x M identically to itself. Let γ ( t ) be the unit speedgeodesic for the metric F , such that ˙ γ ( t ) = ∇ F f ( γ ( t )) and γ (0) = x . Theorem 4.1indicates that ˜ γ ( t ) = Ψ( γ ( t )) is a unit speed geodesic for the metric ˜ F with ˙˜ γ (0) =Ψ ∗ ( ∇ F f ( x )) = ∇ F f ( x ) + V ( x ). Since we have assumed F ( x, − V ( x )) <
1, the strongconvexity of F implies h∇ F f ( x ) + V ( x ) , ∇ F f ( x ) i F ∇ F f ( x ) >
0. So Ψ ∗ : T x M → T x M isan orientation preserving linear isomorphism for each x ∈ M .By Lemma 7.1, we can define the smooth function ˜ f locally around x , with the levelsets ˜ f − ( t ) = f M t = Ψ( M e ct − c ) = Ψ t ( M e ct − c ) . Let γ ( t ) be any integration curve of ∇ F f with γ (0) ∈ M sufficiently close to x and t sufficiently close to zero. Denote the points x = γ ( e ct − c ) ∈ M e ct − c and ˜ x = Ψ( x ) ∈ f M t . Notice that ˜ f (˜ x ) = t and f ( x ) = e ct − c .Theorem 4.1 provides a unit speed geodesic˜ γ ( t ) = Ψ( γ ( e ct − c )) = Ψ t ( γ ( e ct − c ))for the metric ˜ F , so˙˜ γ ( t ) = Ψ ∗ ( ddt γ ( e ct − c )) = Ψ ∗ ( e ct ˙ γ ( e ct − c )) = Ψ ∗ ((2 cf ( x ) + 1) ∇ F f ( x ))is a ˜ F -unit vector which is g ˜ F ˙˜ γ ( t ) -orthogonal to T ˜ x f M t . This implies ∇ ˜ F ˜ f (˜ x ) = ˙˜ γ ( t ) = Ψ ∗ ((2 cf ( x ) + 1) ∇ F f ( x )) . To summarize, we have the following lemma.
Lemma 7.2. If f is a normalized transnormal function around x for the metric F ,then ˜ f is a normalized transnormal function around x for the metric ˜ F , and ∇ ˜ F ˜ f =Ψ ∗ ((2 cf + 1) ∇ F f ) . Comparing dµ F BH ( x ) and dµ ˜ F BH (˜ x ), we get Lemma 7.3. Ψ ∗ ( dµ ˜ F BH ( x )) = (1 + c ( x ))(2 cf ( x ) + 1) − n − dµ F BH ( x ) , in which c ( · ) is asmooth function around x which is constant along each integration curve of ∇ F f . Proof.
Firstly, we have dµ ˜ F BH = dµ F BH by Lemma 3.4, so we only need to concern themetric F in the following discussion.We fix any value of t and consider the tangent map (Φ t ) ∗ for Φ t = Ψ − t ◦ Ψ at x = γ ( e ct − c ). The restriction of (Φ t ) ∗ to T x M e ct − c is the identity map, and (Φ t ) ∗ maps ∇ F f ( x ) = ˙ γ ( e ct − c ) to ˙ γ ( e ct − c ) + e − ct V ( x ). So by Lemma 3.3,Φ ∗ t ( dµ F BH ( x )) = det((Φ t ) ∗ | T x M ) · dµ F BH = h ˙ γ ( e ct − c ) + e − ct V ( x ) , ˙ γ ( e ct − c ) i F ˙ γ ( e ct − c ) · dµ F BH ( x )= (1 + e − ct h V ( x ) , ˙ γ ( e ct − c ) i F ˙ γ ( e ct − c ) ) · dµ F BH ( x )= (1 + e − ct ( c − c · e ct − c )) · dµ F BH ( x )= e − ct (1 + c ) · dµ F BH , Here det((Φ t ) ∗ | T x M ) is the determinant of the matrix for (Φ t ) ∗ | T x M with respect to any g F ˙ γ ( e ct − c ) -orthonormal basis { e = ˙ γ ( e ct − c ) , e , . . . , e n } . The constant c is providedby Lemma 3.1 which depends on the geodesic γ ( t ). So locally around x , we can denoteit as a smooth function c ( x ), which is constant along each integration curve of ∇ F f .By Lemma 3.2, Ψ ∗ t ( dµ F BH (˜ x )) = e − cnt dµ F BH ( x ), in which ˜ x = Ψ( x ). So we haveΨ ∗ ( dµ F BH (˜ x )) = Φ ∗ t Ψ ∗ t ( dµ F BH (˜ x )) = Φ ∗ t ( e − cnt dµ F BH ( x ))= e − c ( n +1) t (1 + c ( x )) · dµ F BH ( x )= (1 + c ( x ))(2 cf ( x ) + 1) − n − dµ F BH ( x ) . This ends the proof of Lemma 7.3.Summarizing above discussion, we can prove the following key lemma.
Lemma 7.4.
Let f be a normalized transnormal function around x , then Ψ ∗ ∆ ˜ F ˜ f = (2 cf + 1)∆ F f − cn. (7.27) OME GEOMETRIC CORRESPONDENCES FOR HOMOTHETIC NAVIGATION 17
Proof.
