DDistortion coefficients of the α -Grushin plane Samuël Borza
Department of Mathematical Sciences, Durham University
November 30, 2020
Abstract
We compute the distortion coefficients of the α -Grushin plane. They are ex-pressed in terms of generalised trigonometric functions. Estimates for the dis-tortion coefficients are then obtained and a conjecture of a curvature-dimensioncondition for the generalised Grushin planes is suggested. Keywords—
Grushin plane, Sub-Riemannian geometry, Distortion coefficients, Syn-thetic curvature
MSC (2010)—
Grushin structures first appeared in the work of Grushin on hypoelliptic operators inthe seventies, see [Gru70] for example. The α -Grushin plane, denoted by G α , consistsof equipping the 2-dimensional Euclidean space with the metric generated by thevector fields X = ∂ x , Y α = | x | α ∂ y .These structures form a class of rank-varying sub-Riemannian manifolds, in aprecise way that will be explained in Section 2.2. In this work, we will focus onthe case α (cid:62)
1. The α -Grushin plane has Hausdorff dimension α + α is an integer. Furhtermore, they constitute a natural gen-eralisation of the traditional Grushin plane, corresponding to the case α =
1. Alongwith the Heisenberg groups H n , they are considered as fundamental examples ofsub-Riemannian geometry, exhibiting key characteristics of the theory.Since the work initiated by Juillet in [Jui09] and [Jui20], it is known that, unlikeRiemannian manifolds, no (ideal) sub-Riemannian manifold satisfies the curvature-dimension condition introduced by Sturm, Lott and Villani. However, it has beenshown by Barilari and Rizzi in [BR19] that they do support interpolation inequalitiesand even a Brunn-Minkowski inequality. For the Heisenberg group, this was firstestablished by Balogh, Kristaly and Sipos in [BKS18]. Distortion coefficients, whichcapture some curvature information, play a key role in these results. In this work,distortion coefficients for the α -Grushin plane are established.A fine study of the geodesics of G α will be important in this work. Because of thelack of a natural connection in sub-Riemannian geometry, geodesics are obtained withthe Pontryagin Maximum Principle. This is the Hamiltonian point of view : a minimizing1 a r X i v : . [ m a t h . M G ] N ov ath between two points can be lifted to one on the cotangent bundle that satisfiesHamilton’s equations.The geodesics of the α -Grushin plane were first studied by Li and Chang in [CL12].They are expressed with a generalisation of trigonometric functions, defined as in-verses of special functions. Section 2.3 and 3.1 are devoted to these topics while inSection 3.2, we use an extended Hadamard technique to find the cut locus of thegeodesics. In what follows, the notation Cut ( q ) stands for the set of cut locci of q ,i.e. the set of points in G α where the geodesics starting at q stop being minimizing. Theorem 1 (Distortion coefficients of the α -Grushin plane) . Let q and q be two pointsof G α such that q / ∈ Cut ( q ) . We denote by ( x ( t ) , y ( t )) : [
0, 1 ] → G α the unique constantspeed minimizing geodesic joining q to q and by ( u ( t ) , v ( t )) ∈ T ∗ ( x ( t ) , y ( t )) ( G α ) its lift. Forall t ∈ [
0, 1 ] , we have β t ( q , q ) = J ( t , x , u , v ) J ( x , u , v ) where J ( t , x , u , v ) = x ( t ) (cid:104) t α u (cid:16) u + v x α (cid:17) (cid:16) ( α − ) u + α ( α + ) v x α (cid:17) (1) +( α − ) u x (cid:105) − u ( t ) (cid:104) t α u (cid:16) u + v x α (cid:17) (cid:16) ( α − ) u + (cid:16) α − α + (cid:17) v x α (cid:17) + tx (cid:16)(cid:16) α − α + (cid:17) u + α ( α + ) u v x α + α ( α + ) v x α (cid:17) +( α − ) u x (cid:105) − x ( t ) u ( t ) ( α − ) (cid:104) tu (cid:16) u + α v x α (cid:17) + u x (cid:105) − u ( t ) x ( t ) ( α − ) u .Because of the analyticity of the geodesic flow, the case v = v →
0. Geometrically, this means that the points q and q are joinedby a straight horizontal line. This limit is performed in Proposition 16.Although the CD condition is not suited for this type of spaces, the weaker mea-sure contraction property introduced independently by Ohta and Sturm (in [Oht07]and [Stu06b]) seems more adapted to sub-Riemannian geometry. Indeed, there are nu-merous examples of sub-Riemannian manifolds that do satisfy a MCP-condition, in-cluding the Heisenberg group H n (cf. [Jui09]) and the Grushin plane G (see [BR19]).We therefore investigate the MCP-condition for G α and we obtain a relevant estimateon the distortion coefficients for singular points, that is to say, those on the y -axis, andfor those lying on the same horizontal line. In light of this, we suggest the followingconjecture. Conjecture 2 (Curvature dimension of the α -Grushin plane) . For α (cid:62) , the α -Grushinplane satisfies the synthetic curvature-dimension condition MCP ( K , N ) if and only if K (cid:54) and N (cid:62) (cid:20) ( α + ) m + m + (cid:21) with m ∈ [ − − ] the unique non zero solution of ( m + ) α ( m + ) − (( α + ) m + ) = x α for ( x ) α . We will provide ev-idence in favour of this conjecture in Section 4.2. As we will see, the MCP ( N ) con-dition is related to a lower bound for the distortion coefficients of the form β t ( q , q ) (cid:62) t N . It will be proven that the lower bound holds for singular points. Furthermore, itis sharp for the points lying on the same horizontal lines. Acknowledgements
I would like to thank Prof Wilhelm Klingenberg for his supervision and precious en-couragements. I am also grateful to Prof Nicolas Juillet, Prof Alpár Mészáros and ProfAlexandru Kristály for carefully reading the manuscript and for providing invaluablefeedback. I benefited greatly from instructive discussions with Prof Shingo Takeuchion the topic of generalised trigonometry. I am thankful to Prof Li Yutian for intro-ducing me to the study of the α -Grushin plane and for his constant support. Thiswork was supported by the UK Engineering and Physical Sciences Research Council(EPSRC) grant EP/N509462/1 (1888382) for Durham University. In this section, we give an overview of metric geometry, synthetic notions of curvatureand distortion coefficients. A metric space ( X , d ) is a length space if the distanceis induced from a length structure; that is to say, d ( x , y ) : = inf { L ( γ ) | γ : [ a , b ] → X is admissible, γ ( a ) = x and γ ( b ) = y } where L : A → R ∩ { + ∞ } is a lengthfunctional on a complete set of admissible paths A ⊆ C ( X ) . A minimal geodesic is anadmissible path γ : [ a , b ] → X in A such that d ( x , y ) = L ( γ ) . We refer to [BBI01] formore on metric geometry.If the space has the property that every two points can be joined by a minimalgeodesic, we will say that ( X , d ) is a geodesic space. We equip the metric space ( X , d ) with a Radon measure m . The structure ( X , d, m ) is called a geodesic metric measurespace. The notion of distortion coefficients fits into this context. Definition 3.
