The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry
Bernard Bonnard, Grégoire Charlot, Roberta Ghezzi, Gabriel Janin
aa r X i v : . [ m a t h . O C ] S e p THE SPHERE AND THE CUT LOCUS AT A TANGENCYPOINT IN TWO-DIMENSIONAL ALMOST-RIEMANNIANGEOMETRY
B. BONNARD, G. CHARLOT, R. GHEZZI, G. JANIN
Abstract.
We study the tangential case in 2-dimensional almost-Riemanniangeometry. We analyse the connection with the Martinet case in sub-Rieman-nian geometry. We compute estimations of the exponential map which allowus to describe the conjugate locus and the cut locus at a tangency point. Weprove that this last one generically accumulates at the tangency point as anasymmetric cusp whose branches are separated by the singular set. Introduction
In a series of recent papers [2,3,8], 2-dimensional almost-Riemannian geometryis investigated under generic conditions, giving rise to Gauss-Bonnet type resultson compact oriented surfaces.Roughly speaking, an almost-Riemannian structure (ARS for short) on an n -dimensional manifold M can be defined locally by the data of n vector fields playingthe role of an orthonormal basis. Where the vector fields are linearly independent,they define a Riemannian metric. But the structure is richer along the set Z wherethey are linearly dependent (see section 3 for a precise definition of ARS).For 2-dimensional ARS, it was proven in [2] that generically the singular set Z is an embedded submanifold of dimension 1 and only 3 types of points exist:the ordinary points where the metric is Riemannian, the Grushin points where thedistribution ∆ generated by the vector fields has dimension 1 and is transverse to Z , and the tangency points where ∆ has dimension 1 and is tangent to Z .The situation around ordinary and Grushin points is well known from the metricpoint of view, even if new considerations about curvature close to the Grushin pointsallow the authors to prove new results in [2,3,8].These metrics have also been studied in [5]. In that paper, the authors deducea global model on the two-sphere of revolution S as a deformation of the roundsphere, the metric being g λ = dϕ + G λ ( ϕ ) dθ , λ ∈ [0 , , with G λ ( X ) = X − λX , where X = sin ϕ , and ( ϕ, θ ) are the spherical coordinates.In this representation, the singularity is located at the equator: ϕ = π/
2. Thismetric appears in orbital transfer and, moreover, the homotopy is important tounderstand the behavior of the curvature. In this framework a short analysis tellsus that for the generic model the symmetries (of revolution and with respect to theequator) cannot be preserved and a non integrable model is obtained.
Mathematics Subject Classification.
Key words and phrases. almost-Riemannian geometry, conjugate and cut loci, sphere of smallradius.G. Janin was supported by DGA/D4S/MRIS, under the supervision of J. Blanc-Talon,DGA/D4S/MRIS, Responsable de Domaine Ing´enierie de l’Information. B. Bonnard and G. Char-lot were supported by the ANR project GCM.
In this paper we analyse the situation at tangency points. The presence of thesepoints is fundamental in the study of 2-dimensional ARS.In [3], the authors provide a classification of oriented ARS on compact orientedsurfaces in terms of the Euler number of the vector bundle corresponding to thestructure (see 2 for definition) in presence of tangency points, generalizing a result of[2]. The construction of Gauss Bonnet type formulae is more intricated in presenceof tangency points because of the geometry of the tubular neighborhoods of Z closeto the tangency points (see [3]).It happens that the geometry close to tangency points is not well known andmore intricated for many reasons. First, the computation of expansions of the wavefront is more complicated and involves elliptic functions. Second, the nilpotentapproximation is far from being generic as defined below. In particular the dis-tribution of the nilpotent approximation is transversal to its singular set at thetangency point.In this paper we focus on two points. First, we analyse the connection be-tween tangency points in 2-dimensional ARS and Martinet points in 3-dimensionalsub-Riemannian structures. This allows us to obtain regularity properties of thedistance function. Second, we compute the jets of the exponential map which al-lows to estimate the conjugate locus and the cut locus at the tangency point. Inparticular we prove that, differing from the nilpotent case, the cut locus generi-cally accumulates at the tangency point as an asymmetric cusp whose branches arelocally separated by the singular set Z (see figure 1). Figure 1.
The sphere (solid line) and the cut locus (dashed line) at atangency point in the generic case together with the singular set (dottedline)
The paper is organised as follows. In section 2 we recall some basic definitionsand results. In section 3 we show the relation between ARS and constant rank sub-Riemannian structures. In section 4 we analyse the special case of the nilpotentapproximation as well as a generic model at a tangency point. In section 5 wecompute the asymptotic expansions of the exponential map at a tangency point.This allows us to estimate, in section 6, the conjugate and cut loci at a tangencypoint, giving rise to a geometric interpretation of the first invariants in terms of theform of the cut locus. 2.
