Lipschitz Carnot-Carathéodory Structures and their Limits

In this paper we discuss the convergence of distances associated to converging structures of Lipschitz vector fields and continuously varying norms on a smooth manifold. We prove that, under a mild controllability assumption on the limit vector-fields structure, the distances associated to equi-Lipschitz vector-fields structures that converge uniformly on compact subsets, and to norms that converge uniformly on compact subsets, converge locally uniformly to the limit Carnot-Carathéodory distance. In the case in which the limit distance is boundedly compact, we show that the convergence of the distances is uniform on compact sets. We show an example in which the limit distance is not boundedly compact and the convergence is not uniform on compact sets. We discuss several examples in which our convergence result can be applied. Among them, we prove a subFinsler Mitchell’s Theorem with continuously varying norms, and a general convergence result for Carnot-Carathéodory distances associated to subspaces and norms on the Lie algebra of a connected Lie group.


Introduction
This paper deals with the following general problem.Let M be a smooth manifold endowed with a family of vector fields and a continuously varying norm on the tangent spaces.Let us consider the length distance associated to the trajectories that infinitesimally follow such a family.What are the weakest notion of convergence and the most general assumptions on the family of vector fields and the norm that ensure the uniform convergence of the associated length distances?Such a question is natural while studying metric geometry.For example, fixed a continuously varying norm on the tangents, a sub-Finsler distance on a manifold is defined as the length distance associated to a bracket-generating family of smooth vector fields.It is a classical fact that a sub-Finsler distance can be approximated from below by increasing sequences of Finsler distances, see, e.g., [9,10].Nevertheless, understanding whether a specific approximation of the vector fields (and, possibly, of the norm) gives raise to the convergence of the associated length distances seems not to have been deeply studied in the literature.A by-product of our study goes in this direction, since it gives an effective tool to approximate sub-Finsler distances with Finsler distances in a controlled way.We refer the reader to the examples discussed in Theorem 4.5 and Theorem 1.6, which are consequences of the main Theorem 1.4 below.
Another situation in which the convergence of the distance associated to converging subFinsler structures emerge is while studying asymptotic or tangent cones of subFinsler structures, see the celebrated works of Mitchell and Bellaiche [12,4] (and the account in [7]), and the work of Pansu [14].We refer the reader to the examples discussed in Proposition 4.1, and Theorem 1.5 below.
We now introduce the language that we will adopt in the paper.Let us fix from now on a finite-dimensional Banach space E, and a smooth manifold M .A Lipschitz-vector-field structure f : M → E * ⊗ T M on M modelled by E is a Lipschitz choice, for every point p ∈ M , of a linear map between E and T p M , see Definition 2.3.We say that a sequence of Lipschitz-vector-field structure {f n } n∈N converges to a Lipschitz-vector-field structure f ∞ if, on every compact subset of M , {f n } n∈N is an equi-Lipschitz family that converges to f ∞ uniformly, see Definition 2.2 for details.Attached to a Lipschitz-vector-field structure f we have the notion of an End-point map, as follows.
Let f be a Lipschitz-vector-field structure on M modelled by E. We say that N : M × E → R is a continuously varying norm on M × E if N is continuous, and N (p, •) is a norm on E for every p ∈ M .We aim now at describing how the couple (f, N ) gives raise to a distance on M .
First, we define the energy and the length functionals working on the space L ∞ ([0, 1]; E), which we sometimes call space of controls.We stress that in our approach the energy and the length are not defined at the level of the curves, but instead on the controls that define them.In particular, if o ∈ M , u ∈ L ∞ ([0, 1]; E), (o, f, u) is in the domain of the End-point map, see Definition 1.1, and N is a continuously varying norm on M × E, we define the length ℓ (resp., the energy J) associated to (o, f, u, N ) to be 1 0 N (γ(t), u(t)) dt (resp., esssup t∈[0,1] N (γ(t), u(t))), where γ(t) := End f o (tu).Given the notion of energy, we can define the distance as follows.
Definition 1.2 (CC distance).Let E be a finite-dimensional Banach space, and M be a smooth manifold.Let f be a Lipschitz-vector-field structure on M modelled by E, and let N : M × E → [0, +∞) be a continuously varying norm.We define the Carnot-Carathéodory distance, or CC distance, between p and q as follows (2) d (f,N ) (p, q) := inf{J(p, f, u, N ) : End f p (u) = q}, where J(p, f, u, N ) is the above defined energy associated to (p, f, u, N ).
In Lemma 3.5 we shall show that a constant-speed reparametrization of a curve always exists, hence we can also equivalently take the infimum of the lengths in (2).We remark that, without any further hypotheses on the couple (f, N ), it might happen that d (f,N ) (p, q) = +∞ for some points p, q ∈ M .
In Corollary 3.19, we shall show that, if f is a Lipschitz-vector-field structure on M modelled by E, and N : M × E → [0, +∞) is a continuously varying norm, then, for every p, q ∈ M , we have also This means that the definition of the distance d (f,N ) working with the controls, i.e., the one given in (2), is equivalent to the definition of the distance as the infimum of the lengths evaluated with respect to the natural sub-Finsler metric associated to (f, N ), i.e., (3).
We now aim at understanding which kind of convergence is expected from the sequence of distances {d (fn,Nn) } n∈N when we have that the sequence {(f n , N n )} n∈N converges.The key hypothesis in order to have the local uniform convergence of the distances is a kind of essential non-holonomicity of the limit vector-fields structure f ∞ , which we next introduce.
First of all we introduce the notion of essentially open map.We say that a continuous map f : M → N between two topological manifolds of the same dimension k is essentially open at p ∈ M at scale U if U is a neighborhood of p homeomorphic to the k-dimensional Euclidean ball, with ∂U homeomorphic to the sphere S k−1 , and there exists V a neighborhood of f (p) homeomorphic to the k-dimensional Euclidean ball, such that f (∂U ) ⊂ V \ f (p) and the map f : ∂U → V \ f (p) induces a nonconstant map between the (k − 1)-homology groups, see Definition 2.10.
Notice that the definition of essential openness at p ∈ M does depend on the scale U ∋ p.Notice also that if the map induced between the local homology of p and f (p) is not trivial, i.e., f ⋆ : H k (M, M \ {p}) → H k (N, N \ f (p)) is nonzero, then f is essentially open at p ∈ M at every sufficiently small neighborhood U of p. Essential opennes at a point p at scale U does not imply openness at p, but only that f (p) is in the interior of f (U ), cf.Remark 2.11.Still a local homeomorphism at p is indeed essentially open at p at some scale U ∋ p.We further stress that Then we are ready to give the following definition.We recall that we denote by Φ t X the flow at time t of a vector field X on the smooth manifold M , whenever it exists.
Definition 1.3 (Essentially non-holonomic).Let M be a smooth manifold of dimension m and let F be a family of Lipschitz vector fields on M .We say that F is essentially non-holonomic if for every T > 0 and every o ∈ M , there are X 1 , . . ., X m ∈ F and t ∈ R m with | t| 1 < T such that there exists a neighborhood Ω t ⊆ B(0, T ) ⊆ R m of t for which the map , is defined on Ω t and, when restricted to Ω t, is an essentially open map at t in a neighborhood of it.
A Lipschitz-vector-field structure f on M modelled by a finite-dimensional Banach space E is said to be essentially non-holonomic if there is a basis (e 1 , . . ., e r ) of E such that F = {f(•, e 1 ), . . ., f(•, e r )} is essentially non-holonomic.
Let us explain the definition above.Equivalently, a set F of Lipschitz vector fields on a smooth manifold M m is essentially non-holonomic at a point p ∈ M whenever there exists a sequence of points p n ∈ M that converges to p such that p n is connected to p with the concatenation, starting at p, of line flows of m vector fields in F for times (t 1 , . . ., t m ), and moreover such concatenation is essentially open around (t 1 , . . ., t m ).We stress that the latter notion is weaker than the bracketgenerating condition in the case the vector fields are smooth, cf.Proposition 2.12.
We are now ready to give the main theorem of the paper.The following theorem is proved at the end of Section 3.5.
Theorem 1.4.Let M be a smooth manifold, and let E be a finite-dimensional Banach space.Let f be an essentially non-holonomic Lipschitz-vector-field structure modelled by E, and let N : M × E → [0, +∞) be a continuously varying norm.Then the following hold.
(i) if M is connected, then d ( f, N ) (p, q) < ∞ for every p, q ∈ M ; (ii) d ( f, N) induces the manifold topology on M ; (iii) Let {f n } n∈N be a sequence of Lipschitz-vector-field structures on M modelled by E, and let {N n } n∈N be a sequence of continuously varying norms on M × E. Let us assume that f n → f in the sense of Lipschitz-vector-field structures (see Definition 2.2), and N n → N uniformly on compact subsets of M × E. Then d (fn,Nn) → d ( f, N) locally uniformly on M , i.e., every o ∈ M has a neighborhood U such that d (fn,Nn) → d ( f, N) uniformly on U × U as n → +∞.
(iv) If in the hypotheses of item (iii) we additionally have that d ( f, N) is a boundedly compact (or equivalently complete) distance, we conclude that uniformly on compact subsets of M × M .Moreover, for every x ∈ M , we have (M, d (fn,Nn) , x) → (M, d ( f, N) , x) in the pointed Gromov-Hausdorff topology as n → +∞.
Let us comment on the latter statements.The first item is a Chow-Rashevskii type result for Lipschitz vector fields that satisfy the essentially non-holonomic condition.It implies in particular the classical Chow-Rashevskii theorem, [1].The item (i) is easy since the fact that two sufficiently near points can be connected by a finite-length curve is guaranteed by Definition 1.3.Item (ii) requires the essentially non-holonomic condition.Indeed, for an arbitrary couple (f, N ), one only has that the topology induced by (f, N ) is finer than the topology of the manifold, see Lemma 3.8 and the discussion after it.Item (iii) is the main convergence result, and it only holds locally around every point.When one adds the hypothesis that the limit distance is boundedly compact, the uniform convergence on compact sets can be obtained, as stated in item (iv).Without the hypothesis on the boundedly compactness of the limit distance, the convergence result might be false, see the example in Remark 3.24.