Let x , y ∈ X. The distortion coefficient from x to y at time t ∈ [
0, 1 ] is β t ( x , y ) = lim sup r → + m ( Z t ( x , B r ( y ))) m ( B r ( y )) where Z t ( x , B r ( y )) stands for the set of t-intermediate points from x to the ball centred in y ofradius r; Z t ( A , B ) : = { γ ( t ) | γ ∈ Geo ( X ) , γ ( ) ∈ A and γ ( ) ∈ B } whenever A and B are m -measurable subsets of X. There is an intuitive physical interpretation of the distortion coefficients: β t ( x , y ) "compares the volume occupied by the light rays emanating from the light source[ x ], when they arrive close to γ ( t ) , to the volume that they would occupy in a flatspace." (see [Vil09, Chapter 14.]). In particular, we can heuristically expect that thesedistortion coefficients would be related to the curvature of the space.3he theory of synthetic curvature was developed by Lott, Sturm, and Villani (see[LV09], [Stu06a], and [Stu06b]). We summarise here what is necessary in this work.We denote by P ( X ) the set of Borel probability measures and by P ( X ) the subsetof those with finite second moment. We write Geo ( X ) for the set of all minimalgeodesics of X parametrised on [
0, 1 ] . For all t ∈ [
0, 1 ] , the evaluation map is definedas e t : Geo ( X ) → X : γ (cid:55)→ γ ( t ) .A dynamical transference plan Π is a Borel probability measure on Geo ( X ) whilea displacement interpolation associated to Π is a path ( µ t ) t ∈ [ ] ⊆ P ( X ) such that µ t = ( e t ) Π for all t ∈ [
0, 1 ] . We equip P ( X ) with the L -Wasserstein distance W :for any µ , ν ∈ P ( X ) , W ( µ , µ ) : = inf π ∈ Π ( µ , ν ) (cid:90) X d ( x , y ) π ( d x d y ) with Π ( µ , ν ) : = { ω ∈ P ( X ) | ( proj ) ω = µ and ( proj ) ω = ν } . For µ , µ ∈ P ( X ) ,the set OptGeo ( µ , µ ) is the space of all measures ν ∈ P ( Geo ( X )) such that ( e , e ) ν realises the minimum for the L -Wasserstein distance. A measure ν ∈ OptGeo ( µ , µ ) is called a dynamical optimal plan. We now need to define the distortion coefficientsof the ( K , N ) -model space. For K ∈ R , N ∈ [ + ∞ ] , θ ∈ ( + ∞ ) and t ∈ [
0, 1 ] , we set τ ( t ) K , N ( θ ) = t N σ ( t ) K , N − ( θ ) − N with σ ( t ) K , N ( θ ) = + ∞ K θ (cid:62) N π sin ( t θ √ K / N ) sin ( θ √ K / N ) if 0 < K θ < N π t if K θ < N = K θ = ( t θ √− K / N ) sinh ( θ √− K / N ) if K θ (cid:54) N > τ K , N is not arbitrary. In fact, the coefficients τ ( t ) K , N are nothing but the distortion coefficients of the model space X ( K , N ) ; that is to say, X ( K , N ) is the N -sphere of constant cuvature K if K > X ( K , N ) is the N -Euclideanspace if K = X ( K , N ) is the N -hyperbolic plane of constant curvature K if K < Definition 4.
Let K ∈ R and N ∈ [ + ∞ ) . A geodesic metric measure space ( X , d, m ) verifies CD ( K , N ) if, for any µ , µ ∈ P ( X , m ) with bounded support, there exists ν ∈ OptGeo ( µ , µ ) and π ∈ P ( X ) a W -optimal plan such that µ t : = ( e t ) ν (cid:28) m and forany N (cid:48) (cid:62) N, E N (cid:48) ( µ t ) (cid:62) (cid:90) X τ ( − t ) K , N ( d ( x , y )) ρ N (cid:48) + τ ( t ) K , N ( d ( x , y )) ρ − N (cid:48) π ( d x d y ) where E N stands for the Rényi functional E N : P ( X ) → [ + ∞ ] : ρ m + µ s (cid:55)→ (cid:90) X ρ − N m ( d x ) .4or an extensive treatment of the CD-condition and more generally of OptimalTransport theory, we refer the reader to [Vil09]. Alongside this notion of curvature,a weaker condition was developed independently by Sturm and Ohta: the MeasureContraction Property (see [Stu06b], [Oht07]). Definition 5.
Let K ∈ R and N ∈ [ + ∞ ) . A geodesic metric measure space ( X , d, m ) verifies MCP ( K , N ) if, for every x ∈ X and measurable set A ⊆ X with m ( A ) ∈ ( + ∞ ) ,there exists ν ∈ OptGeo ( µ A , δ x ) such that for all t ∈ [
0, 1 ] µ A (cid:62) ( e t ) (cid:16) τ ( − t ) K , N ( d ( γ ( ) , γ ( ))) ν ( d γ ) (cid:17) where µ A : = m ( A ) m ∈ P ( X ) is the normalization of µ | A . These two definitions generalise the notion of Ricci curvature bounded from be-low by K ∈ R and dimension bounded from above by N (cid:62)
1. Indeed, if ( M , g ) is a Riemannian manifold and ψ a positive C function on M , the metric measurespace ( M , d g , ψ · vol g ) satisfies the CD ( K , N ) condition if and only if it satisfies theMCP ( K , N ) condition and if and only if Ric g , ψ , N (cid:62) Kg whereRic g , ψ , N : = Ric g − ( N − n ) ∇ g h N − n h N − n .Note that in the case where N = n , it only makes sense to consider constant functions ψ in the definition of the generalised Ricci tensor. These results can be found in[Stu06b], [LV09] for the equivalence with CD and the one with MCP is proved in[Oht07].For general metric measure spaces, the two notions of synthetic curvature are notequivalent. The CD condition does imply the MCP condition however (see [Oht07]and [CS12]). As we will see later, this discordant behaviour already appears for sub-Riemannian manifolds. In what follows, we set up the basics of sub-Riemannian geometry. We rely on[ABB20] for the general theory. We combine the exposition presented in [LD17] (seealso [LLP19]) with the one of [BL05] for geometric controls that are not necessary ofclass C ∞ .A manifold is a set equipped with an equivalence class of differentiable atlasessuch that its manifold topology is connected, Hausdorff and second-countable. Weemphasise here on the theory of sub-Riemannian manifolds of class C r instead of class C ∞ . As we will see later, the α -Grushin plane is a sub-Riemannian manifold that isgenerated by global vector fields that are not smooth. Definition 6.
Let M be a smooth manifold of class C r for r ∈ N (cid:62) ∪ { ∞ } ∪ { ω } . A triple ( E , g , ϕ ) is said to be a sub-Riemannian structure of class C r on M if1. E is a C r -vector bundle on M,2. g is a C r -metric on E,3. ϕ : E → T ( M ) is a C r -morphism of vector bundles. D of C r -horizontal vector fields is defined as D : = { ϕ ◦ u | u is a section of E of class C r } .We also define the distribution at point a p ∈ M with D p : = { v ( x ) | v ∈ D} .The rank of the sub-Riemannian structure at p ∈ M is rank ( p ) : = dim ( D p ) . Observethat in our definition, a sub-Riemannian manifold can be rank-varying; i.e. the maprank might not be constant. Definition 7.
We say that a Lipschitz curve γ : [ T ] → M is horizontal if there exists ameasurable and essentially bounded function u : [ T ] → E such that for all t ∈ [ T ] , wehave u ( t ) ∈ E γ ( t ) and ∂γ ( t ) = ϕ ( u ( t )) . The sub-Riemannian length of γ is defined by L CC ( γ ) = (cid:90) T (cid:107) ∂γ (cid:107) γ ( t ) d twhere (cid:107) v (cid:107) p : = min { σ ( u ) | u ∈ E p and ϕ ( u ) = ( p , v ) } for v ∈ D p and p ∈ M.Remark.