Basic definitions An n -dimensional ARS is the data of a triple ( M, E, f ) where M is an n-dimensional manifold, E is a Euclidean bundle of rank n over M and f is a morphism PHERE AND CUT LOCUS AT TANGENCY POINTS FOR 2-D ARS 3 of vector bundles between E and T M preserving the basis M , such that the eval-uation at any point q ∈ M of the Lie algebra generated by { f ◦ σ | σ section of E } is T q M .From the control theory point of view, an ARS can be defined locally by the dataof n vector fields ( F , . . . , F n ) such that Lie { F , . . . , F n } q = T q M for all q . Theydefine locally the following control dynamical system(21) ˙ q = n X i =1 u i F i ( q ) , n X i =1 u i = 1 , the distance between two points q and q being by definition the minimal timeneeded to join q from q with this control system. We also define the submodule∆ of the module of vector fields on M generated locally by ( F , . . . , F n ) and theflag ∆ k by ∆ = ∆, ∆ k +1 = ∆ k + [∆ , ∆ k ].In the following we deal with 2-dimensional ARS. Let us recall the followingresult proved in [2]. Proposition 1.
The following properties, denoted by (H0) , are generic for2-dimensional ARSs. (1)
The singular set Z is a one-dimensional embedded submanifold of M , (2) the points q ∈ M where ∆ ( q ) is one-dimensional are isolated, (3) ∆ ( q ) = T q M for all q ∈ M .Moreover, if a 2-dimensional ARS satisfies (H0) , then for every point q ∈ M thereexist a neighborhood U of q and an orthonormal frame ( F , F ) of the ARS on U such that, up to a change of coordinates, q = (0 , and ( F , F ) has one of theforms ( F F ( x, y ) = ∂∂x , F ( x, y ) = e φ ( x,y ) ∂∂y , ( F F ( x, y ) = ∂∂x , F ( x, y ) = xe φ ( x,y ) ∂∂y , ( F F ( x, y ) = ∂∂x , F ( x, y ) = ( y − x ψ ( x )) e ξ ( x,y ) ∂∂y , where φ , ψ and ξ are smooth functions such that φ (0 , y ) = 0 and ψ (0) > . Remark. In order to get the same notations as in [7], the normal form ( F
3) will bewritten in the following in coordinates ( y, z )( F F ( y, z ) = ∂∂y , F ( y, z ) = ( z − y ψ ( y )) e ξ ( y,z ) ∂∂z . For a 2-dimensional ARS satisfying (H0) , we say that a point q is • ordinary if ∆( q ) = T q M (normal form ( F • a Grushin point if the dimension of ∆( q ) is one and ∆ ( q ) = T q M (normalform ( F • a tangency point if the dimension of ∆( q ) = ∆ ( q ) is one and ∆ ( q ) = T q M (normal form ( F F y, z ) is a privileged coordinate system with weightsrespectively 1 and 3 (for definitions of privileged coordinates and nilpotent approx-imation we refer the reader to [4]).Consider the change of coordinates ˜ y = y and ˜ z = − z/ (2 ψ (0) e ξ (0) ). Let usstill denote (˜ y, ˜ z ) by ( y, z ). According to the weights the jet up to order 0 of theelements of the orthonormal frame in the normal form ( F
3) is( F F ( y, z ) = ∂∂y , F ( y, z ) = ( εz + y + ε ′ y + o ( y, z )) ∂∂z where ε = e ξ (0) = 0, ε ′ = ψ ′ (0)+ ψ (0) ∂ξ∂y (0)2 ψ (0) and o ( y, z ) is a smooth function of orderhigher than 3 in the variables y and z with respect to their weights. B. BONNARD, G. CHARLOT, R. GHEZZI, G. JANIN Almost-Riemannian geometry and sub-Riemannian geometry
Local desingularization of an n -dimensional ARS. Let us present a clas-sical construction. Consider an n -dimensional ARS on M and let ( F , . . . , F n ) be alocal orthonormal frame on a neighborhood of q . Assume that F i ( q ) = 0 for i > d where d = dim ∆( q ). Define f M = M × R n − d = { ( x, y ) | x ∈ M, y ∈ R n − d } . Denote by π and π the canonical projections on the first and second factor of f M .Then we can define e F i by π ∗ ( e F i ) = F i π ∗ ( e F i ) = 0 if i ≤ d,π ∗ ( e F i ) = ∂∂y i − d if i > d. Then the family { e F , . . . , e F n } has rank n in the neighborhood of q and defines theorthonormal frame of a sub-Riemannian metric on f M . Moreover, if ∆ is bracketgenerating as a submodule of the Lie algebra of vector fields on M , then the sameholds true for e ∆ = span { e F , . . . , e F n } . This metric on f M is invariant with respectto translations in R n − d . Moreover, one can show that the curves between q and q minimizing the almost-Riemannian distance on M are projections of the curvesbetween { q } × R n − d and { q } × R n − d minimizing the sub-Riemannian distanceon f M , a curve in f M and its projection having the same length. This implies thatthe ball centered at q of radius r in M is the projection of any sphere of radius r centered at a point of the type ( q, y ) in f M . Applying the Pontryagin MaximumPrinciple to the corresponding extremals in f M , the transversality conditions to { q } × R n − d and to { q } × R n − d must be satisfied.3.2. Examples.