Let us now describe the main steps of the proof of the Theorem 1.4.The first nontrivial achievement to obtain the proof is that, when f is essentially non-holonomic, the topology of M is finer than the topology induced by d ( f, N ) ; and thus equal taking into account that the other inclusion, i.e., the one in Lemma 3.8, holds in general.The nontrivial inclusion follows from the fact that when f is essentially non-holonomic, the End-point map associated to f is open, see Theorem 2.13, and Lemma 3.21.Such an opennes property is a direct by-product of the essentially non-holonomic condition, see Remark 2.11.
The latter described opennes property is stable along a sequence {f n } n∈N of Lipschitz-vector-field structures that converge to f, and this is the key point to obtain item (iii).Such a stability is the content of Theorem 2.13.Its proof builds on the top of the joint continuity of the End-point map in the three variables o ∈ M , f a Lipschitz-vector-field structure on M modelled by E, and u ∈ L ∞ ([0, 1], E) (with the weak* topology), see Theorem 2.5 and Proposition 2.9, and on the topological Lemma B.1 proved with the aid of degree theory, which tells us that continuous functions that are near to an essentially open function are uniformly surjective.
To conclude the proof of item (iii) one exploits the latter stability property to prove that, on compact sets, the topology of M is uniformly finer than the topologies induced by the distances d (fn,Nn) , see Lemma 3.21.This directly implies that the functions {d (fn,Nn) } n∈N are equicontinuous on compact sets, see Proposition 3.22.To end the proof of item (iii) one then finally uses the previous equicontinuity together with the fact that, d ( f, N ) is locally obtained as a relaxation of d (fn,Nn) , see Proposition 3.20.
Item (iv) is then proved by exploiting Item (iii) and the general metric result in Lemma 3.25 according to which one can pass from the local uniform convergence to the uniform converge on compact sets in a very general setting under mild assumptions, i.e., the metric spaces are length spaces and the limit distance is boundedly compact.
In Appendix C we offer a shorter proof of Theorem 1.4(iv) assuming that the vector-fields structures are smooth.In this case we can argue directly by using Gronwall's Lemma A.1, and the quantitative open-mapping type result in Lemma B.3, which does not need degree theory since we have enough regularity of the flow maps.
Let us now discuss some corollaries of the general convergence result in Theorem 1.4.In Section 4 we list some examples in which a direct application of Theorem 1.4 gives nontrivial consequences.We record here a couple of them.The first is Mitchell's Theorem for subFinsler manifolds with continuously varying norms.
Let us give some preliminary notation and definitions.Let M m be a smooth manifold of dimension m ∈ N, and let X := {X 1 , . . ., X k } be a bracket-generating family of smooth vector fields on M .Let E := R k , and let N : M × E → [0, +∞) be a continuously varying norm.For every p ∈ M and v ∈ T p M , let | • | X ,N be the sub-Finsler metric defined as follows (4) |v| The subFinsler distance between p, q ∈ M is ( 5) We recall that the definition of a regular point can be found, e.g., in [7,Definition 2.4].The following theorem is proved at the end of Section 4.2.
Theorem 1.5 (Mitchell's Theorem for subFinsler manifolds with continuously varying norm).Let M m , X , N be as above.Let us fix o ∈ M .There exists a bracket-generating family of polynomial vector fields X := { X1 , . . ., Xk } on R m such that the following holds.
Let N o be the continuously varying norm on be the subFinsler metric on T R m defined as in (4), and let d ( X ,No) be the subFinsler distance on R m , defined as in (5), by using the subFinsler metric Then, the Gromov-Hausdorff tangent of (M, We remark that the construction of X , which is usually called the homogeneous nilpotent approximation of X at o, can be made explicitely with respect to X , compare with [7, Section 2.1], and does not depend on N . Giving for granted the construction of privileged coordinates, for which we give precise references in Section 4.2, the proof of the Theorem 1.5 is a direct consequence of the application of Theorem 1.4(iv), see Section 4.2.
The last consequence we want to discuss is in the setting of Lie groups.We introduce some notation.Let G be a connected Lie group, and g its Lie algebra.Given a vector subspace H ⊆ g of g, and a norm b on H, we associate to (H, b) a left-invariant subFinsler structure (D, b) as follows ( 6) where L p (q) := p • q for p, q ∈ G.
To each (H, b), one attaches the (possibly infinite-valued) distance between p, q ∈ G as Notice that if H is a bracket-generating vector subspace of g, the sub-Finsler structure (G, D, b) satisfies the bracket-generating condition and thus d (H,b) is finitevalued.Let us denote by Gr g (k) the Grassmannian of k-planes endowed with the usual topology of the Grassmannian of a vector space.The proof of the following result is at the end of Section 4.3 and it is an immediate consequence of Theorem 1.4(iv).
Theorem 1.6 (Convergence of distances on Lie groups).Let G be a connected Lie group with Lie algebra g, and let H ⊆ g be a bracket-generating vector subspace of dimension k.
Let {H n } n∈N be a sequence of vector subspaces of g that converges in the topology of Gr g (k) to H, and let {b n } n∈N be a sequence of norms on g that converges uniformly on compact sets to a norm b on g.Then, being d (Hn,bn) and d (H,b) the distances defined as in (7), we have d (Hn,bn) → d (H,b) , uniformly on compact subsets of G × G.
We briefly describe the structure of the paper and we refer the reader to the introductions of the sections for more details.
In Section 2 we introduce the main notation and definitions of the paper, we study the continuity property of the End-point map, the notion of essentially open map and essentially non-holonomic set of vector fields, and we finally prove the open property of the End-point map associated to an essentially non-holonomic structure.
In Section 3 we define and study the length, the energy and the Carnot-Carathéodory distance associated to a Lipschitz-vector-field structure and a continuously varying norm.We thus study how the distances behave under taking limit of the corresponding structures, and we prove the main Theorem 1.4.
In Section 4 we discuss some examples in which Theorem 1.4 applies, namely the fact that the asymptotic cone of the Riemannian Heisenberg group is the subRiemannian Heisenberg group; the subFinsler Mitchell's Theorem with continuously varying norms; a general convergence result for Carnot-Carathéodory distances on connected Lie groups; and a general convergence result for Carnot-Carathéodory distances on manifolds.
In the Appendix we give a self-contained proof of the version of the Gronwall Lemma that we need, we prove some ancillary lemmas about open maps, and we finally give a shorter and more direct proof of Theorem 1.4(iv) in the case in which the vector fields are smooth.according to which the curves satisfying the Cauchy system (1) starting from a converging sequence The latter result relies on a general convergence criterion for flow lines of (not necessarily smooth) vector fields, see Propostion 2.6.
In Section 2.3 we introduce the notion of essentially open map between two topological manifolds of the same dimension, see Definition 2.10.We recall that such a notion is used to give the notion of essentially non-holonomic distribution of vector fields, see Definition 1.3.
In Section 2.4 we finally show the second main result of this section.Namely, we show that whenever a Lipschitz-vector-field structure f modelled by E is essentially non-holonomic, the End-point map of f enjoys a uniform openness property, see Theorem 2.13 for further details.Such a uniform openness result, which follows both from the continuity result proved in Theorem 2.5, and from the ancillary Lemma B.1, strongly requires the essentially non-holonomicity property.Eventually, the latter uniform openness result will be of key importance in proving Lemma 3.21, which is one of the main steps to prove the main result of the next section, see Theorem 1.4(iv).We refer the reader to the introduction of the next section for further details.
2.1.Lipschitz-vector-field structures on a manifold.In this subsection we study basic facts about Lipschitz-vector-field structures.Definition 2.1 (Uniformly locally Lipschitz sections LipΓ).Let M be a smooth manifold and E → M a vector bundle.We say that a family X of sections of E is uniformly locally Lipschitz if for every p ∈ M there exist a coordinate neighborhood of p that trivializes E and a constant L > 0 so that every element of X is L-Lipschitz in this trivialization.We denote by LipΓ(E) the collection of all locally Lipschitz sections of E, i.e., sections X so that the singleton {X} is uniformly locally Lipschitz.
It is a direct consequence of the definition that a family X of sections is uniformly locally Lipschitz if and only if it is uniformly Lipschitz on compact sets on every trivialization of E. Definition 2.2 (Sequential topology on uniformly locally Lipschitz sections).We define a sequential topology on LipΓ(E) as follows.We say that a sequence {X n } n∈N ⊂ LipΓ(E) converges to X ∞ ∈ LipΓ(E) if and only if {X n } n∈N is uniformly locally Lipschitz and X n → X uniformly on compact subsets of M , that is, if every p ∈ M has a coordinate neighborhood that trivializes E on which X n converge to X uniformly on compact sets.Definition 2.3 (Lipschitz-vector-field structure).Let E be a finite-dimensional Banach space and M a smooth manifold.We denote by E * ⊗ T M the vector bundle on M whose fibers are E * ⊗ T p M .A section f of E * ⊗ T M is a choice, for each p ∈ M , of a linear map f| p : E → T p M .An element f ∈ LipΓ(E * ⊗ T M ) will be called a Lipschitz-vector-field structre on M modelled by E.
The definition of the Banach spaces L p ([0, 1]; E) does not depend on the choice of a Banach norm on E. The predual of We denote by D End ⊂ M × LipΓ(E * ⊗ T M ) × L ∞ ([0, 1]; E) the domain of the End-point map, as defined in Definition 1.1.
Remark 2.4 (Concatenation of flows of vector fields).Let X 1 , . . ., X ℓ be locally Lipschitz vector fields on M .Take E = R ℓ with the standard Euclidean norm, and with the canonical basis {e 1 , . . ., e ℓ }, and define f ∈ LipΓ(E * ⊗ T M ) by extending linearly f p (e i ) := X i (p), for all p ∈ M and i ∈ 1, . . ., ℓ.The Cauchy system in Equation ( 1), for any u : Fix j ∈ N, (t 1 , . . ., t j ) ∈ R j and e i1 , . . ., e ij ∈ {e 1 , . . ., e ℓ }.If we define the controls where T := j i=1 |t i |, and whenever one of the terms exists.We recall that Φ t X (o) is the flow line at time t of the vector field X starting at o.