It can be proven that (cid:107)·(cid:107) p is well-defined and is a norm on D p , induced byan inner product (cid:104)· , ·(cid:105) p .In the case where every two points can be joined by a horizontal curve, we have awell defined distance function on M . Definition 8.
Let M be a sub-Riemannian manifold. The sub-Riemannian distance d CC ofM, also called the Carnot-Caratheodory distance, is defined by d CC ( x , y ) : = inf { L CC ( γ ) | γ : [ T ] → M is horizontal and γ ( ) = x and γ ( T ) = y } .Traditionally, the definition of a sub-Riemannian structure demands that D is a C ∞ -distribution and that it satisfies the Hörmander condition; that is to say, Lie p ( D ) = T p ( M ) for all p ∈ M . This is motivated by the following well-known result. Theorem 9 (Chow–Rashevskii theorem) . Let M be a sub-Riemannian manifold such thatits distribution D is C ∞ and satisfies the Hörmander condition. Then, ( M , d CC ) is a metricspace and the manifold and metric topology of M coincides. We refrain from this convention here, as the Grushin planes that we will study donot always satisfy this property. However, we will assume from now on that everytwo points of the sub-Riemannian manifold M can be joined by a horizontal curve,making d CC a distance of M , and that the metric and manifold topology do coincide.Finally, the horizontal distribution of a sub-Riemannian manifold M is defined byH ( M ) : = (cid:71) p ∈ M D p .Now that we have turned our sub-Riemannian manifold into a metric space, wewould like to study the geodesics associated with d CC . These would be horizontalcurves that are locally a minimiser for the length functional L CC . Because of the lackof a torsion-free metric connection, we cannot study geodesics through a covariantderivative. Rather, we will characterise these geodesics via Hamilton’s equations.6 efinition 10. Let M a sub-Riemannian manifold and ( X , . . . , X m ) a generating family ofvector fields. The Hamiltonian of the sub-Riemannian structure is defined byH : T ∗ ( M ) → R : λ (cid:55)→ H ( λ ) : = m ∑ i = (cid:104) λ , X i (cid:105) .We therefore approach the problem via the cotangent bundle T ∗ ( M ) , on whichthere is a natural symplectic form σ . We can now characterise length minimisers of asub-Riemannian manifold. Theorem 11.
Let γ : [ T ] → M be a horizontal curve which is a length minimiser andparametrised by constant speed. Then, there exists a Lipschitz curve λ ( t ) ∈ T ∗ γ ( t ) ( M ) suchthat one and only one of the following is satisfied:(N) ∂λ = (cid:126) H ( λ ) , where (cid:126) H is the unique vector field in Γ ( T ∗ ( M )) such that σ ( · , (cid:126) H ( λ )) = d λ H for all λ ∈ T ∗ ( M ) ;(A) σ ( ∂λ ( t ) , T λ ( t ) ( D ) ⊥ ) = for all t ∈ [ T ] . If λ satisfies ( N ) (resp. ( A ) ), we will say that λ is a normal extremal (resp. ab-normal extremal) and γ is a normal geodesic (resp. abnormal geodesic). Note that ageodesic may be both normal and abnormal.If γ is a normal geodesic associated with a normal extremal λ , then ( N ) is nothingbut Hamilton’s equation for H in the natural coordinates of the cotangent bundle: ∂ x i = ∂ H ∂ p i ∂ p i = − ∂ H ∂ x i . (2)The study of abnormal geodesics is an area of intensive research. It does happen thata sub-Riemannian structure do not have any non trivial abnormal geodesic. In thiscase, a complete sub-Riemannian manifold is said to be ideal.As explained in the introduction, no ideal sub-Riemannian manifold satisfies theCD ( K , N ) condition (see [Jui20]). However, it is known that they often satisfy an MCPcondition: the Heisenberg groups (see [Jui09]), generalised H-type groups, Sasakianmanifolds (see [BR19, Section 7.]), etc. We conclude this section with the followingtheorem that relates an MCP condition to a lower bound on the distortion coefficientsof a ideal sub-Riemannian manifold. Theorem 12 ([BR19, Theorem 9.]) . Let M be an ideal sub-Riemannian manifold equippedwith a measure µ . Let N (cid:62) . Then, the following conditions are equivalent:1. β t ( q , q ) (cid:62) t N for all q , q / ∈ Cut ( M ) and t ∈ [
0, 1 ] ;2. the measure contraction property MCP ( N ) is satisfied, i.e. for all non-empty Borelsets B ⊆ M and q ∈ M we have µ ( Z t ( q , B )) (cid:62) t N µ ( B ) . In this section, we give an account of ( p , q ) -trigonometry. The generalised sine andcosine functions will be essential in the study of the geometry of the α -Grushin plane,as shown by Li in [CL12]. Generalised trigonometry has a long history. The theory as7resented here was pioneered by Edmunds in [EGL12]. For recent developments, wepoint out the work of Takeuchi (see [Tak17] and the references therein).Consider F p , q : [
0, 1 ] → R : t (cid:55)→ (cid:90) x p √ − t q d t .The map F p , q being strictly increasing, we may define its inversesin p , q : (cid:104) π p , q (cid:105) → R : x (cid:55)→ F − p , q ( x ) where the ( p , q ) -pi constant is defined as π p , q : = (cid:90) p √ − t q d t = B (cid:18) p , 1 − q (cid:19) .Here the function B ( · , · ) stands for the complete beta function.We will extend the ( p , q ) -sine function to the whole real line. We first note thatsin p , q ( ) = p , q ( π p , q /2 ) = 1. For x ∈ (cid:2) π p , q /2, π p , q (cid:3) , we set sin p , q ( x ) = sin p , q ( π p , q − x ) . Then, the ( p , q ) -sine is extended to the whole R by 2 π p , q -periodicity.We then define the ( p , q ) -cosine by setting cos p , q = ( sin p , q ) (cid:48) . These two functions areof class C . In fact, they are also of class C ∞ except at the points x = k π p , q for k ∈ Z .We have the following identities: | sin p , q | q + | cos p , q | p = ( sin p , q ) (cid:48)(cid:48) = ( cos p , q ) (cid:48) = − qp | cos p , q | − p | sin p , q | q − sin p , q .Therefore, the ( p , q ) -sine function can be alternatively defined as the solution to thefollowing ordinary differential equation ∂ f = − q ( p − ) p | f | q − f , f ( ) = ∂ f ( ) =
1. (3)Unlike in the case of classical trigonometric functions, additions formulas are notknown for sin p , q ( x + y ) and cos p , q ( x + y ) . This problem basically boils down to find-ing a function F p , q ( x , y ) that solves the integral equation (cid:90) F p , q ( x , y ) p √ − t q d t = (cid:90) x p √ − t q d t + (cid:90) y p √ − t q d t .We would then have sin p , q ( x + y ) = F ( sin p , q ( x ) , sin p , q ( y )) . This is a very hard prob-lem, even for integers value for p and q . For ( p , q ) = (