By Lemma 3.4, Lemma 7.2 and Lemma 7.3,Ψ ∗ ∆ ˜ F ˜ f = Ψ ∗ div dµ F BH (Ψ ∗ ((2 cf + 1) ∇ F f )) = div Ψ ∗ dµ F BH ((2 cf + 1) ∇ F f )= div ((1+ c ( x ))(2 cf +1) − n − dµ F BH ) ((2 cf + 1) ∇ F f )= L ((2 cf +1) ∇ F f ) ((1 + c ( x ))(2 cf + 1) − n − dµ F BH )(1 + c ( x ))(2 cf + 1) − n − dµ F BH = ( ∇ F f )(2 cf + 1) + (2 cf + 1) · L ∇ F f ((1 + c ( x ))(2 cf + 1) − n − dµ F BH )(1 + c ( x ))(2 cf + 1) − n − ) dµ F BH = 2 c + (2 cf + 1)∆ F f + (2 cf + 1)( ∇ F f )(ln(2 cf + 1) − n − ) + ln(1 + c ( x )))= (2 cf + 1)∆ F f − cn, (7.28)in which c ( x ) does not appear in the last line because ( ∇ F f )( c ( x )) ≡ f is isoparametric for ( F, dµ F BH ), ∆ F f is constant on each M t . ByLemma 7.4, ∆ ˜ F ˜ f is constant on each f M t , i.e., ˜ f is a normalized isoparametric functionaround x for ( ˜ F , dµ ˜ F BH ).To summarize, we have proved Theorem 7.5.
Let ˜ F be the Finsler metric defined by navigation from the datum ( F, V ) in which V is a homothetic vector field with dilation c = 0 . Assume x is a point where F ( x , − V ( x )) < . Then for any normalized isoparametric function f for ( F, dµ F BH ) around the point x , the function ˜ f defined by ˜ f − ( t ) = Ψ t ( f − ( e ct − c )) is a normalizedisoparametric function for ( ˜ F , dµ ˜ F BH ) around x . Nextly, we consider a normalized isoparametric function ˜ f for ( ˜ F , dµ ˜ F BH ) around x ∈ M . We can construct a smooth function f locally around x , such that ˜ f − ( t ) =Ψ t ( f − ( e ct − c )). By similar argument as for Lemma 7.2, we can prove f is a normalizedtransnormal function around x . Using Lemma 7.4 again, it is easy to see that when∆ ˜ F ˜ f is constant on each level set of ˜ f , ∆ F f is constant on each level set of f , i.e., f is anormalized isoparametric function for ( F, dµ F BH ). So Theorem 7.5 can be strengthenedas following. Theorem 7.6.
Keep all assumptions and notation in Theorem 7.5. Then we havea one-to-one correspondence from f to ˜ f between normalized isoparametric functionsaround x , with respect to ( F, dµ F BH ) and ( ˜ F , dµ ˜ FBH ) respectively. Above argument also works in the case that V is a Killing vector field. In this case,we only need to modify Ψ such that Ψ | M t = Ψ t , and make a few more minor changesaccordingly. The correspondence between normalized isoparametric functions around x is then from f to ˜ f = (Ψ − ) ∗ f . To avoid iteration, we skip the details. Proof of Theorem 1.5.
For any isoparametric hypersurface N for either F or ˜ F ,locally around x where F ( x , − V ( x )) <
1, we can find a normalized isoparametricfunction accordingly, such that N is the level set for the zero value. By Theorem 7.6and its Killing navigation version, N is also isoparametric for the other metric.Finally, we remark that Theorem 1.5 helps us find abundant examples of non-homogeneous isoparametric hypersurfaces in Finsler geometry. Let (
M, F ) be a Finsler manifold admitting the cohomogeneity one isometric actionof a connected Lie group G , such that each G -orbit is closed in M . Then principal G -orbits are homogeneous isoparametric hypersurface for ( F, dµ F BH ) [23]. Denote ˜ F the metric defined by navigation from the datum ( F, V ) in which V is a Killing orhomothetic vector field. Then the non-empty intersection between any principal G -orbit G · x and U = { x ∈ M | F ( x, − V ( x )) < } provides isoparametric hypersurfacesfor ( ˜ F , dµ ˜ F BH ). Generally speaking, the connected isometry group of ( M, ˜ F ) is smallerthan that of ( M, F ). So very likely, many homogeneous isoparametric hypersurfacesfor (
F, dµ F BH ) lose their homogeneity after the navigation. See Theorem 5.4 in [23] forthe case that ( M, ˜ F ) is a Randers sphere of constant flag curvature. Acknowledgements.
The first author is supported by National Natural ScienceFoundation of China (No. 11821101, No. 11771331), Beijing Natural Science Foundation(No. 00719210010001, No. 1182006), Research Cooperation Contract (No. Z180004),and Capacity Building for Sci-Tech Innovation – Fundamental Scientific Research Funds(No. KM201910028021). The second author is support by DFG (projects MA 2565/4and MA 2565/6). He would also like to thank Capital Normal University in BeijingChina for hospitality during the preparation of this paper. All the authors would liketo thank Qun He sincerely for her precious suggestions.
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E-mail address : [email protected] Vladimir MatveevInstitut f¨ur Mathematik, Fakult¨at f¨ur Mathematik und Informatik, Friedrich-Schiller-Universit¨at Jena, Germany
E-mail address : [email protected] Ke YanSchool of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P.R.China
E-mail address : Shaoxiang Zhang, the correspondence authorSchool of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P.R.China
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