2.2.
Continuity of the End-point map.In this section we shall prove the following continuity theorem for the End-point map.On D End ⊂ M × LipΓ(E * ⊗ T M ) × L ∞ ([0, 1]; E) we consider the topology that is the product of the manifold topology on M , the sequential topology defined in Definition 2.2 on LipΓ(E * ⊗T M ), and the weak* topology on L ∞ ([0, 1]; E).Theorem 2.5.Let M be a smooth manifold and let E be a finite-dimensional Banach space.Then the domain D End of the End-point map is open and the function End : D End → M is continuous.
Moreover, given (ô, f, û) ∈ D End , and given a weak*-compact neighborhood U of û such that (ô, f, u) ∈ D End for every u ∈ U , we have that that the limit (9) lim Inspired by [11,Proposition 3.6], we prove the following ancillary proposition.
Remark 2.7.Suppose that the hypothesis of Proposition 2.6 except (10) are satisfied.We claim that, if X n → X ∞ pointwise a.e. on [0, 1] × M , then (10) Proof.We denote by •|• the pairing of a Banach space with its dual space.We need to show that, for every α ∈ L 1 ([0, 1]; E * 2 ), (11) lim Since A is continuous in s, the operator norm of the adjoint operators A * s is bounded uniformly in s ∈ [0, 1], by R say.Hence . We conclude that, for every hence (11) indeed holds for every α ∈ L 1 ([0, 1]; E * 2 ).The next proposition will be at the core of the proof of Theorem 2.5.

Then the following hold:
(1) There is N ∈ N such that {(o n , f n , u n )} n≥N ⊂ D End , that is, for every n ≥ N the Cauchy system (1) has an integral curve γ (on,fn,un) defined on [0, 1]; (2) The sequence of integral curves γ (on,fn,un) uniformly converge to γ (o∞,f∞,u∞) .
Proof.The convergence in the assumptions means that uniformly on compact sets of every trivialization of E * ⊗ T M , and {f n } n is uniformly locally Lipschit on compact sets of every trivialization of E * ⊗ T M .For n ∈ N ∪ {∞}, denote by γ n : [0, T n ) → M the maximal integral curve of the Cauchy system (1) with control u n and initial point o n .Notice that γ n can be extended to T n if and only if Let ρ be a complete Riemannian metric on M and let ι : M ֒→ R N be a Riemannian isometric embedding.The ρ-length of curves in M is thus equal to their Euclidean length in R N .We denote by d ρ and diam ρ the Riemannian distance and the corresponding diameter on M defined by ρ.Notice that, for every r ≥ 0 and p ∈ M , the closed ρ-ball of radius r and center p in M , Bρ (p, r), is a compact subset of R N .
By means of the isometric embedding ι, we interpret the sections We claim that, for every t To prove the claim, first notice that there is ), which is compact, and thus and thus we deduce that ŝ = t + D HR .We conclude that γ n is defined on HR is a fixed positive quantity, the above claim readily implies the first part of the lemma.For the second part of the proof, we assume that γ n is defined on [0, 1] for all n.
We claim that, for every t To prove the latter claim, notice that from the previous claim we have that there is ).We next apply Proposition 2.6, whose hypotheses must be met.Define the vector fields where We have thus shown that the non-autonomous vector fields X n satisfy all conditions in Proposition 2.6 on . The claim is proven.Finally, since the constant D RH does not depend on t, we can subdivide [0, 1] into intervals of length less than D RH and apply the above claim iteratively to each interval, concluding the proof of the proposition.
Proof of Theorem 2.5.Proposition 2.9.(1) implies that D End is open, while Proposition 2.9.(2) implies that End is continuous.The uniform limit ( 9) is a direct consequence of the continuity of End simultaneously in the three variables (o, f, u), and a standard compactness argument.

Essentially non-holonomic Lipschitz distributions.
In this section we discuss the notions of essentially open maps and essentially non-holonomic distributions of vector fields.

Definition 2.10 (Essentially open map). A continuous map
, and such that f | ∂B(0,r1) : ∂B(0, r 1 ) → B(0, r 2 ) \ {0} induces a non-constant group homomorphism between the (k − 1)-th homology groups.Hence, arguing as in the first few lines of the proof of Lemma B.2, the map f : has nonzero degree.Then we can apply Lemma B.1 with the constant sequence f n := f to obtain that f (0) is contained in the interior of f (B(0, r 1 )).
We notice that by virtue of Lemma B.1, the essential openness at a point p at some scale, implies a surjectivity property at p which is stable with respect to uniform convergence.This stability is of crucial importance in Definition 1.3.Indeed, if we naïvely only require that φ in Definition 1.3 is open at t, it is not possible to prove the uniform openness in Theorem 2.13, because the property of being open is not stable under uniform convergence.For example, consider the functions f t (x) := tx defined on [0, 1] and let t → 0.
We show in the next proposition that the condition of essential non-holonomicity (as in Definition 1.3) is a weakening of the bracket-generating condition for smooth vector fields.
Proposition 2.12.Every bracket-generating family of smooth vector fields is essentially non-holonomic.
Proof.Let F ⊂ Γ(T M ) be a bracket-generating family of smooth vector fields.Due to [1, Lemma 3.33], cf. also [2, Section 5.4], we have that for every T > 0 and every o ∈ M , there are X 1 , . . ., X m ∈ F such that the map Then an application of Lemma B.2 gives the sought conclusion.

2.4.
Uniform openness of the End-point map.In this final part of Section 2, we prove that whenever a Lipschitz-vector-field f is essentially non-holonomic, and a point o ∈ M is given, then the End-point map End f o (•) is open at 0 uniformly as f varies in a neighborhood of f.Theorem 2.13.Let f ∈ LipΓ(E * ⊗ T M ) be essentially non-holonomic, let o ∈ M , and R > 0. Then there are neighborhoods F of f and U of o such that Let m be the dimension of the manifold M .Arguing by contradiction, suppose that there are f n → f and . Let e 1 , . . ., e r be a basis of E as in Definition 1.3.Given σ : {1, . . ., m} → {1, . . ., r} and f ∈ LipΓ(E * ⊗ T M ), define, for every j = 1, . . ., m, X σ,f j (p) = f(p, e σ(j) ), and, for every (s 1 , . . ., s m , t 1 , . . ., t m ) ∈ R 2m , let us define and there is Since f is essentially non-holonomic, there are t ∈ R m , a neighborhood Ω of t, and σ such that 2C|t| 1 < R for all t ∈ Ω, and Now, the maps f n = ψ σ,fn ( t, •) are continuous on Ω and converge uniformly to f , by Theorem 2.5.By reasoning as at the beginning of Remark 2.11, we have that at some scale around t, f satisfies the hypothesis on the degree required to apply Lemma B.1.Hence, using Lemma B.1, there is an open neighborhood