2, 2 ) , the classical addition for-mula for sin emerges. When ( p , q ) = (
2, 4 ) , the corresponding addition formula is theone for the lemniscate function that Euler investigated in [Eul61]: let sl ( x ) : = sin ( x ) (resp. cl ( x ) : = cos ( x ) ) stand for the sinlem function (resp. the coslem function),then we have sl ( x + y ) = sl ( x ) cl ( y ) + sl ( y ) cl ( x ) + sl ( x ) sl ( y ) with an analogous formula for cl ( x + y ) . 8 Geometry of the α -Grushin plane α -Grushin plane For α ∈ [ + ∞ ) , the α -Grushin plane G α is defined as the sub-Riemannian structureon R generated by the global vector fields X = ∂ x and Y = | x | α ∂ y . This genetatingfamily of vector fields are C (cid:98) α (cid:99) if α is not an integer and C ∞ otherwise. The horizontalspace at p ∈ G α is H p ( G α ) = span { X ( p ) , Y α ( p ) } and the horizontal bundle is thedisjoint union of these H α = (cid:116) p ∈ G α H p ( G α ) . The rank of D = span { X , Y α } is notconstant: it is a singular distribution if x = g α to be the one that makes ( X , Y α ) an orthonormal familyof vector fields: g α ( x , y ) ( v , w ) = v + x α w . This turns the structure ( G α , H α , g α ) intoa sub-Riemannian manifold. It is easy to see that it does not satisfy the Hörmanderunless α ∈ N \ { } .Let I be a non-empty interval of R . A path γ : I → G α is said to be horizontal if ∂γ ( t ) ∈ H γ ( t ) ( G α ) for almost every t ∈ I . We can compute the length of a horizontalcurve L ( γ ) = (cid:82) I (cid:112) g α ( ∂γ ( t ) , ∂γ ( t )) d t . We denote by d α the Carnot-Caratheodorydistance associated with L α . This means that a path γ is horizontal if and only if ∂γ ( t ) = u ( t ) X ( γ ( t )) + v ( t ) Y α ( γ ( t )) for some measurable maps u , v : I → R and foralmost every t ∈ I . Equipping the α -Grushin plane with the Lebesgue measure L ,we obtain a metric measure space ( G α , d α , L ) .The theory of sub-Riemannian geometry informs us that the geodesics of the spaceare found by solving Hamilton’s equations. Here, the Hamiltonian is H : T ∗ ( G α ) → R with H ( x , y , u , v ) = ( u + v x α ) . A simple calculation shows that there are no non-trivial abnormal geodesics in the α -Grushin plane. Consequently, the sub-Riemannianmanifold G α is ideal. In this context, Hamilton’s equations (2) become ∂ x = u ∂ y = vx α ∂ u = − α v x ( α − ) x ∂ v =
0. (4)We observe that ∂ x = − α v x ( α − ) , which is the equation (3) for ( p , q ) = (
2, 2 α ) .In what follows, we will therefore denote sin α for sin α (and respectively cos α ). Theorem 13.
Let γ : I → G α be horizontal path with initial value γ ( ) = ( x , y ) andinitial covector λ ( ) = ( u , v ) . In the case where v (cid:54) = and ( x , u ) (cid:54) = , the curve γ is ageodesic if and only if x ( t ) = A sin α ( ω t + φ ) y ( t ) = y + v A α ( α + ) ω (cid:0) ω t + ω cos α ( φ ) sin α ( φ ) − ω cos α ( ω t + φ ) sin α ( ω t + φ )) u ( t ) = A ω cos α ( ω t + φ ) v ( t ) = v (5) for uniquely determined parameters A > ω ∈ R \ { } and φ ∈ [
0, 2 π α ) satisfyingA ω = u + v x α , ω = v A ( α − ) , x = A sin α ( φ ) and u = A ω cos α ( φ ) . (6)9 f v = or ( x , u ) = , the geodesic is ( x ( t ) , y ( t )) = ( u t + x , y ) with its lift beingconstant: ( u ( t ) , v ( t )) = ( u , v ) .Remark. Since the right-hand side of the equation is continuous with respect to theinitial condition v , we an obtain the γ ( t | x , y , u , 0 ) from lim v → γ ( t | x , y , u , v ) . Proof.
The case when v = ( x , u ) = v (cid:54) = ( x , u ) (cid:54) =
0. For A > ω ∈ R \ { } and φ ∈ [
0, 2 π α ) , we have ∂ ( A sin α ( ω t + φ )) = ∂ ( A ω cos α ( ω t + φ ))= − α A ω sin α ( ω t + φ ) ( α − ) sin α ( ω t + φ )= − α ω A ( α − ) ( A sin α ( ω t + φ )) ( α − ) ( A sin α ( ω t + φ )) .So by uniqueness of solutions to the differential equation (4), we get x ( t ) = A sin α ( ω t + φ ) and u ( t ) = A ω cos α ( ω t + φ ) if we set ω = v A ( α − ) , x = A sin α ( φ ) and u = A ω cos α ( φ ) . Considering theintegral of motion u + v x α at t = u + v x α = ( A ω cos α ( φ )) + ω A ( α − ) ( A sin α ( φ )) α = A ω .Since ∂ x = − α v x ( α − ) x , we deduce that x α = − x ∂ x / α v and thus, integrating bypart, we have (cid:90) t x α = (cid:90) t − x ∂ x α v = − α v (cid:18) [ x ∂ x ] t − (cid:90) t ( ∂ x ) (cid:19) = − α v (cid:18) [ xu ] t − (cid:90) t u (cid:19) .Now we remember that A ω = u + v x α . So, u = A ω − v x α and (cid:90) t x α = − α v (cid:18) x ( t ) u ( t ) − x ( ) u ( ) − (cid:90) t A ω + (cid:90) t v x α (cid:19) = A α v (cid:16) ω t + ω cos α ( φ ) sin α ( φ ) − ω cos α ( ω t + φ ) sin α ( ω t + φ ) − v A (cid:90) t x α (cid:33) .Finally, we isolate (cid:82) t x α and integrate ∂ y = v x α to get y ( t ) = y + v A α ( α + ) ω (cid:16) ω t + ω cos α ( φ ) sin α ( φ ) − ω cos α ( ω t + φ ) sin α ( ω t + φ ) (cid:17) .It remains to prove that there is a one-to-one correspondence between { ( x , v , v ) ∈ ( R ) ∗ × ( R ∗ ) } and { ( A , ω , φ ) ∈ R > × R ∗ × [
0, 2 π α ) } ( A , ω , φ ) to ( x , u , v ) is clear. The other direction is given by A = (cid:32) u + v x α v (cid:33) α , ω = v (cid:32) u + v x α v (cid:33) ( α − ) /2 α , (7)sin α ( φ ) = x (cid:32) v u + v x α (cid:33) α , cos α ( φ ) = sgn ( v ) u ( u + v x α ) . (8)By simply differentiating the relations (6) with respect to x , u and v , we find thefollowing identities: A x = − cos α ( φ ) sin α ( φ ) , A u = cos α ( φ ) αω , A v = − cos α ( φ ) A α v ; (9) ω x = ( α − ) (cid:16) ω A (cid:17) (cid:18) − cos α ( φ ) sin α ( φ ) (cid:19) , ω u = (cid:18) α − α (cid:19) cos α ( φ ) α A , ω v = ω v (cid:18) − (cid:18) α − α (cid:19) cos α ( φ ) (cid:19) ; (10) φ x = cos α ( φ ) A , φ u = − sin α ( φ ) αω A and φ v = sin α ( φ ) cos α ( φ ) α v . (11)We mention here the work of Li and Chang (cf. [CL12]). They obtained thegeodesics joining every two points in the α -Grushin plane by solving the boundaryvalue problem corresponding to the differential equation in Theorem 13. We notethat their results are stated for α ∈ N \ { } . However, if we carefully define sub-Riemannian manifolds of class C k as it was done in Section 2.2, we can see that theirconclusions remain valid in the case α (cid:62)
1. In particular, their detailed study of thegeodesics was used to derive an expression for the Carnot-Caratheodory distance of G α between every two points. As a consequence, the α -Grushin plane is a uniquely geodesic space, that is to say, there is a unique minimizing geodesic joining almostevery two pairs of points in G α . α -Grushin plane When we look at the the geodesics of G α , we observe three types of behaviours: thestraight lines corresponding to an initial covector v =