The Grushin plane and the Heisenberg group.
The 2-dimensional ARS de-fined by the orthonormal frame n F = ∂∂x , F = x ∂∂y o in R is the Grushin plane. Itis the first example of almost-Riemannian structure with non empty singular locus,namely the y -axis. Moreover, it is the nilpotent approximation at any Grushinpoint of a 2-dimensional ARS (for a precise definition of nilpotent approxima-tion of a system we refer the reader to [4]). If we apply the desingularizationprocedure, we find the sub-Riemannian metric defined by the orthonormal frame n F = ∂∂x , F = x ∂∂y + ∂∂z o on R , which is the Heisenberg metric. The Heisen-berg metric is the nilpotent approximation at any point of contact of a rank-2 sub-Riemannian structure defined on a 3-dimensional manifold, that is at any point p where the rank-2 distribution satisfies [∆ , ∆]( p ) = T p M . The Hamiltonian associ-ated with the Grushin metric is H = 12 ( p x + x p y )while the one related to the Heisenberg metric is H = 12 ( p x + ( xp y + p z ) )where p x , p y , p z are the dual coordinates to x , y and z in the cotangent bundle.Geodesics of the Heisenberg group projecting to geodesics of Grushin are thosewith p z = 0, respecting the transversality condition to the vertical lines given bythe Pontryagin Maximum Principle.In general, the relation between the cut and conjugate loci for the sub-Rie-mannian metric on f M and almost-Riemannian one on M is not clear, the projection PHERE AND CUT LOCUS AT TANGENCY POINTS FOR 2-D ARS 5 π introducing singularities. The Grushin plane is a good illustration of this fact.The two geodesics for the Grushin metric starting at (0 ,
0) with the initial covectors( p x = 1 , p y ) and ( p x = − , p y ) are x ( t ) = p x sin( p y t ) p y ,y ( t ) = 2 p y t − sin(2 p y t )(2 p y ) . Hence, these two geodesics first intersect for p y ¯ t = π and one can prove that ¯ t corresponds to the cut time along them. Moreover, computing the Jacobian of theexponential mapping, one proves that the conjugate time ˜ t satisfies p y ˜ t = tan( p y ˜ t ).Lifting to the corresponding geodesics in the Heisenberg space starting at (0 , , p x , p y , p z = 0), one finds as third coordinate z ( t ) = p x − cos( p y t ) p y . Hence, the two lifted geodesics do not intersect anymore at ¯ t and the computationof the Jacobian of the exponential mapping shows that the conjugate time for bothlifted curves satisfies p y ˜ t = tan( p y ˜ t ). It corresponds to the second conjugate timein the Heisenberg case.3.2.2. The nilpotent approximation at a tangency point and the Martinet flat case.
Consider the normal form ( F
3) at a tangency point as presented in section 2.Recall that the weight of the variable y is 1 and the weight of z is 3. Hence thenilpotent approximation can be given in ( y, z ) coordinates by the orthonormal frame n ∂∂y , y ∂∂z o , corresponding to the metric g = dy + ( y ) − dz and the Hamiltonian H = 12 ( p y + y p z ) . Applying the desingularization procedure, one finds the orthonormal frame (cid:26) ∂∂y , y ∂∂z + ∂∂x (cid:27) in ( y, z, x ) coordinates in R , whose corresponding Hamiltonian is H = 12 ( p y + ( p x + y p z ) ) . This lifted structure is the Martinet flat case of sub-Riemannian geometry. It isthe nilpotent approximation at any Martinet point of a rank-2 sub-Riemannianstructure defined on a 3-dimensional manifold, that is at any point p where therank-2 distribution satisfies [∆ , ∆]( p ) = ∆( p ) and [[∆ , ∆] , ∆]( p ) = T p M . The setof these points is called Martinet surface. Being the nilpotent approximation, theMartinet flat case will provide the starting point to analyse the general tangentialcase and it will allow to make some preliminary estimates of the sphere and thedistance function using previous computations as in [7].4. Local analysis at a tangency point
In this section we focus on the following models in order to study the localsituation around tangency points for a generic 2-dimensional ARS.(1)
The nilpotent approximation (of order -1) g − = dy + ( y − dz (2) The generic model of order 0 g = dy + ( εz + y ε ′ y ) − dz where ε = ε ′ = 0 gives the nilpotent approximation. Analysis of the nilpotent model.