Sub-Finsler distances for Lipschitz-vector-field structures
In this section we shall fix a smooth manifold M and a Banach space E. In M we shall fix a Lipschitz-vector-field structure f modelled by E, as in Definition 2.3.
In Section 3.2, given a continuously varying norm N on M × E, we are going to define the energy and the length functionals associated to controls u ∈ L ∞ ([0, 1]; E), we prove that they are lower semicontinuous, see Proposition 3.4, and we prove that every curve has a constant speed reparametrization, see Lemma 3.5.
In Section 3.3, we define the distance d (f,N ) (p, q) between p, q ∈ M to be the infimum of the energy (equivalently the length) of a control that gives raise to a curve that connects the two points, see Definition 3.6.We then prove the local existence of geodesics, see Proposition 3.9, and in Proposition 3.10 we give a criterion, ultimately based on the growth of (f, N ), to show that (M, d (f,N ) ) is a complete, boundedly compact, geodesic metric space.We prove that the topology generated by d (f,N ) is larger than the topology of M , see Lemma 3.8.We shall then show that they are equal if f is essentially non-holonomic, see Theorem 1.4(ii).
In Section 3.4 we shall address the problem of linking the definition of the distance given in Section 3.2 with the Lagrangian one given directly on the curves.In particular, having a couple (f, N ) on M , one can define a sub-Finsler metric on T M , see (32).We show that a curve γ : [0, 1] → M is d (f,N ) -Lipschitz if and only if γ ′ has a bounded sub-Finsler metric as in (32), see Lemma 3.16.As a consequence, on d (f,N ) -Lipschitz curves, the length associated to d (f,N ) coincides with the one associated to the sub-Finsler metric, see Proposition 3.17.We thus conclude that the distance d (f,N ) (p, q) is the infimum of the sub-Finsler lengths of the curves connecting the points p, q, see Corollary 3.19.
In Section 3.5 we finally investigate what happens when we take the limit of CC distances.We first prove a relaxation result which tells us that when N∞) on compact sets can be recovered as the relaxation of the distances d (fn,Nn) , see Lemma 3.20.Hence we prove that when ) and f ∞ is essentially non-holonomic, then {d (fn,Nn) } n∈N are equi-continuous functions on compact sets, see Lemma 3.21 and Proposition 3.22.The equi-continuity together with the relaxation property allows us to show that if , and f ∞ is essentially non-holonomic, then d (fn,Nn) converges to d (f∞,N∞) locally uniformly, see Theorem 1.4(iii).Finally, when (M, d (f∞,N∞) ) is boundedly compact, the local uniform convergence can be upgraded to a uniform convergence on compact sets by means of the metric Lemma 3.25, see Theorem 1.4(iv).The boundedly compact hypothesis is necessary to obtain such a uniform convergence on compact sets, see the example in Remark 3.24.
3.1.Continuous norms along a segment.In this section we give a preliminary and technical discussion about norms depending on a parameter that we will later use.
Let N : [0, 1] × E → [0, ∞) be a continuous function so that, for every t ∈ [0, 1], the restriction N t := N (t, •) is a norm on the finite-dimensional vector space E. We define, for a measurable u : Notice that, since N is continuous, u N,1 is a Banach norm on L 1 ([0, 1]; E) and u N,∞ is a Banach norm on L ∞ ([0, 1]; E), and both are bi-Lipschitz equivalent to the standard Banach norms on those spaces.
If • is a Banach norm on a vector space, we denote by • * its Banach dual norm on the dual space.With this convention in mind, we denote by

and
(15) Notice that ( 14) is a pair of genuine identities of norms, because The second pair of identities (15) is understood via the standard embedding of a Banach space into its bi-dual, because L 1 is not reflexive, i.e., the dual of The fact that u N,1 is a maximum is an application of Remark 3.1.On the contrary, u N,∞ is only a supremum and we can see this phenomenon with E = R and N 1 ≡ | • |: indeed, on the one hand, u(x) := x is an element of L ∞ ([0, 1]; R) and esssup x |u(x)| = 1; on the other hand, there is no function w ∈ L 1 ([0, 1]; R) so that Proof of Lemma 3.2.Notice that in both ( 14) and (15), the two identities are equivalent because E has finite dimension.
Let us now prove the second identity in (14).Fix u ∈ L ∞ ([0, 1]; E).Up to choosing a representative, we can assume u is defined on the whole [0, 1].Notice that, by definition of dual norm, , it is sufficient to prove that, for every ε > 0, there exists and notice that L 1 (B ε ) > 0. Let {w j } j∈N be a dense subset of E * and define From how w ε is defined, we obtain that Next, we prove the second identity in (15): Fix u ∈ L 1 ([0, 1]; E) and notice that For the opposite inequality, it is sufficient to prove that, for every ε > 0 there exists and then taking ε → 0. Indeed, let {w j } j∈N be a dense subset of E * and, for ε > 0 and each j ∈ N, define the sets Both suprema in ( 16) and ( 17) are a direct consequence of the previous identities.The fact that the supremum in ( 16) is attained, is an application of Remark 3.1, since we know that the dual space of (L ), thanks to the first identity in (14).
To prove the last statement of the lemma, let w be an argument of the maximum in (16).First, we claim that N * t (w(t)) = 1 for almost every t ∈ [0, 1].If this were not the case, then there would be a set A ⊂ [0, 1] with positive measure such that N * t (w(t)) ≤ λ < 1, for some λ > 0 and all t ∈ A. Define w(t) := χ [0,1]\A (t)w(t) + χ A (t)w(t)/λ.Then w N * ,∞ ≤ 1 and w(t)|u(t) > w(t)|u(t) for t ∈ A, in contradiction to the maximality of w.
Finally on the one hand from (18) we consequently have w(t)|u(t) ≤ N t (u(t)) for almost every t ∈ [0, 1].On the other hand, by maximality of w we have

Energy and length.
As in Section 2, we are in the setting where M is a smooth manifold and E is a finite-dimensional Banach space.Let N be the space of all continuously varying norms on ) that is a norm on fibers.We endow N with the topology of uniform convergence on compact sets.We now define the energy and the lengths associated to controls, i.e., elements of L ∞ ([0, 1]; E), and we study some of their basic properties.For the next result, we recall that D End is the domain of the End-point map.
Proposition 3.4 (Semi-continuity of energy and length).Both functions J and ℓ from D End ×N to R are lower-semicontinuous.In other words, if ) is lower-semicontinuous with respect to the weak* convergence.Notice that the previous one is a Banach norm on L ∞ ([0, 1]; E).In particular, it is the dual norm of the norm defined on Since the weak* convergence on L ∞ ([0, 1]; E) does not depend on the choice of biLipschitz equivalent Banach norms on the predual L 1 ([0, 1]; E * ), and since every dual norm is weakly* lower-semicontinuous, we get the sought claim.
Let us now prove the first inequality in (19).Notice that, for every n ∈ N and t ∈ [0, 1] we have (20) where, for j ∈ N ∪ {∞}, and t ∈ [0, 1], γ j (t) := End fj oj [tu j ].As a consequence of Proposition 2.9 we have that , and since the u n are uniformly bounded, the second term in the lower bound (20) goes to 0 as n → ∞, uniformly in t.
The first inequality in (19) thus follows from the following computation Similarly, to prove that ℓ is lower-semicontinuous, we need, in addition to (20), to prove that u → Moreover, if we set S * = {v ∈ L ∞ ([0, 1]; E * ) : v(t) ∈ S * (t) for almost every t ∈ [0, 1]}, then, by Lemma 3.2, for every We can finally conclude, by using the fact that since u n , u ∞ ∈ L ∞ ([0, 1]; E) we have a fortiori that u n , u ∞ ∈ L 1 ([0, 1]; E), with the following chain of inequalities With an abuse of language, we say that a curve γ has constant N -speed or simply constant speed if γ = γ (o,f,u) with t → N (γ(t), u(t)) almost everywhere constant for t ∈ [0, 1].That a curve γ has constant speed, really depends on all the data (o, f, u, N ).
Proof.Let us assume that ℓ := ℓ(o, f, u, N ) = 0, otherwise the result is trivial.Define where does not exist}) has measure zero by the area formula.Notice that for every s ∈ [0, 1] \ E there exists a unique t ∈ [0, 1] such that ψ(t) = s, and thus the formula and then, since ) is constant, then it is equal to both the energy and the length of the curve γ.

3.3.
Carnot-Carathéodory distances.In this section we define the Carnot-Carathéodory distance, CC distance for brevity, associated to (f, N ) and we investigate some of its properties.We use the notation LipΓ from Section 2.1, and N from Section 3.2.
Definition 3.6 (CC distance).Given f ∈ LipΓ(E * ⊗ T M ) and N ∈ N , we define the Carnot-Carathéodory distance, or CC distance, Moreover, the following hold for every Proof.The existence of L that satisfies (26) is a consequence of the compactness of Bρ (K, R) × K × K, and the continuity of f and For part (d), we see from (25) and item (b) that, if d (f,N ) (p, q) < R/L 2 , then the infimum in (25) can be taken on curves laying in Bρ (K, R).This shows the equality in (28); the first inequality in (28) is then a direct consequence of (24) and item (c).Proof.Let U ∈ τ M and p ∈ U .We need to show that there is r > 0 such that B (f,N ) (p, r) ⊂ U .Fix a complete Riemannian metric ρ on M and let L ≥ 1 be a constant that satisfies (26 In the setting of the above Lemma 3.8, the two topologies may not be equal.As an example, consider a structure f defined by a constant line field on R 2 : the integral lines of such a structure are open sets in τ (f,N ) but not in the standard topology of R 2 .We will later show that the two topologies do agree under an essentially non-holomic condition on f, see Theorem 1.4.Proposition 3.9 (Local existence of geodesics).Let K ⊂ LipΓ(E * ⊗T M ), K ⊂ M and K ⊂ N be compact sets.Then there is a constant C > 0 so that, for every f ∈ K , every N ∈ K, every p ∈ K, and every q ∈ M with In particular, the curve γ (p,f,u) : [0, 1] → (M, d (f,N ) ) is a homothetic embedding, i.e., a d (f,N ) -length minimizing curve.
In the setting of Lemma 3.7, we can take C = R L 2 , for R, L > 0 satisfying (26).Proof.In the setting of Lemma 3.7, fix R > 0 and the corresponding L > 0 and set and γ (p,f,un) (t) ∈ B ρ (K, R) for all t ∈ [0, 1] and all n.
From Lemma 3.7.(c),|||u n ||| ∞ is uniformly bounded in n.Therefore, up to passing to a subsequence, u n weakly* converge to some By Lemma 3.7.(a),we thus have (p, f, u ∞ ) ∈ D End .Finally, From Theorem 2.5 we get End f p (u ∞ ) = q, while from Proposition 3.4 we get and thus We claim that N (γ(t), u ∞ (t)) is constant for almost every t ∈ [0, 1].Indeed, if not, we have that ℓ(p, f, u ∞ , N ) < J(p, f, u ∞ , N ).Moreover, since we have (24), (25), and the trivial inequality ℓ ≤ J, we get that on controls that realize the distance one has ℓ = J, which gives the sought contradiction.Therefore, from the minimality, we get N (γ(t), u ∞ (t)) = d (f,N ) (p, q) for almost every t ∈ [0, 1].Notice that, for curves of constant speed, energy and length are equal.
Proof.By Lemma 3.7.(d), the bound (28) holds for every p, q ∈ M .Therefore, a d (f,N ) -closed set is also ρ-closed, and a d (f,N ) -bounded set is also ρ-bounded.In particular, a d (f,N ) -closed and d (f,N ) -bounded set is compact.Hence closed bounded sets in (M, d (f,N ) ) are compact, and thus d (f,N ) -Cauchy sequences converge.Proposition 3.9 readily implies that (M, d (f,N ) ) is a geodesic space.Remark 3.11.We know that for every R > 0 there are L R > 0 so that the two conditions in (26) hold, with two independent constants.One can modify Proposition 3.10 by requiring that the growth of L R as R → ∞ are slow enough, although not bounded.However, we don't need such a finer analysis.