0; the geodesics for which x = x (cid:54) = α -Grushinplane. The techniques used here were developed in [ABS08, Section 3.2], [Riz18,Appendix A] and [ABB20, Section 13.5].The case when v = γ starting from a singular point x = v = ± s with s >
0. From (5), we see that the geodesic γ ( ·| y , u , s ) is areflection of γ ( ·| y , u , − s ) with respect to the y -axis. Furthermore, these two inter-sect at the y -axis a first time when t = π α / | ω | . Therefore, γ must lose its optimality11fter t (cid:62) π α / | ω | . From [CL12, Theorem 12], we know that there is one and only oneminimizing geodesic joining the singular point ( y ) to a point ( x , y ) with x (cid:54) = γ is minimizing before t = π α / | ω | .It remains to study the case of a geodesic γ starting at a Riemannian point ( x , y ) ,i.e. with x (cid:54) =
0. We will use an extended Hadamard technique, as described in[ABB20, Section 13.4].We firstly observe that γ (cid:18) π α | ω | (cid:12)(cid:12)(cid:12)(cid:12) A , ω , φ (cid:19) = γ (cid:18) π α | ω | (cid:12)(cid:12)(cid:12)(cid:12) A , ω , π α − φ (cid:19) .This means that the points (cid:32) − x , y + sgn ( ω ) (cid:18) x sin α ( φ ) (cid:19) α + π α ( α + ) (cid:33) are joined by two distinct geodesics if φ (cid:54) = π α /2, 3 π α /2.This leads us to conjecture that the cut time should still be t ∗ cut ( u , v ) = π α / | ω | and that the cut locus should beCut ∗ ( q ) = (cid:26) ( − x , y ) ∈ G α (cid:12)(cid:12)(cid:12)(cid:12) | y − y | (cid:62) | x | α + π α ( α + ) (cid:27) .We know that A ω = u + v x α = κ where the positive parameter κ is the con-stant speed of the geodesic γ . We can then parametrise u , v and the correspondingparameters A and ω with respect to φ ∈ (
0, 2 π α ) \ { π α } : u = κ cos α ( φ ) , v = κ sin α ( φ ) x (cid:12)(cid:12)(cid:12)(cid:12) sin α ( φ ) x (cid:12)(cid:12)(cid:12)(cid:12) α − , A = (cid:12)(cid:12)(cid:12)(cid:12) x sin α ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) and ω = κ sin α ( φ ) x .For t ∈ [ t ∗ cut ( u , v )] and φ ∈ (
0, 2 π α ) \ { π α } , the expression of the geodesics fromTheorem 13 can thus be written as x ( t , φ ) = (cid:12)(cid:12)(cid:12)(cid:12) x sin α ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) sin α (cid:18) κ sin α ( φ ) x t + φ (cid:19) y ( t , φ ) = y + ( α + ) (cid:12)(cid:12)(cid:12)(cid:12) x sin α ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) α + (cid:20) κ sin α ( φ ) x t + cos α ( φ ) sin α ( φ ) − cos α (cid:18) κ sin α ( φ ) x t + φ (cid:19) sin α (cid:18) κ sin α ( φ ) x t + φ (cid:19)(cid:21) This gives us the exponential map and we compute the determinant of its differential: D ( t , φ ) = κ sin α ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) x sin α ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) α (cid:20) x sin α (cid:18) κ sin α ( φ ) x t + φ (cid:19) cos α ( φ ) − sin α ( φ )( x + κ t cos α ( φ )) cos α (cid:18) κ sin α ( φ ) x t + φ (cid:19)(cid:21) One can check that lim φ → D ( t , φ ) = lim φ → π α D ( t , φ ) = α =
1, in which case wehave lim φ → D ( t , φ ) = | x | κ t (cid:32) κ t x + κ tx + (cid:33) φ → π α D ( t , φ ) = | x | κ t (cid:32) κ t x − κ tx + (cid:33) .We now claim that there are no conjugate points before t = π α / | ω | . Indeed,we firstly observe that D ( φ ) vanishes for every φ . Secondly, with the help of thederivative of D with respect to t ; ∂ t D ( t , φ ) = ακ x ( x + κ t cos α ( φ )) (cid:12)(cid:12)(cid:12)(cid:12) x sin α ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) α sin ( α − ) α (cid:18) κ sin α ( φ ) x t + φ (cid:19) sin α (cid:18) κ sin α ( φ ) x t + φ (cid:19) ,we see that ∂ t D ( t , φ ) = t = − x κ cos α ( φ ) or t = x κ sin α ( φ ) ( l π α − φ ) , l ∈ Z .The former is a local minimum that is positive while the later is a local maximum thatis also positive. Thirdly, we observe that D ( t ∗ cut , φ ) = κ π α sin α ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) x sin α ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) α + cos α ( φ ) which is zero if and only if φ = π α /2. So, the function D is never zero on ( t ∗ cut ) .Therefore, there are no conjugate points in [ t ∗ cut ) .Finally, we need to make some topological considerations in order to conclude.Consider the sets N : = { t φ | t ∈ [ t ∗ cut ) , φ ∈ [
0, 2 π α ] } = { ( u , v ) ∈ T ∗ ( G α ) | | v | < π α } ; M : = exp ( N ) = { ( x , y ) ∈ G α | ( x , y ) / ∈ Cut ∗ ( q ) } .The map exp : N → exp ( M ) is clearly proper. Since M is simply-connected, we canconclude that exp is a diffeomorphism by [ABB20, Corollary 13.24] and the conjec-tured cut locii and time are thus the true ones by the extended Hadamard technique.Summarizing the findings of this section, we have proven the following result. Theorem 14.
Let α (cid:62) and γ ( t | A , ω , φ ) = ( x ( t ) , y ( t )) be a geodesic of G α with initialvalue γ ( ) = ( x , y ) and initial covector λ ( ) = ( u , v ) , such as described in Theorem 13.If v = , then there are no conjugate points along γ , t cut [ γ ] = + ∞ and Cut ( x , y ) = ∅ . Ifv (cid:54) = , the cut time is t cut [ γ ] = π α | ω | while the cut locus is Cut ( y ) = { ( y ) | y ∈ R } and, if x (cid:54) = , Cut ( x , y ) = (cid:26) ( − x , y ) ∈ G α (cid:12)(cid:12)(cid:12)(cid:12) | y − y | (cid:62) | x | α + π α ( α + ) (cid:27) .The cut locii and geodesics of G α are illustrated at the Figure 1. With that in mind,we now turn to the analysis of distortion coefficients on the α -Grushin plane.13igure 1: Geometry of G α Illustration of the geodesics of the α -Grushin plane from a singular point (on the left)and from a Riemannian point (on the right). The shaded area represents a ballaround the starting point and the thick line is the cut locus. α -Grushin plane We present here our main result: an explicit computation of the distortion coefficientof G α . To this aim, we use the techniques initiated by Balogh, Kristaly and Siposin [BKS18] and generalised by Barilari and Rizzi in [BR19]. In the latter, the au-thors prove interpolation inequalities of optimal transport for ideal sub-Riemannianmanifolds. They are expressed in terms of the distortion coefficients for which theexpression is obtained through a fine analysis of sub-Riemannian Jacobi fields. Theorem 15.