In this case, the desingularization pro-cedure gives the orthonormal frame on R F = ∂∂x + y ∂∂z , F = ∂∂y which generates the distribution∆ = ker( dz − y dx ) . Proposition 2.
Consider the almost-Riemannian metric g = dy + y dz on R . The y -axis is the union of the two geodesics starting at the origin with initialcovectors ( p y = ± , p z = 0) . The geodesics with initial covector ( p y = ± , p z = λ =0) are given by y ( t ) = − p y √ p | λ | cn ( K + t p | λ | ) ,z ( t ) = sign( λ )3 | λ | / [ t p | λ | + 2sn ( K + t p | λ | )cn ( K + t p | λ | )dn ( K + t p | λ | )] , where K is the complete elliptic integral of the first kind Z π/ dϕ q − / ϕ and cn , sn , dn denote the Jacobi elliptic functions of modulus k = √ . Moreoverthe following properties hold true. (1) The almost-Riemannian spheres centered at the origin are subanalytic. (2)
For λ = 0 , the cut point coincides with the first return to the z -axis thatoccurs at t = 2 K/ p | λ | , where two extremals with the same length intersect.The cut locus from the origin is { ( y, z ) | y = 0 } \ { (0 , } . (3) For λ = 0 , the conjugate point corresponds to t ∼ K/ p | λ | . The conjugatelocus from the origin accumulates at the origin as a set of the form { ( y, z ) | z = αy } ∪ { ( y, z ) | z = − αy } \ { (0 , } , with α = 0 . Proof.
Define F = ∂∂z and P i = < p, F i ( q ) > , i = 1 , ,
3. Using the PontryaginMaximum Principle, the equations for normal extremals of the sub-Riemannianstructure are given by the Hamiltonian system associated with H = ( P + P ),i.e. ˙ x = P , ˙ P = yP P , ˙ y = P , ˙ P = − yP P , ˙ z = y P , ˙ P = 0 . There are three first integrals, namely p x , p z = λ, H . The normalization condition H = 1 / t = 0 gives P (0) + P (0) = 1 , hence, we set P (0) = sin ϕ , P (0) = cos ϕ . The set of extremals is invariant underthe action of the group generated by the diffeomorphisms ( x, y, z ) ( x, − y, z )and ( x, y, z ) ( − x, y, − z ). Therefore, it is sufficient to integrate the system withinitial point (0 , ,
0) and covector (sin ϕ, cos ϕ, λ ) with λ ≥ ϕ ≥
0. Recallthat the normal extremals for the almost-Riemannian structures are projections on
PHERE AND CUT LOCUS AT TANGENCY POINTS FOR 2-D ARS 7 the ( y, z ) coordinates of the geodesics for the sub-Riemannian metric satisfying thetransversality condition p x = 0. Hence, for λ = 0 we get y ( t ) = cos( ϕ ) t, z ( t ) ≡ λ > k, k ′ such that k = − sin ϕ , 0 < k, k ′ < k + k ′ =1. Then we find˙ y = (1 − P )(1 + P ) = (1 − p x − y p z )(1 + p x + y p z )= (2 k − y λ )(2 k ′ + y λ ) . Setting y( t ) = √ λ k y ( t ), the evolution equation for y is˙y λ = (1 − y )( k ′ + k y ) , that can be integrated, with ˙y(0) >
0, as y( t ) = − cn ( K ( k ) + t √ λ, k ), where K ( k ) = Z π/ dϕ p − k sin ϕ . Hence y ( t ) = − k √ λ cn ( K ( k ) + t √ λ, k ) . Remark that the extremals that project on geodesics for the ARS satisfy thetransversality condition p x = 0 which implies k = 1 /
2. Thus the y coordinateof the geodesic with p x = 0 , p y = 1 , λ > y ( t ) = − √ √ λ cn ( K + t √ λ ) , where, to simplify notations, we denote K ( √ /
2) by K and we omit the dependenceof the Jacobi function cn on the modulus. To compute the z coordinate along thesame geodesic, we use the primitive R cn ( K + u ) du = [ u + sn ( K + u ) cn ( K + u ) dn ( K + u )] (where k = √ / z ( t ) = 13 λ / [ t √ λ + 2 sn ( K + t √ λ ) cn ( K + t √ λ ) dn ( K + t √ λ )] . Using the symmetries of the system we find the required expressions for the geodesicsstarting at the origin for the almost-Riemannian metric. In particular, y and z arequasi-homogeneous with respective weights 1 and 3. The cut instant of a geodesiccoincides with the first return to y = 0 that occurs at t = 2 K/ √ λ , thus the cutlocus is the z -axis. The conjugate time satisfies t ∼ K/ √ λ , whence the conjugatelocus can be approximated by the parametric curve y = − √ λ / , z = Kλ / (for the detailed proof see [1]). (cid:4) Remark that for the desingularized structure, i.e., the sub-Riemannian Martinetflat case, the sub-analyticity of the sphere is lost in the abnormal direction for which k →
1. This does not arise for the almost-Riemannian structure, since geodesicssatisfy k = 1 / B. BONNARD, G. CHARLOT, R. GHEZZI, G. JANIN
Analysis of the generic model of order 0.