CC distances and sub-Finsler lengths of curves. We use the notation
LipΓ from Section 2.1, and N from Section 3.2.In this section we prove that the distance d (f,N ) is obtained as the infimum of the length of curves, where the length element is the natural sub-Finsler structure on T M associated to (f, N ), see (32).
For f ∈ LipΓ(E * ⊗ T M ) and N ∈ N , define ) is compact.We fix a complete Riemannian metric ρ on M and apply (28) with K = γ([0, 1]), and R = 1, obtaining that the curve γ : [0, 1] → (M, ρ) is locally Lipschitz and thus an absolutely continuous curve in the manifold M .Without losing in generality, we can assume that γ is 1-Lipschitz, that is, for every t, s ∈ [0, 1], By Proposition 3.9 (with K = {f}, K = γ([0, 1]) and K = {N }), for every n ∈ N large enough and 0 ≤ j ≤ 2 n − 1 integer, there is a control u ) is a geodesic parametrized with constant speed, with end point γ N (γ Thus, from Lemma 3.7.(b)and (c), we obtain that there exists L such that, for n large enough, u where we mean that u Since the controls u (n) are uniformly bounded in n, there is a subsequence {u (n k ) } k that weakly* converges to some u (∞) .
We claim that u (∞) is a control for γ.Let γ n be the integral curve of u (n) with starting point γ n (0) = γ(0).Notice that γ n is the reparametrization on [0, 1] of the concatenation of the γ (n) j for j from 0 to 2 n − 1.In particular, γ n ( j 2 n ) = γ( j 2 n ) for all n and j.
For every n large enough and every ĵ < 2 n, there exist ε n, ĵ > 0 for which the control u (∞) can be integrated on the interval ĵ 2 n , ĵ 2 n + ε n, ĵ .Namely, there exists a curve . By Proposition 2.9 (and by taking an affine reparametrization), the restrictions uniformly converge to η n, ĵ on compact subsets.By continuity of η n, ĵ , and by the previous convergence, it follows that η n, ĵ (t) = γ(t) for all t in the respective domains, and thus γ is an integral curve of u (∞) .
In the rest of this section we will not need the following Lemma 3.13, since it will be enough to use Lemma 3.12.We decided to keep this result here for an independent interest because the proof is different and because it says something more precise, that is, that γ is tangent to the image of f exactly at all points of differentiability of γ.
where γ is differentiable.
Using the fact that f ∈ LipΓ(E * ⊗T M ), that N is continuous, and that f(0, v) n = 0 for every v ∈ E, we deduce that there are r > 0 and C > 0 such that, for every p ∈ R n with |p| ≤ r and every v ∈ E, By definition of d (f,N ) , for every t there is u t ∈ L ∞ ([0, 1]; E) such that End f 0 (u t ) = γ(t) and J(0, f, u t , N ) ≤ 2d (f,N ) (0, γ(t)).Let σ t : [0, 1] → M be the curves with control u t and starting point 0.
By construction, we have that for every y ∈ Y the following holds and, since n∈N Z n is dense in X, we have lim n→∞ d(g n (y), f −1 (y)) = 0. Therefore, taking into account (34), every y ∈ Y gives a Cauchy sequence {g n (y)} n with lim n→∞ d(g n (y), f −1 (y)) = 0. Since X is complete, we can define As g is the pointwise limit of a sequence of Borel functions, g is also Borel.Moreover, d(g(y), f −1 (y)) = 0, i.e., there is a sequence x j ∈ f −1 (y) converging to g(y).Hence, by the continuity of f , we have f (g(y)) = lim j→∞ f (x i ) = y.
If v is only measurable, then there is a Borel function v ′ that is equal to v almost everywhere, and so we can apply the proposition from the Borel setting.
Proof.The implication (i) ⇒ (iv) is proven in Lemma 3.12.The implication (iv) ⇒ (i) is a direct consequence of the definition of d (f,N ) .
Notice that the statement (iv) in Lemma 3.16 does not depend on N : that a curve is Lipschitz does not depend on the particular norm we choose.We can thus say that a curve γ : [a, b] → M is f-Lipschitz if, up to an affine reparametrization, statement (iv) in Lemma 3.16 holds.
We can define the length of f-Lipschitz curves in three ways, which we will show being equivalent.Proposition 3.17.Let f ∈ LipΓ(E * ⊗ T M ) and N ∈ N .For every f-Lipschitz curve γ : [a, b] → M the following three quantities are equal: Moreover, the infimum in the definition of L 3 (γ) is a minimum.
Proof.Notice that all three quantities L j (γ) are invariant under affine reparametrizations of γ, so we may assume a = 0, and b = 1.
The quantities L 2 (γ) and L 3 (γ) are equal by Lemma 3.16, which also shows that the infimum in the definition of L 3 (γ) is a minimum.
Let us now show that is Lipschitz, we have that its metric derivate exists for almost every t ∈ (0, 1), and L 1 (γ) = where in the last inequality we have used the definition of the length L 1 (γ).Taking ε → 0 in the previous inequality we get L 1 (γ) ≥ L 2 (γ), which is the sought inequality.
Thus we finally get that L 1 (γ) = L 2 (γ), and the proof is concluded.
Proof.It is a direct consequence of Proposition 3.17 Proof.It is a direct consequence of Proposition 3.17, according to which L 1 = L 2 , and [6, Proposition 2.4.1.].
3.5.Limits of CC distances.We use the notation LipΓ from Section 2.1, and N from Section 3.2.In this section we investigate what happens when one takes the limit of Carnot-Carathéodory distances associated to (f, N ).We prove a relaxation property of the limit distance, and finally the main theorem of this section, i.e., Theorem 1.4.
Moreover, if K ⊂ M is a compact set, there exists C > 0 such that whenever p ∈ K and q ∈ M with d (f∞,N∞) (p, q) ≤ C, one has that (36) d (f∞,N∞) (p, q) = inf lim inf n→∞ d (fn,Nn) (p n , q n ) : p n → p, q n → q .
In the setting of Lemma 3.7, we can take C = R L 2 , for R, L > 0 satisfying (26) on the compact set Bρ (K, 1) in place of K and with Proof.Let us first prove (35).If d (f∞,N∞) (p, q) = ∞, there is nothing to prove, so we assume Let p n ≡ p and q n = End fn p (u) that exists for n big enough thanks to Proposition 2.9 (1).By Theorem 2.5, q n → q, and moreover, by the definition of the distance, Let γ n be the curve γ n (t) := End fn p (tu).By Proposition 2.9, γ n → γ ∞ uniformly.Hence, by possibly passing to subsequences, We conclude that lim inf n→∞ d (fn,Nn) (p, q n ) ≤ J(p, f ∞ , u, N ∞ ) ≤ d (f∞,N∞) (p, q)+ε.Since ε can be taken arbitrarily small, we get the sought claim.
Let us now prove (36).In the notation of Lemma 3.7, take K := {f n } n∈N ∪{f ∞ }, K := {N n } n∈N ∪ {N ∞ }, and R > 0. Let L be the constant of Lemma 3.7 associated to these choices on the compact set Bρ (K, 1) in place of K, and set and K ⊂ M be a compact set.Assume that f ∞ is essentially non-holonomic and fix a complete Riemannian metric ρ on M .
We are now ready to prove the main result of the paper, namely Theorem 1.4.
Proof of Theorem 1.4.In order to prove item (i), we define for p ∈ M Notice that, if U (p) ∩ U (q) = ∅ then U (p) = U (q).Therefore, {U (p)} p∈M is a partition of M .Moreover, by Lemma 3.21 applied to the constant sequence ( f, N ) and the compact set {p}, p is always in the interior of U (p).Now, if p ′ ∈ U (p) then p ′ is in the interior of U (p ′ ) = U (p) and thus we proved that U (p) is open for every p ∈ M .Since M is connected, we conclude that U (p) = M .The proof of item (ii) has two parts and uses an auxiliary complete Riemannian distance ρ on M .First, if U ⊂ M is d ( f, N ) -open and p ∈ U , then there is ε > 0 so that B ( f, N ) (p, ε) ⊂ U .By Lemma 3.21 applied to the constant sequence ( f, N ) and the compact set {p}, there is Item (iii) is proven as follows.We show that every o ∈ M has a compact neighborhood U so that d (fn,Nn) → d ( f, N ) pointwise on U ×U .By Proposition 3.22, {d (fn,Nn) } n∈N ∪ {d ( f, N ) } is an equicontinuous family of functions U × U → R, hence pointwise convergence would imply the uniform convergence on U × U .
Let us show the pointwise convergence.Let r > 0 so that B( f, N) (o, r) is compact.This can be done thank to the item (ii) that we previously proved.By Proposition 3.20 there is C > 0 so that for every p ∈ B( f, N ) (o, r) and q ∈ M with d ( f, N) (p, q) ≤ C, one has (36).We may assume r < C/2, that is, that (36) holds for every p, q ∈ B( f, N) (o, r).So, let p, q ∈ B( f, N ) (o, r) and p n → p and q n → q so that d ( f, N) (p, q) = lim n→∞ d (fn,Nn) (p n , q n ). Then We need to show that the previous limit is zero.Let ε > 0. By Proposition 3.22, there are n 0 > 0 and δ > 0 so that, if n > n 0 , we have ρ(p n , p) + ρ(q n , q) < δ, and then d (fn,Nn) (p n , q n ) − d (fn,Nn) (p, q) < ε.This shows that the limit is zero.Item (iv) is a consequence of item (iii) together with the forthcoming metric Lemma 3.25.Remark 3.24 (About the completeness assumption in Theorem 1.4(iv)).In this remark we show that the assumption of the completeness of d ( f, N ) in Theorem 1.4(d) is necessary in order to have the uniform convergence lim (f,N )→( f, N ) d (f,N ) = d ( f, N ) on compact subsets.In the following example we show that one may not even have pointwise convergence.Thus Corollary 3.23 cannot be improved in general.
Let us fix M := R × (−1, 1) ⊆ R 2 and p := (−2; 0), q := (2; 0).Let us take, for every n ∈ N, a smooth function g n : R 2 → [1, +∞) such that Let d n be the Riemannian distance associated to the Riemannian tensor g n ( dx ⊗ dx + dy ⊗ dy) on M .We can take g n such that we have g n → g ∞ uniformly on compact subsets of M , where g ∞ is a smooth function with g ∞ = 10 inside [−1, 1] × (−1, 1).Let d ∞ the Riemannian distance associated to the Riemannian tensor g ∞ ( dx ⊗ dx + dy ⊗ dy) on M .We have that d n → d ∞ locally uniformly on M , i.e., every p ∈ M has a neighborhoof U such that d n → d ∞ uniformly on U × U .Nevertheless, d n does not converge uniformly to d ∞ on compact subsets of M .Indeed, we have that d n (p, q) ≤ 6, while d ∞ (p, q) ≥ 10.Lemma 3.25.Let Λ be endowed with a sequential topology.Let X be a set.For t ∈ Λ, let d t be a length metric on X. Assume that for some t 0 ∈ Λ we have that (X, d t0 ) is boundedly compact.Assume that for every point x ∈ X and every sequence t n → t 0 there exists a d t0 -neighborhood U of x such that (40) sup as n → +∞.Hence for every d t0 -compact set K we have Moreover,we have that, for every x ∈ X, and for every sequence in the pointed Gromov-Hausdorff sense.
Proof.Let us first prove (41).Fix K a d t0 -compact set.Let D := diam dt 0 K and let K ′ := B dt 0 (K, D + 3) be the closed Suppose by contradiction that (41) does not hold for some sequence t n → t 0 .Hence, up to passing to subsequences, we have that for some 0 < ε < 1 and every n ∈ N the following holds (43) sup For every s ∈ K ′ there exists U s a d t0 -neighborhood of s such that (40) holds for the sequence {t n } n∈N .Since K ′ is compact, we can extract a finite covering of K ′ from {U s } s∈K ′ .Hence there exists m ∈ N and s 1 , . . ., s m ∈ K ′ such that For the sake of simplicity we rename U si =: U i for every i = 1, . . ., m.Let us take N big enough such that for every n ≥ N and every i = 1, . . ., m we have (44) sup We now aim at showing that for every p, q ∈ K and every n ≥ N we have Since d t0 is a length distance, given p, q ∈ K, there exists a curve γ : For every α ∈ [0, 1] we have that and hence γ ⊆ int(K ′ ).We now aim at finding on γ a finite number i, with i ≤ m, of points p = p 1 , p 2 , . . ., p i = q such that for every j = 1, . . ., i − 1 we have that that p j , p j+1 are in the same U kj , for some k j ∈ {1, . . ., m}.We define such a sequence inductively.First, since p ∈ K, there exists a k 1 ∈ {1, . . ., m} such that p ∈ U k1 .Let us suppose that the sequence p = p 1 , . . ., p ℓ has been defined for some ℓ ∈ N, in such a way that (i) for every j = 1, . . ., ℓ, there exist k j ∈ {1, . . ., m} that are pairwise distinct such that (ii) p j , p j+1 ∈ U kj for every j = 1, . . ., ℓ − 1, and (iii) Obviously we have p ℓ+1 ∈ U k ℓ .If α ℓ+1 = 1 the process ends and q = p ℓ+1 ∈ U k ℓ+1 with k ℓ+1 distinct from every k 1 , . . ., k ℓ by the inductive definition of the α's.If not, we now show that p ℓ+1 ∈ U k ℓ+1 for some k ℓ+1 ∈ {1, . . ., m} different from every k 1 , . . ., k ℓ .This is true since for every η > 0 small enough we have that γ(α + η) ∈ K ′ and hence γ(α + η) ∈ U kη , where k η ∈ {1, . . ., m}.Since k η ranges in a finite set, there exists k ℓ+1 ∈ {1, . . ., m} such that γ(α + η j ) ∈ U k ℓ+1 for a sequence η j → 0. Moreover k ℓ+1 has to be different from every k 1 , . . ., k ℓ , since it is inductively defined by means of (47).This eventually proves that, after at most m steps, we end the process at q, since also q ∈ U ki for some k i ∈ {1, . . ., m}.Hence the claim is shown.
Hence we now want to obtain (45).Fix p, q ∈ K, n ≥ N , and take the chain of points p = p 1 , . . ., p i = q previously defined.Hence where the first inequality is an application of the triangle inequality; the second inequality comes from (44), the fact that p ℓ , p ℓ+1 ∈ U k ℓ for some k ℓ ∈ {1, . . ., m}, and the fact that i ≤ m; the third inequality comes from the definition of length; and the fourth is a consequence of (46).
With a slight variation of the previous argument, we now aim at showing that for every p, q ∈ K and every n ≥ N we have (49) d t0 (p, q) ≤ d tn (p, q) + ε.
Given p, q ∈ K and n ≥ N , since d tn is a length distance, there exists a curve We do not know a priori if γ tn ⊆ K ′ , but nevertheless we may argue as before, paying attention to one more detail.Again, we aim at finding on γ tn a finite number i, with i ≤ m, of points p = p 1 , p 2 , . . ., p i = q such that for every j = 1, . . ., i − 1 we have that that p j , p j+1 are in the same U kj , for some k j ∈ {1, . . ., m}.We proceed by induction.Since p ∈ K, there exists a k 1 ∈ {1, . . ., m} such that p ∈ U k1 .Let us suppose that the sequence p = p 1 , . . ., p ℓ has been defined for some ℓ ∈ N, in such a way that items (i), (ii), and (iii) above hold.Hence define (51) Clearly p ℓ+1 ∈ U k ℓ .We now first show that p ℓ+1 ∈ int(K ′ ).Indeed where the first inequality is a consequence of the triangle inequality; the second is a consequence of (44) and the fact that the chain of points has cardinality not greater than m; the third inequality is a consequence of the definition of length; the fifth is a consequence of (50); and the sixth is a consequence of (45).Now, arguing exactly as before, we can show that p ℓ+1 ∈ U k ℓ+1 with k ℓ+1 ∈ {1, . . ., m} different from all k 1 , . . ., k ℓ .Now to obtain (49) one argues exactly as before.Namely, for p, q ∈ K, and n ≥ N we fix a chain of points p = p 1 , . . ., p i = q inductively constructed as above, and we repeat the estimate (48) exchanging the roles of d tn and d t0 .Hence, (45) and (49) give the sought contradiction with (43), thus proving (41).
Taking into account the definition of pointed Gromov-Hausdorff convergence, see [6, page 272], to prove (42) it is sufficient to use (41) and that, if we fix x ∈ X, we have that, for every t n → t 0 and for every R, The previous inequality is a direct consequence of a slight variation of the second argument above.Indeed, arguing as before, one can show that for every sequence t n → 0 and every R > 0 there exists N sufficiently big such that for every n ≥ N we have Bdt n (x, R) ⊆ Bdt 0 (x, R + 1).