Let q , q ∈ G α such that q / ∈ Cut ( q ) . Assume that q and q do not lie on thesame horizontal line. Then, for all t ∈ [
0, 1 ] , we have β t ( q , q ) = J ( t , A , ω , φ ) J ( A , ω , φ ) where J ( t , A , ω , φ ) = sin α ( ω t + φ ) cos α ( φ ) (cid:16) ω t α ( α ( α + ) − ( α − ) cos α ( φ )) (12) +( α − ) sin α ( φ ) cos α ( φ ) (cid:17) + cos α ( ω t + φ ) (cid:16) ( ω t ) α cos α ( φ )( − + α − α + ( α − ) cos α ( φ )) − ω t sin α ( φ )( α ( α + ) − ( α − ) cos α ( φ )) − ( α − ) sin α ( φ ) cos α ( φ ) (cid:17) − sin α ( ω t + φ ) cos α ( ω t + φ )( α − ) cos α ( φ ) (cid:16) ω t (( α − ) cos α ( φ ) − α ) − sin α ( φ ) cos α ( φ )) − sin α ( ω t + φ ) cos α ( ω t + φ )( α − ) cos α ( φ ) . Remark.
We consider geodesics parametrised by constant speed on [
0, 1 ] . Conse-quently, since Theorem 14 states that t cut = π α / | ω | , we always have | ω | (cid:54) π α .14 roof. We let λ = ( u , v ) ∈ T ∗ q ( G α ) be the covector corresponding to the uniqueminimizing geodesic joining q = ( x , y ) to q = ( x , y ) in G α . The assumption that q and q do not lie on the same horizontal line means that v (cid:54) = ( T ∗ ( G α )) ; E = ∂ u , E = ∂ v , F = ∂ x , F = ∂ y , Lemma 44 in [BR19] yields that β t ( q , q ) = J ( t ) / J ( ) where the function J is the determinant of the exponential map ( u , v ) → exp ( x , y ) ( u , v ) in these coordinates, computed at ( u , v ) .Taking the derivatives of (5), we find ∂ u x ( t ) = A u sin α ( ω t + φ ) + A ( ω u t + φ u ) cos α ( ω t + φ ) ∂ v x ( t ) = A v sin α ( ω t + φ ) + A ( ω v t + φ v ) cos α ( ω t + φ ) and ∂ u y ( t ) = [( α A u ω − A ω u ) ( sin α ( φ ) cos α ( φ ) − sin α ( ω t + φ ) cos α ( ω t + φ ))+ A ω ( α + )( φ u cos α ( φ ) − ( ω u t + φ u ) cos α ( ω t + φ ))+ αω t ( A u ω + A ω u )] v A ( α − ) A ( α + ) ω ; ∂ v y ( t ) = [( αω v A v + A ( ω − v ω v )) ( sin α ( φ ) cos α ( φ ) − sin α ( ω t + φ ) cos α ( ω t + φ ))+ A ω v ( α + )( φ u cos α ( φ ) − ( ω v t + φ v ) cos α ( ω t + φ ))+ ω t ( α v ω A v + A ( ω + α v ω v ))] v A ( α − ) A ( α + ) ω .To make things clearer, we have used the subscript notation to indicate a partialderivative and we also set [ f , g ] : = f u g v − f v g u .With this in mind, we calculate [ x , y ]( t ) = A α ( α + ) ω (cid:104) sin α ( ω t + φ ) cos α ( φ ) ( v [ A , ω ] − ω A u )+ α v ω sin αα ( ω t + φ ) sin α ( ω t + φ ) ([ A , ω ] t + [ A , φ ])+ sin α ( ω t + φ ) (cid:16) sin α ( φ ) cos α ( φ )( ω A u − v [ A , ω ]) − α v ω sin αα ( φ )[ A , φ ]+ ω ( ω A u t + v cos α ( φ )[ A , φ ]) (cid:17) + sin α ( ω t + φ ) cos α ( ω t + φ ) ( t (( α − ) v ω [ A , ω ] − A ωω u )+( α − ) v ω [ A , φ ] + Av [ φ , ω ] − A ωφ u )+ cos α ( ω t + φ ) ( sin α ( φ ) cos α ( φ )( A ωφ u + α v ω [ φ , A ] + Av [ ω , φ ])+ ω t ( α v [ ω , A ] + A ω u ) − ω t ( sin α ( φ ) cos α ( φ )( α v [ A , ω ]) − A ω u )+ ω ( α v [ A , φ ]) − A φ u + α v A sin αα ( φ )[ ω , φ ] + α v cos α ( φ )[ φ , ω ] (cid:17)(cid:105) .15sing the identities sin αα ( x ) + cos α ( x ) =
1, (9), (10) and (11), we find that [ x , y ]( t ) = A α ( α + ) ω (cid:34) sin α ( ω t + φ ) cos α ( φ ) (cid:16) ω t α ( α ( α + ) − ( α − ) cos α ( φ )) (13) +( α − ) sin α ( φ ) cos α ( φ ) (cid:17) + cos α ( ω t + φ ) (cid:16) ( ω t ) α cos α ( φ )( − + α − α + ( α − ) cos α ( φ )) − ω t sin α ( φ )( α ( α + ) − ( α − ) cos α ( φ )) − ( α − ) sin α ( φ ) cos α ( φ ) (cid:17) − sin α ( ω t + φ ) cos α ( ω t + φ )( α − ) cos α ( φ ) (cid:16) ω t (( α − ) cos α ( φ ) − α ) − sin α ( φ ) cos α ( φ ) (cid:17) − sin α ( ω t + φ ) cos α ( ω t + φ )( α − ) cos α ( φ ) (cid:35) .By simplifying, rearranging and performing β t ( q , q ) = [ x , y ]( t ) (cid:14) [ x , y ]( ) , we finallyobtain the desired expression.We can transform (12) from the set of coordinates A , ω and φ to x , u and v viathe identities (7) and (8). Performing β t ( q , q ) = [ x , y ]( t ) (cid:14) [ x , y ]( ) concludes the proofof the Theorem 1.It is interesting to look at the limit α →
1. In this case, the α -Grushin plane is thetraditional Grushin plane while sin α and cos α are the usual sine and cosine functions.The formula (12) simplifies to [ x , y ]( t ) = t A ω (cid:34) sin ( ω t + φ ) cos ( φ ) − ω cos ( ω t + φ ) (cid:16) sin ( φ ) + t cos ( φ ) (cid:17)(cid:35) = t ( u + tu v x + v x ) sin ( tv ) − tu v cos ( tv ) v and thus, we find what was already established in [BR19, Proposition 61]: the distor-tion coefficients of the usual Grushin plane are β t ( q , q ) = t ( u + tu v x + v x ) sin ( tv ) − tu v cos ( tv )( u + u v x + v x ) sin ( v ) − u v cos ( v ) , for all t ∈ [
0, 1 ] .We now want to investigate the behaviour of β t ( q , q ) when q and q do lie on thesame horizontal line, that is to say, when v → Proposition 16.
When v → , we have β t ( q , q ) = t ( u t + x ) α ( u t + x ) − x α x ( u + x ) α ( u + x ) − x α x . Proof.