The objective of this sectionis to lift the generic model of order 0 into a constant rank sub-Riemannian modelin order to analyse the role of the invariants in the optimal dynamics. A geometricinterpretation will be given in section 6 in terms of the form of the cut locus.Recall that from [7] the sub-Riemannian Martinet model of order zero is nor-malized to (1 + αy ) dx + (1 + βx + γy ) dy , where the distribution has the standard Martinet form2 dz = y dx. In this normal form the parameters α, β, γ are related to the geometric propertiesof the sphere with small radius and appear in the pendulum interpretation of theextremals. More precisely, for β = 0 the extremal system is integrable while if β isnon zero we have dissipation. In the integrable case the important parameter is α and if it is non zero the abnormal direction is strict. The role of the parameter γ is unimportant and it can be absorbed by reparameterization.Consider the almost-Riemannian metric on R given by the orthonormal frame(42) F = ( εz + y / ε ′ y ) ∂∂z , F = ∂∂y , where ε = 0. This metric can be seen as the generic model of order 0 for an ARSin a neighborhood of a tangency point (use the normal form ( F
3) and weights ofcoordinates). Next proposition gives a possible lifting of the model of order 0 ata tangency point for an almost-Riemannian metric, showing the relation with themodel of order 0 of a Martinet type distribution for a sub-Riemannian metric.
Proposition 3.
The generic model of order for an ARS in a neighborhoodof a tangency point lifts into the sub-Riemannian Martinet model of order zero dx ( ε (1+ x )) + dy (1+2 ε ′ y + o ( y )) on the distribution dz − y dx = 0 . Proof.
Applying the desingularization procedure (see section 3.2.2) to thealmost-Riemannian metric defined by (42), we get the sub-Riemannian metric in R defined by the orthonormal frame, still denoted by F , F , F = ∂∂x + ( εz + y ε ′ y ) ∂∂z , F = ∂∂y . One gets [ F , F ] = − y (1 + 3 ε ′ y ) ∂∂z , [[ F , F ] , F ] = (1 + 6 ε ′ y ) ∂∂z , [[ F , F ] , F ] = εy (1 + 3 ε ′ y ) ∂∂z , hence the Martinet surface is the set { ( x, y, z ) | y = 0 } . Moreover the singularcontrol in the Martinet surface is defined by u det( F , F , [[ F , F ] , F ]) + u det( F , F , [[ F , F ] , F ]) = 0which implies u = 0. The corresponding trajectories are solutions of˙ x = u , ˙ y = 0 , ˙ z = u εz. In order to build a coordinate system (˜ x, ˜ y, ˜ z ) in which the distribution has thenormal form D = ker ω , ω = d ˜ z − ˜ y / d ˜ x , we normalize the singular flow to PHERE AND CUT LOCUS AT TANGENCY POINTS FOR 2-D ARS 9 lines parallel to the ˜ x -axis and lying in the Martinet surface. We consider thediffeomorphism ˜ x = e − εx − ε − , ˜ y = y p ε ′ y = y + ε ′ y + o ( y ) , ˜ z = ze − εx . The orthonormal frame in the new coordinate system becomes F = − ε (1 + ˜ x ) ∂∂ ˜ x − ε (1 + ˜ x ) ˜ y ∂∂ ˜ z , F = (1 + 2 ε ′ ˜ y + o (˜ y )) ∂∂ ˜ y . Hence the distribution is in the normal form d ˜ z = ˜ y d ˜ x and the metric is given by g = d ˜ x ε (1 + ˜ x ) + d ˜ y (1 + 2 ε ′ ˜ y + o (˜ y )) . (cid:4) Introducing F = ∂∂z and P i = < p, F i > , the extremal flow is given by˙ X = − ε (1 + X ) P , ˙ Y = (1 + 2 ε ′ Y + o ( Y )) P , ˙ Z = − ε (1 + X ) Y P , ˙ P = − ε (1 + X ) Y (1 + 2 ε ′ Y + o ( Y )) P P , ˙ P = ε (1 + X ) Y (1 + 2 ε ′ Y + o ( Y )) P P , ˙ P = 0 . Setting P = λ and using the time parameter τ such that dτ = (1 + X ) dt , we canwrite dXdτ = − εP ,dYdτ = (1 + 2 ε ′ Y + o ( Y ))1 + X P ,dZdτ = − ε Y P ,dP dτ = − λεY (1 + 2 ε ′ Y + o ( Y )) P ,dP dτ = λεY (1 + 2 ε ′ Y + o ( Y )) P . Define θ in R / π Z by P = cos( θ ) and P = sin( θ ). It satisfies dθdτ = λεY (1 + 2 ε ′ Y + o ( Y ))and then d θdτ = λε ε ′ Y + o ( Y )1 + X sin( θ ) , which can be approximated by d θdτ = λε (1 − X + 4 ε ′ Y + o ( Y )) sin( θ ) . According to [7], this corresponds to a dissipative pendulum, the non nullity of theparameter ε ′ inducing a coupling with the y -coordinate. Note that more computa-tions are necessary to get the sub-Riemannian Martinet metric in the normal formof order 0, leading to a true dissipative pendulum equation with no coupling withthe x and y variables, see [7].5. Asymptotics of the wave front
In this section we use the techniques and results from [7], developped in the sub-Riemannian Martinet case, to compute asymptotics of the front from the tangencypoint for the generic model of order 0 for ARS. Remark that the higher orderterms in the expansion of the elements of the orthonormal frame play no role in theestimation of the front and, consequently, in the estimation of the cut and conjugateloci from the tangency point (see section 6), as one can check easily.