Examples
In this section we discuss several examples in which we can apply our main convergence result Theorem 1.4.
In Section 4.1 we use Theorem 1.4 to directly prove that the asymptotic cone of the Riemannian Heisenberg group is the sub-Riemannian Heisenberg group, see Proposition 4.1.The same reasoning can be easily generalized to arbitrary Carnot groups.
In Section 4.2 we state and prove Mitchell's Theorem in the sub-Finsler cathegory for a continuously varying norm on the manifold, see Theorem 1.5.We give for granted the construction of privileged coordinates and of the nilpotent approximation, for which we refer the reader to standard and well-established references, see [4], [7, Section 2.1], [1, Sections 10.4-10.5-10.6],or the recent [13].Hence we exploit Theorem 1.4 to directly prove the final convergence part of Mitchell's Theorem in such a general setting.
In Section 4.3 we use Theorem 1.4 to directly prove Theorem 1.6.Namely, we prove that on a connected Lie group the CC distances associated to bracketgenerating sub-spaces and norms that converge are uniformly convergent on compact subsets.The latter result has been used in the very recent [8].
In Section 4.4 we record a general approximation theorem for sub-Finsler distances associated to converging vector fields on a manifold.Notice that Theorem 4.5 can be used to produce Finsler approximation of sub-Finsler manifolds.
4.1.Asymptotic cone of the Riemannian Heisenberg group.The first application we discuss is the well-known fact that the asymptotic cone of the Riemannian Heisenberg group is the sub-Riemannian Heisenberg group.
Using exponential coordinates of the first kind, we identify the first Heisenberg group H 1 with the manifold R 3 endowed with left-invariant frame Let •, • be the left-invariant Riemannian tensor on H 1 that makes the above frame orthonormal, and d R the corresponding distance.Let d sR be the sub-Riemannian distance defined by X, Y , namely for every p, q ∈ M , where the infimum is taken over absolutely continuous curves γ.
Proposition 4.1.The asymptotic cone of Then, , we have that d 1 = d R and d 0 = d sR .Since f 0 is totally non-holonomic, see Proposition 2.12, we obtain from Theorem 1.4(iv) that d ε → d 0 as ε → 0, uniformly on compact subsets of What is only left to show is that (H 1 , d ε ) is isometric to (H 1 , εd R ).Notice that εd R is the Riemannian distance defined by the orthonormal frame (X/ε, Y /ε, Z/ε).Notice also that the map δ ε (x, y, z) := (εx, εy, ε 2 z) satisfies Mutatis mutandis the statement of Proposition 4.1 works, with the same proof, for arbitrary Carnot groups.Actually, with some additional work, one could also recover the well-known fact that the asymptotic cone of a sub-Finsler nilpotent Lie group is a sub-Finsler Carnot group.4.2.Tangents to sub-Finsler manifolds.In this section we discuss the celebrated Mitchell's Theorem.We take for granted the existence of a system of priviliged coordinates, for which we refer the reader, e.g., to the complete discussion in [7, Section 2.1].We stress that, up to the authors' knowledge, this is the first time that such a theorem is stated in the generality of sub-Finsler cathegory.Standard references for the sub-Riemannian Mitchell's Theorem are [7, Theorem 2.5], [12,4], and [2, Sections 10-4-10.5-10.6].Our aim, here, is to give a complete and detailed proof of the final convergence part of the proof.
Hereafter we follow the terminology of Jean's book [7, Section 2.1].We fix a smooth manifold M m of dimension m ∈ N, k ∈ N, and k smooth vector fields X 1 , . . ., X k .We say that the family X := {X 1 , . . ., X k } is a non-holonomic system if it is bracket-generating.Notice that we do not assume that the rank of X is constant.
Let us fix E := R k , with canonical basis {e 1 , . . ., e k }, and let us fix N : M × E → [0, +∞) a continuously varying family of norms on E. Attached to the previously defined non-holonomic system X and to N there is a notion of a sub-Finsler metric | • | X ,N , see (4).In the specific case in which N (p, •) is the sandard Euclidean norm for every p ∈ M , the previous sub-Finsler metric is exactly the one considered in Jean's book [7,Equation (1.4)].
The sub-Finsler metric | • | (X ,N ) gives raise to a length distance d (X ,N ) , see ( 5) and compare with [7,Definition 1.3].Notice that, by definition, and by exploiting Lemma 3.7(d) and Theorem 1.4(ii), we get that any two distances d (X ,N1) and d (X ,N2) are locally bi-Lipschitz equivalent.As a consequence we stress that the notion of non-holonomic order of a smooth function/vector field at a point p ∈ M , see [7,Section 2.1], can be equivalently given by using any of such distances.
We define the Lipschitz-vector-field structure on M modelled by R k Proof of Theorem 1.5.Let o ∈ M be as in the statement.There exists a neighborhood U ⊆ M m of o and a neighborhood V ⊆ R m of 0 such that (x 1 , . . ., x m ) : U → V is a system of priviliged coordinates, see [7,Definition 2.5] and [7, pages 22-23].Hereafter we will identify U ⊆ M with V ⊆ R m by means of the coordinates (x 1 , . . ., x m ).
for almost every t ∈ [0, 1], we let γ ε := δ ε • γ.We notice that γ ε has support contained in V and, from the definition of f ε , we have Since we have that γ ε ⊆ V , by taking into account the previous computation, the definition of the norms N ε , see (56), and the definition of the distance, see (24), we finally get (58) for every p, q ∈ δ −1 ε V .The latter reasoning implies that for every R > 0 there exists ε 0 small enough such that for every ε < ε 0 one has and the isometry is given by δ ε .Thus, the latter, together with the convergence in (57), and (55), directly implies that the Gromov-Hausdorff tangent of (M, d ( f1,N ) ) at o is R n equipped with the sub-Finsler distance induced by the vector fields X1 , . . ., Xk and the norm N (o, •), which is what we wanted.
Remark 4.3.We stress that our convergence result Theorem 1.4 holds in the Lipschitz cathegory, provided the essentialy non-holonomicity of the limit.Hence, whenever one has some analogues of priviliged coordinates while dealing with less regular vector fields, the proof of Theorem 1.5 is very likely to be adapted.