We aim to perform lim v → J ( t ) (cid:14) J ( ) where J is defined by (1). We alreadyknow from Theorem 13 have that lim v → x ( t ) = u t + x and lim v → u ( t ) = u . Let16s make the following preliminary calculations: x v ( t ) = A v sin α ( ω t + φ ) + A ( ω v t + φ v ) cos α ( ω t + φ )= A α v (cid:16) cos α ( ω t + φ ) (cid:104) sin α ( φ ) cos α ( φ ) + ω t ( α − ( α − ) cos α ( φ )) (cid:105) − sin α ( ω t + φ ) cos α ( φ ) (cid:17) = (cid:2) t ( u + α v x α ) + u x (cid:3) · u ( t ) + u · x ( t ) α v ( u + v x α ) and u v ( t ) = − α A ω ( ω v t + φ v ) sin ( α − ) α ( ω t + φ ) sin α ( ω t + φ )+ cos α ( ω t + φ )( ω A v + A ω v )= A ωα (cid:16) sin ( α − ) α ( ω t + φ ) sin α ( ω t + φ ) (cid:104) ω t (( α − ) cos α ( φ ) − α ) − sin α ( φ ) cos α ( φ ) (cid:105) + cos α ( ω t + φ )( − cos α ( φ )) (cid:17) = v x α · u ( t ) − (cid:2) t ( u + α v x α ) + u x (cid:3) · x ( t ) ( α − ) x ( t )( u + v x α ) .Since simply replacing v with 0 in β t ( q , q ) leads to 0/0, we will use l’Hospital theo-rem, as many times as needed, we find: β t ( q , q ) = lim v → J ( t , x , u , v ) J ( x , u , v ) = lim v → ∂ v J ( t , x , u , v ) ∂ v J ( x , u , v )= lim v → ∂ v J ( t , x , u , v ) ∂ v J ( x , u , v ) = t ( u t + x ) α ( u t + x ) − x α x ( u + x ) α ( u + x ) − x α x . Now that we have the expressions for the distortion coefficients, we would like to findappropriate bounds on them. In [Jui20], Juillet proved that an ideal sub-Riemannianmanifold never satisfies the CD ( K , N ) condition. This means that we do not have β t ( q , q ) (cid:62) τ ( t ) K , N for whatever choice of K ∈ R and N (cid:62)
1. However, there is achance that the weaker curvature-dimension condition MCP ( K , N ) could hold for the α -Grushin plane.In particular, the traditional Grushin plane, equivalent to G α when α =
1, isMCP ( K , N ) if and only if N (cid:62) K (cid:54)
0. We expect the α -Grushin plane tosatisfy the MCP property for a minimal value of N that would depend on α . ByTheorem 12, the related bound on the distortion coefficients should be of the form β t ( q , q ) (cid:62) t N . In this section, we provide a bound in the case where q and q lie onthe same horizontal line and when q is a Grushin point.For α >
1, it is easy to see that the equation ( m + ) α ( m + ) − (( α + ) m + ) =
0. (14)has a unique non-zero solution m ∈ ( − − ) . If α =
1, the value of the root is m = −
3. 17 roposition 17.
Let q , q ∈ G α lying on the same horizontal line. We have that β t ( q , q ) (cid:62) t N if and only if N (cid:62) (cid:20) ( α + ) m + m + (cid:21) with m ∈ ( − − ) being the unique solution non zero solution of (14) .Proof. We are looking for the best N ∈ [ + ∞ ] such that t ( u t + x ) α ( u t + x ) − x α x ( u + x ) α ( u + x ) − x α x (cid:62) t N (15)for all t ∈ [
0, 1 ] and x , u ∈ R ; that is to say, ( u t + x ) α ( u t + x ) − x α x ( u + x ) α ( u + x ) − x α x (cid:62) t N − .If we take the logarithm of the above, we find that the inequality is equivalent to (cid:90) u u t ∂ z log (cid:104) ( z + x ) α ( z + x ) − x α x (cid:105) d z (cid:54) ( N − ) (cid:90) u u t ∂ z log | z | d z .The latter is equivalent to the same inequality for the integrands: ∂ z log (cid:104) ( z + x ) α ( z + x ) − x α x (cid:105) (cid:54) ( N − ) ∂ z log | z | .This leads to ( α + )( z + x ) α ( z + x ) α ( z + x ) − x α x (cid:54) ( N − ) z and thus N (cid:62) ( ( α + ) z + x )( z + x ) α − x x α ( z + x ) α ( z + x ) − x α x .We are therefore looking for the global maximum of the map f : R → R : ( x , y ) (cid:55)→ ( ( α + ) x + y )( x + y ) α − yy α ( x + y )( x + y ) α − yy α .Firstly, let us compute the critical points of f . We find ∂ x f ( x , y ) = ( α + ) y ( x + y ) α (cid:2) ( x + y )( x + y ) α − (( α + ) x + y ) y α (cid:3) ( x + y ) [( x + y )( x + y ) α − yy α ] and ∂ y f ( x , y ) = − ( α + ) x ( x + y ) α (cid:2) ( x + y )( x + y ) α − (( α + ) x + y ) y α (cid:3) ( x + y ) [( x + y )( x + y ) α − yy α ] x + y (cid:54) = ∂ x f ( x , y ) = ∂ y f ( x , y ) = { ( x , y ) ∈ G α | x + y = } ∪ { ( x , y ) ∈ G α | ( x + y )( x + y ) α − (( α + ) x + y ) y α = } .The case x + y = f ( x , y ) = N (cid:62) y =
0, or else we can set m = x / y and it followsthat ( m + ) α ( m + ) − (( α + ) m + ) = f ( x , y ) = ( α + ) at the limit and N (cid:62) ( α + ) while the latermeans that f ( x , y ) = ( m + ) α ( ( α + ) m + ) − ( m + ) α ( m + ) − = (cid:20) ( α + ) m + m + (cid:21) ;the function is constant at these points. Since m (cid:62) − α (cid:62)
1, we get2 (cid:20) ( α + ) m + m + (cid:21) (cid:62) ( α + ) (cid:62) ( x , y ) ∈ R ( ( α + ) x + y )( x + y ) α − yy α ( x + y )( x + y ) α − yy α = (cid:20) ( α + ) m + m + (cid:21) .As pointed out previously, this maximum provides the desired optimal N in the in-equality (15).It seems that Grushin structures behave in such a way that the distortion coeffi-cients for points q and q lying on the same horizontal line provides the sharpest N such that β t ( q , q ) (cid:62) t N . This is also what happens when α = N obtained in Proposition 17 is sharp. We are ableto verify this suggestion for singular points, i.e. when q = ( y ) . Proposition 18.