Proposition 4.
Consider the ARS on R defined by the orthonormal framegiven in (42). The extremals satisfying initial condition ( y, z, p y , p z ) | t =0 = (0 , , ± , λ ) with | λ | ∼ + ∞ can be expanded as y ( t ) = ηY ( t/η ) + η Y ( t/η ) + o ( η ) ,z ( t ) = η Z ( t/η ) + η Z ( t/η ) + o ( η ) , where η = √ | λ | , ˙ Y = P Y , ˙ P Y = − ( P Z ) ( Y ) , (53) ˙ Z = P Z ( Y ) , ˙ P Z = 0 , with initial condition ( Y , Z , P Y , P Z ) | t =0 = (0 , , ± , ± and ˙ Y = P Y , (54) ˙ Z = 14 P Z ( Y ) + P Z (( Y ) Y + εZ ( Y ) + ε ′ ( Y ) ) , ˙ P Y = − P Z P Z ( Y ) − ( P Z ) ( 32 ( Y ) Y + εZ Y + 52 ε ′ ( Y ) ) , ˙ P Z = −
12 ( P Z ) ε ( Y ) , with initial condition ( Y , Z , P Y , P Z ) | t =0 = (0 , , , . Remark. Computations in this case are similar to the ones of the Martinet sub-Riemannian case. System (54) represents a variational equation whose integrationis related to the second-order equation¨ Y + ( 32 P Z Y ) Y = K ( Y )where Y is a periodic elliptic function. Proof.
Recall that y, z have weight 1 and 3, respectively. In order to have thestandard Darboux form of order 1, we fix the weight 0 for p y and − p z .The Hamiltonian is H = 12 ( p z ( εz + y / ε ′ y ) + p y )and the extremal flow is˙ y = p y , ˙ p y = − p z ( εz + y / ε ′ y )( y + 3 ε ′ y ) , ˙ z = p z ( εz + y / ε ′ y ) , ˙ p z = − p z ( εz + y / ε ′ y ) ε. PHERE AND CUT LOCUS AT TANGENCY POINTS FOR 2-D ARS 11
According to the weights we set y = ηY, p y = P Y ,z = η Z, p z = P Z η , where η is a parameter. The evolution equations for ( Y, Z, P Y , P Z ) are˙ Y = P Y η , ˙ Z = P Z ( Y η + εZY + ε ′ Y + ε ′ ηY + 2 εε ′ ηZY + ε ηZ ) , ˙ P Y = − P Z ( Y η + εZY + 52 ε ′ Y + 3 ηε ′ Y ( εZ + ε ′ Y )) , ˙ P Z = − P Z ε ( Y εηZ + ε ′ ηY ) . Considering the expansions with respect to ηY = Y + ηY + o ( η ) , P Y = P Y + ηP Y + o ( η ) ,Z = Z + ηZ + o ( η ) , P Z = P Z + ηP Z + o ( η ) , by identification we find that the leading terms satisfy(55) ˙ Y = P Y η , ˙ P Y = − ( P Z ) ( Y ) η , ˙ Z = P Z ( Y ) η , ˙ P Z = 0 . In particular P Z is constant. Setting λ = p z (0), for λ = 0 we can fix η = 1 / p | λ | and then P Z is normalized to 1 or -1.Introducing the time parameter s = t p | λ | the equations (55) for the first-orderapproximation become(56) dY ds = P Y , dP Y ds = − ( P Z ) ( Y ) , dZ ds = P Z ( Y ) , P Z ≡ ± . System (56) coincides with the Hamiltonian system for the nilpotent model thathas been integrated in Proposition 2, using elliptic functions with modulus k suchthat k = 1 /
2. The solution is given in Proposition 2. Y ( s ) = − P Y (0) √ K + s ) ,Z ( s ) = P Z s + 2sn ( K + s )cn ( K + s )dn ( K + s )) ,P Y ( s ) = P Y (0) + ( P Y (0)) ( − √ K + s )sn ( K + s )) ,P Z ( s ) ≡ ± . Using s = t/η , the system for ( Y, Z, P Y , P Z ) becomes dYds = P Y ,dZds = P Z ( Y η ( εZY + ε ′ Y ) + η ( ε ′ Y + 2 εε ′ ZY + ε Z )) ,dP Y ds = − P Z ( Y η ( εZY + 52 ε ′ Y ) + 3 η ε ′ Y ( εZ + ε ′ Y )) ,dP Z ds = − P Z ε ( η Y η ( εZ + ε ′ Y )) . Hence, identifying terms of order 0, one gets dY ds = P Y ,dZ ds = 14 P Z ( Y ) + P Z (( Y ) Y + εZ ( Y ) + ε ′ ( Y ) ) ,dP Y ds = − P Z P Z ( Y ) − ( P Z ) ( 32 ( Y ) Y + εZ Y + 52 ε ′ ( Y ) ) ,dP Z ds = − P Z εY . (cid:4) Geometric estimates of the conjugate and cut loci
The conjugate locus.