4.3.
Left-invariant CC distances on Lie groups.Let G be a connected Lie group, and let g be its Lie algebra.Given a vector subspace H ⊆ g of g, and a norm b on H, we associate to (H, b) a left-invariant sub-Finsler structure (D, b) as in (6).Moreover, we define d (H,b) as in (7).
Let us denote by k the dimension of H. Choose a basis {v 1 , . . ., v k } of H and define X i to be the left-invariant extension of v i , for every i = 1, . . ., k.
), for all p, q ∈ G.
We now give the proof of Theorem 1.6.
Proof of Theorem 1.6.Let us choose a bracket-generating basis {v 1 , . . ., v k } of H.By the fact that H n → H, we have that, for every n ∈ N, there exists a basis {v n 1 , . . ., v n k } of H n such that v n i → v i as n → +∞ for every i = 1, . . ., k.Let X i be the left-invariant extension of v i for every i = 1, . . ., k, and let X n i be the left-invariant extension of v n i for every i = 1, . . ., k, and every n ∈ N. Let us fix E := R k with basis {e 1 , . . ., e k }.
For every n ∈ N, we define the Lipschitz-vector-field structure f n modelled by R k as (f n ) p (e i ) := X n i (p), for all p ∈ G, for all i = 1, . . ., k, and the Lipschitz-vector-field structure f modelled by R k as f p (e i ) := X i (p), for all p ∈ G, for all i = 1, . . ., k.
By the convergence v n i → v i in g, we have that f n → f in the sense of Definition 2.2.For every n ∈ N we define the continuously varying norm N n : G × E as Remark 4.4.As a special case of Theorem 1.6, when b n ≡ b is a norm coming from a scalar product, it follows that the Condition 3.9 conjectured in [8] is always true.

4.4.
Limit of sub-Finsler distances on a manifold.In this section we prove a general convergence result that is a consequence of Theorm 1.4.Theorem 4.5.Let M m be a smooth connected manifold of dimension m.Let k ∈ N.For every λ ∈ [0, 1) consider X λ := {X λ 1 , . . ., X λ k } a family of smooth vector-fields such that (1) X λ i are locally equi-Lipschitz for every i = 1, . . ., k, and every λ ∈ [0, 1); (2) X λ i → X 0 i uniformly on compact sets as λ → 0, for every i = 1, . . ., k.Let us assume that {X 0 1 , . . ., X 0 k } is a bracket-generating set of vector fields.Let E := R k with basis {e 1 , . . ., e k } and, for every λ ∈ [0, 1), let N λ : M × E → [0, +∞) be a continuously varying norm.Assume that N λ → N 0 uniformly on compact sets.For each λ, let | • | λ the sub-Finsler metric defined by Proof.For every λ ∈ [0, 1), let f λ ∈ LipΓ(E * ⊗ T M ) be defined as f λ (p)(e i ) := X λ i (p), for all p ∈ M , and for every i = 1, . . ., k. From the hypotheses we get that (f λ , N λ ) → (f 0 , N 0 ) as λ → 0, where the convergence of the first components has to be intended in the sense of Definition 2.2, and the convergence of the second components has to be intended in the uniform sense on compact sets.Moreover, as a consequence of Corollary 3.19, we have that d λ = d (f λ ,N λ ) for every λ ∈ [0, 1).Hence the results follow from Theorem 1.4(iii) and Theorem 1.4(iv).has nonzero degree.Let f n : Ω → R m be continuous functions that converge uniformly on Ω to f ∞ .
Then there exist δ > 0 and N ∈ N such that for all n ≥ N , we have Proof.Let B := B(0, r) ⋐ Ω and, for every n ∈ N ∪ {∞}, define S n := f n (∂B).Since S ∞ is compact and it does not contain 0 by assumption, we have that δ := d(S ∞ , 0) > 0. Hence, there exists N ′ such that for n ≥ N ′ we have This means that for every x ∈ ∂B and every n ≥ N ′ , we have from which we deduce that d(S n , 0) ≥ δ/2 for every n ≥ N ′ .Hence, for every n ∈ N ∪ {∞} such that also n ≥ N ′ , we can define g n : ∂B → S m−1 as From the hypothesis we have that g n → g ∞ uniformly on ∂B.Hence, for some N ≥ N ′ , we have that, for every n ≥ N , the map Fix n ≥ N and suppose by contradiction that (64) is not true.Then there exists p ∈ B(0, δ/2) \ f n (B).For every 0 ≤ η ≤ 1 let us define φ η : ∂B → S m−1 as which is well defined since p / ∈ f n (B).Clearly, as η varies from 0 to 1, the maps φ η are an homotopy between the constant map φ 0 ≡ p/|p| and φ 1 : therefore, deg φ 1 = 0.Moreover, for every 0 ≤ η ≤ 1, we can define ψ η : ∂B → S m−1 as Notice that ψ η is well-defined because |f n (x)| ≥ δ/2 > η|p| for every 0 ≤ η ≤ 1 and for every x ∈ ∂B.Since the maps ψ η are an homotopy from ψ 0 = g n to ψ 1 = φ 1 , we obtain deg g n = deg(φ 1 ) = 0, which is in contradiction with the fact that deg(g n ) = 0 for every n ≥ N .Therefore, (64) must be true, and it directly implies the assertion.
In order to apply the previous result, the following lemma gives a criterium to check the hypothesis of Lemma B.1.