Let q = ( x , y ) ∈ G α with x = and q / ∈ Cut ( q ) . For all t ∈ [
0, 1 ] , β t ( q , q ) (cid:62) t N for all N (cid:62) (cid:20) ( α + ) m + m + (cid:21) where m < is such that (14) holds.Proof. In the case of x =
0, the Jacobian determinant (13) becomes [ x , y ]( t ) = A α ( α + ) ω (cid:16) α ( α − ) ω t − ( α − ) sin α ( ω t ) cos α ( ω t ) (cid:17) ( sin α ( ω t ) − ω t cos α ( ω t ))= α ( α + ) | u | v (cid:16) α ( α − ) u t − ( α − ) x ( t ) u ( t ) (cid:17) ( x ( t ) − tu ( t )) (16)19f we let x =
0, then we get φ = φ = π α . In both cases, it follows from (16) that β t ( q , q ) = g ( ω t ) h ( ω t ) g ( ω ) h ( ω ) . (17)where we have set g ( z ) = α ( α − ) z − ( α − ) sin α ( z ) cos α ( z ) and h ( z ) = sin α ( z ) − z cos α ( z ) . We first note that g ( ) = h ( z ) =
0. Then, we check that g (cid:48) ( z ) = α z sin ( α − ) α ( z ) sin α ( z ) > h (cid:48) ( z ) = α ( α − ) − ( α − ) (cid:104) cos α ( z ) − α sin αα ( z ) (cid:105) = ( α + ) (cid:104) α − ( α − ) cos α ( z ) (cid:105) > z ∈ [ π α ] , | cos α | (cid:54) α >
1. Therefore, the functions g and h are strictlyincreasing and positive.We want to prove that (17) is greater than t N . We will find N , N (cid:62) h ( ω t ) h ( ω ) (cid:62) t N and g ( ω t ) g ( ω ) (cid:62) t N .Let us look at N first. Similarly as we did in the proof of Proposition 17, we knowthat the desired inequality happens if and only if we have H ( z ) : = N h ( z ) − zh (cid:48) ( z ) (cid:62) z ∈ [ π α ] .We can see that H ( ) = H (cid:48) ( z ) = α z sin ( α − ) α ( z ) [( N − ) sin α ( z ) − ( α − ) z cos α ( z )] (cid:62) α ( α − ) z sin ( α − ) α ( z ) [ sin α ( z ) − z cos α ( z )] (cid:62) N (cid:62) α +
1, in which case, H ( z ) (cid:62)
0. Now, let us find a condition on N . We have N f ( z ) − z f (cid:48) ( z ) = α ( N ( α − ) − α ( α + )) z − ( α − ) N sin α ( z ) cos α ( z )+ ( α − ) ( α + ) cos α ( z ) (cid:62) α ( N ( α − ) − α ( α + )) z − ( α − ) N sin α ( z ) cos α ( z ) (cid:62) α ( N ( α − ) − α ( α + )) z − ( α − ) Nz = ( α + )( N ( α − ) − α ) z (cid:62) N (cid:62) α α − β t ( q , q ) (cid:62) t α + t α α − (cid:62) t N (cid:62) t N since N (cid:62) (cid:20) ( α + ) m + m + (cid:21) (cid:62) α + + α α − ( α − ) m + α ( α − ) + (cid:62) m (cid:62) − Proposition 19.
Let q = ( x , y ) ∈ G α with x (cid:54) = and q / ∈ Cut ( q ) . Assume that theunique minimal geodesic joining q to q has an initial covector ( u , v ) with u = whilev (cid:54) = . Then, for all t ∈ [
0, 1 ] , β t ( q , q ) (cid:62) t N for all N (cid:62) (cid:20) ( α + ) m + m + (cid:21) where m < is such that (14) holds.Proof. The condition u = φ = π α /2 or 3 π α /2. In both cases, this yields β t ( q , q ) = t cos α ( ω t + φ ) cos α ( ω + φ ) = t sin α ( ω t ) sin α ( ω ) (cid:62) t (cid:62) t N .Indeed, N (cid:48) sin α ( z ) − z cos α ( z ) (cid:62) N (cid:48) (cid:62) N (cid:62) (cid:20) ( α + ) m + m + (cid:21) (cid:62) x (cid:54) = φ (cid:54) = , π α /2, 3 π α /2. We therefore suggest the Conjecture 2. A proof of thiscould require further work, potentially requiring a more comprehensive study of the (
2, 2 α ) -trigonometric functions. 21 References [ABB20] Andrei Agrachev, Davide Barilari, and Ugo Boscain.
A comprehensive in-troduction to sub-Riemannian geometry , volume 181 of
Cambridge Studies in Ad-vanced Mathematics . Cambridge University Press, Cambridge, 2020. From theHamiltonian viewpoint, With an appendix by Igor Zelenko.[ABS08] Andrei Agrachev, Ugo Boscain, and Mario Sigalotti. A Gauss-Bonnet-likeformula on two-dimensional almost-Riemannian manifolds.
Discrete Contin.Dyn. Syst. , 20(4):801–822, 2008.[BBI01] Dmitri Burago, Yuri Burago, and Sergei Ivanov.
A course in metric geometry ,volume 33 of
Graduate Studies in Mathematics . American Mathematical Society,Providence, RI, 2001.[BKS18] Zoltán M. Balogh, Alexandru Kristály, and Kinga Sipos. Geometric inequal-ities on Heisenberg groups.
Calc. Var. Partial Differential Equations , 57(2):PaperNo. 61, 41, 2018.[BL05] Francesco Bullo and Andrew D. Lewis.
Geometric control of mechanical systems ,volume 49 of
Texts in Applied Mathematics . Springer-Verlag, New York, 2005.Modeling, analysis, and design for simple mechanical control systems.[BR19] Davide Barilari and Luca Rizzi. Sub-Riemannian interpolation inequalities.
Invent. Math. , 215(3):977–1038, 2019.[CL12] Der-Chen Chang and Yutian Li. SubRiemannian geodesics in the Grushinplane.
J. Geom. Anal. , 22(3):800–826, 2012.[CS12] Fabio Cavalletti and Karl-Theodor Sturm. Local curvature-dimension condi-tion implies measure-contraction property.
J. Funct. Anal. , 262(12):5110–5127,2012.[EGL12] David E. Edmunds, Petr Gurka, and Jan Lang. Properties of generalizedtrigonometric functions.
J. Approx. Theory , 164(1):47–56, 2012.[Eul61] Leonhard Euler. Observationes de comparatione arcuum curvarum irrecti-ficibilium.
Novi commentarii academiae scientiarum Petropolitanae , pages 58–84,1761.[Gru70] V. V. Grušin. A certain class of hypoelliptic operators.
Mat. Sb. (N.S.) , 83(125):456–473, 1970.[Jui09] Nicolas Juillet. Geometric inequalities and generalized Ricci bounds in theHeisenberg group.
Int. Math. Res. Not. IMRN , 2009(13):2347–2373, 2009.[Jui20] Nicolas Juillet. SubRiemanniann structures do not satisify Riemannian Brunn–Minkowski inequalities. arXiv e-prints , page arXiv:2002.01170, February 2020.[LD17] Enrico Le Donne. Lecture notes on sub-riemannian geometry. preprint , 2017. https://sites.google.com/site/enricoledonne/lecture_notes .[LLP19] Enrico Le Donne, Danka Luˇci´c, and Enrico Pasqualetto. Universal in-finitesimal Hilbertianity of sub-Riemannian manifolds. arXiv e-prints , pagearXiv:1910.05962, October 2019.[LV09] John Lott and Cédric Villani. Ricci curvature for metric-measure spaces viaoptimal transport.
Ann. of Math. (2) , 169(3):903–991, 2009.[Oht07] Shin-Ichi Ohta. On the measure contraction property of metric measurespaces.
Comment. Math. Helv. , 82(4):805–828, 2007.[Riz18] Luca Rizzi. A counterexample to gluing theorems for MCP metric measurespaces.
Bull. Lond. Math. Soc. , 50(5):781–790, 2018.22Stu06a] Karl-Theodor Sturm. On the geometry of metric measure spaces. I.
ActaMath. , 196(1):65–131, 2006.[Stu06b] Karl-Theodor Sturm. On the geometry of metric measure spaces. II.
ActaMath. , 196(1):133–177, 2006.[Tak17] Shingo Takeuchi. Arithmetric properties of the generalized trigonometricfunctions (qualitative theory of ordinary differential equations and related ar-eas).
RIMS Kokyuroku , 2032:76–100, 2017.[Vil09] Cédric Villani.
Optimal transport , volume 338 of