The following result gives a description of the conju-gate locus from a tangency point of a 2-dimensional ARS.
Proposition 5.
Consider an ARS on R defined by the orthonormal frame F = ( εz + y / ε ′ y + o ( y, z )) ∂∂z , F = ∂∂y . Then there exists a constant α = 0 such that the conjugate locus from (0 , accu-mulates at (0 , as the set { ( y, z ) | z = αy } ∪ { ( y, z ) | z = − αy } \ { (0 , } . Proof.
Applying Proposition 4, the exponential map at (0 ,
0) is given by( η, s ) ( ηY ( s ) + o ( η ) , η Z ( s ) + o ( η )) , where s = t p p z (0), η parametrizes the initial covector as ( p y (0) = ± , p z (0) = P Z /η ), and Y ( s ) = − P Y (0) √ K + s ) ,Z ( s ) = P Z ( s + 2sn ( K + s )cn ( K + s )dn ( K + s )) . The conjugate time is the first zero of the Jacobian of the exponential map. TheJacobian is equal, up to a multiplicative constant, to η ( Y d Z d s − Z d Y d s ) + o ( η ) . It was proven in [7] that the function j ( s ) = Y ( s ) d Z d s − Z d Y d s has its first positive zero at s = s ∼ K and that j ′ ( s ) = 0. Hence, the conjugatetime is of the form s + o (1) where o (1) is a continuous map going to zero when η goes to zero.In terms of the singularity theory, this computation proves that the exponentialmap of a general two-dimensional ARS can be seen as a small deformation ofthe exponential map of the nilpotent case. In the nilpotent case, the exponentialmap has only stable singularities (folds) corresponding to the first conjugate locus.Hence, in the general case, the first conjugate locus also corresponds to folds andaccumulates at (0 ,
0) as the set { ( y, z ) | ( z − αy )( z + αy ) = 0 } , where α = 0,see Proposition 2. (cid:4) PHERE AND CUT LOCUS AT TANGENCY POINTS FOR 2-D ARS 13
The cut locus.
In this section we provide a description of the cut locus at atangency point for a generic ARS. As one can infer from the proof of the followingproposition, the shape of the cut locus is determined only by the terms of orderup to 0 in the expansion of the elements of the orthonormal frame. Higher orderterms do not contribute to the estimation of the way the cut locus approaches tothe tangency point. One can see in figure 2 small spheres for different values of ε and ε ′ . Figure 2.
The sphere of small radius for the nilpotent approxi-mation (dotted line) and for an example with ǫ ′ = 0 (dashed line)are symmetric. The two spheres are not C at their intersectionwith the cut locus, which in both cases is the vertical axis. Thesphere of small radius for the generic model of order zero in which ε ′ = 0 (solid line) loses the symmetry. In this case, the cut locusis different from the previous cases (see Proposition 6). Proposition 6.
Consider the ARS on R defined by the orthonormal frame F = ( εz + y / ε ′ y + o ( y, z )) ∂∂z , F = ∂∂y . Then, if ε ′ = 0 , there exist non zero constants α , α such that the cut locus from (0 , accumulates at (0 , as the set { ( y, z ) | z > , z − α y = 0 } ∪ { ( y, z ) | z < , z − α y = 0 } . Proof.