Lemma B.2.
Let Ω ⊂ R m be an open set containing 0, and let f : Ω → R m a topological embedding with f (0) = 0. Let r > 0 such that B(0, r) ⋐ Ω and define φ : Proof.The function φ is the composition Notice that the maps x → rx and x → x |x| are retracts of R m to S m−1 , and thus they induce isomorphisms between the homology groups.Since f is an embedding and f (0) = 0, also f induces an isomorphism between the homology groups of Ω \ {0} and those of f (Ω) \ {0}.Finally, since f (Ω) is open, every homology class of R m has a representative inside f (Ω) and thus the immersion f (Ω) ֒→ R m defines a surjective morphism of the corresponding homology groups.
We conclude that the induced group morphism φ * : H n (S m−1 ) → H n (S m−1 ) is a surjective group morphism from Z to Z and thus deg(φ) ∈ {−1, 1}.
Proof.Fix such ρ and K. Let us denote, for λ ∈ Λ and p ∈ M , X λ i (p) := f(λ, p, e i ), where {e 1 , . . ., e k } is the standard basis of E = R k .
We know that {X λ0 i } k i=1 is a bracket-generating set of vector fields on M .Hence, there exists a compact neighborhood I 1 of λ 0 such that {X λ i } k i=1 is a bracketgenerating set of vector fields on B ρ (K, 1).
From Claim 1 and a routine compatness argument, as already done at the end of Lemma 3.21, we have that for every ε > 0 there exists δ > 0 such that for every x ∈ K and every λ ∈ I 1 we have (72) B ρ (x, δ) ⊆ B d λ (x, ε).
From (72) the proof of the lemma follows with the following argument.Let K be a path-connected compact set containing K. For instance, K can be chosen to be a closed ρ-balls of sufficiently large radius.The inequality in (68) is trivially satisfied if we define β as β(s) = sup{d λ (p, q) : p, q ∈ K, λ ∈ I 1 , ρ(p, q) ≤ s}, for every s ∈ (0, +∞).
We now give the proof of Theorem C.2.The strategy is different with respect to the proof of Theorem 1.4(iv).There, we first proved the local uniform convergence, relying on the relaxation result in Proposition 3.20, and then we upgrade it to a uniform convergence on compact sets thanks to Lemma 3.25.Here, instead, we directly obtain the uniform convergence on compact sets by making a careful use of Gronwall's Lemma A. By continuity, there exists a constant C > 0 such that d λ0 (p, q) ≤ C for every p, q ∈ K. Let K ′ := B λ0 (K, C + 1) the closed tubular neighborhood of K of radius C + 1.Since (M, d λ0 ) is boundedly compact, we deduce that K ′ is compact.
Let β be the function, and I λ0 be the compact neighborhood of λ 0 , associated to K ′ given by Lemma C.3.We have that, for some ϑ > 0, ρ(p, q) ≤ ϑ|p − q| for every p, q ∈ K ′ .Thus, up to renaming β, for every p, q ∈ K ′ , and for every λ ∈ I λ0 , Since N (λ, p, •) is a norm for every λ ∈ I λ0 and every p ∈ M , and since N is continuous, we get that there exists a compact set K ′′ ⊆ E such that (73) if N (λ, x, v) ≤ β(diam |•| K ′ ) + 1 for some λ ∈ I λ0 and x ∈ K ′ , then v ∈ K ′′ .Moreover, by definition of continuously varying CC-structures, (67), we have that there exists L > 0 such that for every λ ∈ I λ0 and v ∈ K ′′ the map Because of continuity of the functions N and f we get that there exist 0 < δ 1 < ε and a compact neighborhood where a is chosen such that a e L −1 L < δ 1 .We now prove the following claim.Claim 1.For every λ ∈ I ′ λ0 and every p, q ∈ K, we have d λ0 (p, q) ≤ d λ (p, q) + 2ε + β(ε).
From Claim 1 and Claim 2 jointly with the fact that β(ε) → 0 as ε → 0 we get the proof of the Theorem.

1 0 1 0 1 0
|w(x)| dx ≤ 1 and xw(x) dx = 1.One can show the latter claim from the fact that it would hold 1 = xw(x) dx ≤ 1 0 |w(x)| dx ≤ 1, from which it would follow that 1 0 xw(x) dx = 1 0 |w(x)| dx.Writing w as the difference between its positive and negative part, one gets a cotradiction.

f 1
(p)(e i ) := X i | p , for every p ∈ M , and i = 1, . . ., k.The distance d ( f1,N ) as in Definition 3.6, coincides with the length distance induced by | • | X ,N as above, by virtue of Corollary 3.19.
Let E := R k with the canonical basis {e 1 , . . ., e k }.Define a Lipschitz-vector-field structure f modelled by R k as f p (e i ) := X i (p), for all p ∈ G, and i = 1, . . ., k. Define the continuously varying norm N : G × E → R as N (p, w) := b k i=1 w i v i , for all p ∈ G, and w = k i=1 w i e i ∈ E. By virtue of Corollary 3.19 we deduce that (62)

N
n (p, w) := b n k i=1 w i v i , for all p ∈ G, and w = k i=1 w i e i ∈ E, and the continuously varying norm N as N (p, w) := b k i=1 w i v i , for all p ∈ G, and w = k i=1w i e i ∈ E.By the fact that b n → b uniformly on compact sets it follows that N n → N uniformly on compact sets.Hence we showed that(f n , b n ) → (f, b) as n → +∞.Let us finally check that the remaining hypotheses of Theorem 1.4 are met.Indeed, f is essentially nonholonomic due to Proposition 2.12.Moreover, (G, d (H,b) ) is boundedly compact because, by homogeneity and Theorem 1.4(ii), there exists ε > 0 such that for every p ∈ G the closed ball B d (H,b) (p, ε) is compact.Hence an application of Theorem 1.4(iv), together with the equality (62), gives the sought conclusions.
v ∈ T p M , and p ∈ M .Let d λ be the sub-Finsler distance associated to the sub-Finsler metric | • | λ (cf.(5)).Hence, d λ → d 0 locally uniformly on compact sets of M × M , as λ → 0.Moreover, if (M, d 0 ) is a complete metric space, we have that d λ → d 0 uniformly on compact sets of M × M , as λ → 0, and for every p ∈ M , we have that (M, d λ , p) → (M, d 0 , p) in the pointed Gromov-Hausdorff topology as λ → 0.
has a regular point at t and sends t to x.Now, let ω := ω 1 . . .ω D be a word of D letters such that it contains as a subword every string of 2n elements chosen among {1, . . ., k}.Notice now that the map (70) Ψ : (λ, t 1 , . . ., t D ) → Φ tD X together with the maps D T Ψ, D 2 T Ψ -where D T denotes the differential with respect to the components in R D -are continuous and well defined onI 2 × B |•|1 (0, ξ), where B |•|1 (0, ξ) is the ball in R D with respect to the ℓ 1 -norm |•| 1 centeredat 0 and with a sufficiently small radius ξ, and I 2 ⊆ I 1 is a sufficiently small compact neighborhood of λ ′ .The last assertion is a consequence of an iterated application of Gronwall's Lemma (see Lemma A.1), and the fact that we have the continuity property in Definition C.1(1).Define the compact set K := Ψ(I 2 × B(0, ξ)) in M .Hence, by continuity of the norm N , there exists L > 0 such that (71) N (λ, p, v) ≤ L|v| 1 , for all λ ∈ I 2 , p ∈ K, and v ∈ E.Let us now conclude the proof of the claim.Fix ε > 0. Letν := min{ξ/2, ε/(2DL)},where ξ is defined above.Notice that, by what we noticed above, we have that there exists t ∈ R D with | t| 1 < ν such that Ψ(λ ′ , •) has a regular point at t and Ψ(λ ′ , t) = x.In addition to this, Ψ(λ, t) → Ψ(λ ′ , t) as λ → λ ′ , uniformly when t ∈ B |•|1 (0, ξ) ⊆ R N , and the same convergence holds with D T Ψ, D 2 T Ψ.The last assertion is a consequence of the fact that the maps Ψ, D T Ψ, D 2 T Ψ are continuous, and thus uniformly continuous on compact sets.

1 .
Proof of Theorem C.2.We embed M smoothly isometrically into some R N , on which we denote with | • | the standard norm.Let us fix a compact set K and a Riemannian metric ρ on M .Notice that on every compact set of M , ρ and | • | are biLipschitz equivalent.Let us fix 0 < ε < 1/2.