In the following, we restrict our analysis to the upper half plane z > z <
0. First of all, recall that for thenilpotent model ( ε = ε ′ = 0), the cut locus is { ( y, z ) | y = 0 } \ { (0 , } and the cuttime on the geodesic with initial covector (1 , λ ) is 2 K/ √ λ which corresponds to thefirst intersection with the symmetric geodesic whose initial covector is ( − , λ ), seeProposition 2. Denote by ( Y , Z , P Y , P Z ) the geodesic with initial condition (0 , , ,
1) for theARS with ε = ε ′ = 0, i.e., Y ( s ) = −√ K + s ) ,Z ( s ) = 13 ( s + 2 sn ( K + s ) cn ( K + s ) dn ( K + s )) ,P Y ( s ) = √ K + s ) sn ( K + s ) ,P Z ≡ . Moreover, denote by Y , Z , P Y , P Z the terms of order 0 in the expansion of thegeodesic for the ARS with ε ′ = 0, i.e., solutions of the system (54) with ε ′ = 0 andinitial condition (0 , , , Y , Z , P Y , P Z ) to be the terms of order 0 in theexpansion of the geodesic for the ARS with ε ′ = 0, i.e., solutions of system (54)with the initial condition (0 , , , s , g , g , g and g , by Y = Y + ε ′ g , P Y = P Y + ε ′ g , Z = Z + ε ′ g , P Z = P Z + ε ′ g . Combining the equations satisfied by Y , Z , P Y , P Z and by Y , Z , P Y , P Z ,we find that ( g , g , g , g ) satisfy the following system of ODEs˙ g = g ˙ g = −
32 ( Y ) g −
52 ( Y ) ˙ g = g ( Y ) + ( Y ) g ≡ , where ˙ g i = dg i /ds , with the initial conditions g (0) = g (0) = g (0) = 0.Remark that if ( Y , Z , P Y , P Z , Y , Z , P Y , P Z , g , g , g )is solution of ((53),(54),(67)) with the initial condition (0 , , , , , , , , , , − Y , Z , − P Y , P Z , − Y , Z , − P Y , P Z , g , − g , g )is also solution with the initial condition (0 , , − , , , , , , , , g (2 K ) ∼ − π , g (2 K ) ∼ − π and g (2 K ) ∼ t = 2 Kη close to the initial condition η , thatis for η = η + cη + o ( η ). Making Taylor expansions in terms of η , one finds forthe front corresponding to the initial conditions p y (0) = 1 and p z (0) = 1 /η y = ηY ( Kη η ) + η ( Y (2 K ) + ε ′ g (2 K )) + o ( η ) ,z = η Z ( Kη η ) + η ( Z (2 K ) + ε ′ g (2 K )) + o ( η ) . Hence we obtain y = η ( Y (2 K ) + ε ′ g (2 K ) + 2 Kc ) + o ( η ) ,z = η Z (2 K ) + η ( Z (2 K ) + ε ′ g (2 K ) + 3 cZ (2 K )) + o ( η ) , since Y (2 K ) = ˙ Z (2 K ) = 0 and ˙ Y (2 K ) = −
1. For the front corresponding tothe initial conditions p y (0) = − p z = 1 / ¯ η where ¯ η = η + c ′ η + o ( η ) onefinds y = η ( − Y (2 K ) + ε ′ g (2 K ) − Kc ′ ) + o ( η ) ,z = η Z (2 K ) + η ( Z (2 K ) − ε ′ g (2 K ) + 3 c ′ Z (2 K )) + o ( η ) . These expressions are affine with respect to parameters c and c ′ , up to order 2for y and 4 for z in the variable η . The two geodesics with initial covectors p y (0) = 1 , p z (0) = 1 /η and p y (0) = − , p z (0) = 1 / ¯ η intersect for c + c ′ = − Y (2 K ) K + o (1) ,c ′ − c = ε ′ g (2 K ) K + o (1) PHERE AND CUT LOCUS AT TANGENCY POINTS FOR 2-D ARS 15 where o (1) denotes any function going to 0 with η . Hence the intersection is for c = − ε ′ g (2 K ) + Y (2 K )2 K + o (1)and c ′ = ε ′ g (2 K ) − Y (2 K )2 K + o (1) , which implies that the cut point is y cut = η ε ′ ( g (2 K ) − g (2 K )) + o ( η ) ,z cut = η
30 2 K + o ( η ) . Hence, if ε ′ = 0, the upper branch of the cut locus from (0 ,
0) accumulates as theset { ( y, z ) | z > , z = α y } , where α = 4 K ε ′ ( g (2 K ) − g (2 K )) ∼ − K ε ′ π . Similar computations show that the lower branch of the cut locus from (0 ,
0) accu-mulates as the set { ( y, z ) | z < , z = α y } , where α = 4 K ε ′ ( g (2 K ) + g (2 K )) ∼ − K ε ′ π . (cid:4) Remark that the case of generic ARS with ε ′ = 0 is rather different from thesub-Riemannian Martinet case (see [7]), in which a similar argument cannot apply.Indeed, in the latter situation, the asymptotic expansions for small time cannotbe used, since there exists an abnormal direction corresponding to the case where k → K ( k ) → ∞ . Hence, we should use the asymptotic expansion for a timeparameter tending to + ∞ , which is clearly not valid. References
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