Tangent lines and Lipschitz differentiability spaces

We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces. We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line. Metric differentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz differentiability spaces. We show that any tangent space of a Lipschitz differentiability space contains at least $n$ distinct tangent lines, obtained as the blow-up of $n$ Lipschitz curves, where $n$ is the dimension of the local measurable chart. Under additional assumptions on the space, such as curvature lower bounds, these $n$ distinct tangent lines span an $n$-dimensional part of the tangent space.


Introduction
During the past few years there has been growing interest towards studying the infinitesimal structure of "nice" metric measure spaces. One class of nice metric measure spaces is formed by the ones in which Lipschitz functions are differentiable almost everywhere with respect to Lipschitz charts covering the space. The study of such spaces originates from the work of Cheeger [10] and the spaces are now often called Lipschitz differentiability spaces (following Bate [7]). Cheeger proved that a doubling condition on the reference measure and the validity of a local Poincaré inequality (as defined by Heinonen and Koskela [13]) are sufficient for the space to be a Lipschitz differentiability space. Although there are quite wild examples of doubling metric measure spaces supporting a local Poincaré inequality [9,22,33], these assumptions still have strong geometric implications, [10,34,21]. In particular, there are lots of rectifiable curves joining any two points in such a space.
A general Lipschitz differentiability space might not contain any rectifiable curves besides the trivial ones. However, they always contain sufficiently many broken curves in different directions so that the reference measure can be expressed by independent Alberti representations that completely characterize derivatives of Lipschitz functions, see the work of Bate [7]. On the other hand, when we perform a Gromov-Hausdorff blow-up of a broken biLipschitz curve γ : Dom (γ) → X of the metric space X at a density point of the domain Dom (γ) the broken curve approaches, after passing to a subsequence, a limit curve defined on the whole R.
We first define metric differentiability, see Definition 3.3, and then prove that, at points of metric differentiability, this limit curve is a line-segment, see Proposition 3. 10. By a result of Kirchheim [20], we observe that metric differentiability coincides with the metric speed at almost every point of Dom (γ). Therefore we deduce that a Lipschitz curve γ is metrically differentiable at almost every point (see also Proposition 3.8 for an alternative proof of this fact). Thus broken biLipschitz curves always converge to a line-segment at almost every point of their domain.
Given an n-dimensional Lipschitz chart on a Lipschitz differentiability space we know from the work of Bate [7] that there exist n independent Alberti representations. Noticing that metric differentiability is Borel measurable, see Lemma 4.1, one can use it in the context of Lipschitz differentiability spaces to deduce that (see Proposition 4.3) at almost every point the blow-up will give n distinct tangent lines. If one also assumes the Lipschitz differentiability space to be doubling, then one can find n distinct tangent lines at almost every point of the tangent space. Theorem 1.1 (Theorem 4.5). Let (X, d, m) be a doubling Lipschitz differentiability space and (U, ϕ) be an n-dimensional chart. Then for m-almost everyx ∈ U , there exist v 1 , . . . , v n ∈ R n linearly independent such that for any element (X ∞ , d ∞ ,x ∞ ) ∈ Tan(X, d,x) and for each z ∈ X ∞ there exist ι z 1 , . . . , ι z n : R → X ∞ so that i) ι z j (0) = z, for any j = 1, . . . , n; ii) d ∞ (ι z j (t), ι z j (s)) = |t − s|, for any j = 1, . . . , n, for all s, t ∈ R; iii) d ∞ (ι z j (t), ι z k (t)) ≥ C|t| · |v j − v k |, for any j, k = 1, . . . , n, for all t ∈ R; for some positive constant C. For each z ∈ X ∞ , each line ι z i is obtained as the blow-up of a Lipschitz curve, with the blow-up depending on z.
The question is then how and what kind of subspace of the tangent space these tangent lines form. Since the Heisenberg group is a Lipschitz differentiability space and purely 2-unrectifiable [4], we know that the tangent lines do not always span an n-rectifiable set. However under the additional assumption that the space is Ahlfors n-regular with n being the dimension of the chart, at almost every point there is a tangent space biLipschitz equivalent to R n , see [12]. We are interested in finding other conditions that would provide information on the tangents.
Our considerations originate from the study of another class of nice metric measure spaces -namely of those with Ricci curvature lower bounds. There are many notions of Ricci curvature lower bounds on metric measure spaces. For the most strict one, the RCD * (K, N ) spaces (defined in [3,1,14,5]), it is known that they infinitesimally look like Euclidean spaces, [16,25]. Moreover, the tangents in an RCD * (K, N ) space are almost everywhere spanned by the tangent lines obtained from the Lipschitz charts as described above, see Section 5 for details. Thus the infinitesimal structure of RCD * (K, N ) spaces is already well understood.
We would like to understand the structure of spaces with Ricci curvature lower bounds with the more general definitions. Most of the definitions are known to imply doubling condition on the measure and a local Poincaré inequality. Thus these spaces are Lipschitz differentiability spaces and Theorem 1.1 holds. One line of investigation is to continue from the proof in [16]. There the fact that RCD * (K, N ) spaces have at least one Euclidean tangent space was proven following the idea of Preiss [28] (and its adaptation to metric spaces by Le Donne [23]) of iterated tangents. The proof essentially used only the fact that the tangent spaces split off any part that is isometric to R.
Taking into consideration also the Lipschitz charts, the isometric splitting property implies the existence of R n in each of the tangent at almost every point, where n is again the dimension of the chart. Theorem 1.2 (Theorem 5.1). Suppose that (X, d, m) is a doubling Lipschitz differentiability space with the splitting of tangents property. Let (U, ϕ) be an n-dimensional chart of (X, d, m). Then for m-a.e.
For the more general CD(K, N ) spaces (see [35,36,24] for the definitions) isometric splitting is impossible since already R n with any norm and the Lebesgue measure satisfies CD(0, n). On the other hand, Ohta has recently shown that a version of splitting theorem holds for Finsler manifolds [27]. Such weaker versions might be enough to give some information on the infinitesimal structure. For example, if the existence of a tangent line would always imply that the tangent could be written to be biLipschitz equivalent to a product R × Y for some metric space Y , the n dimensional Lipschitz chart could result in a piece of the tangent biLipschitz equivalent to R n .
Let us note that for the even more general notion MCP(K, N ) of Ricci curvature lower bound (see [36,26] for the definitions) the above splitting result does not hold even in a topological sense [19]. Moreover, it is not known if a local Poincaré inequality holds in MCP(K, N ) spaces without the nonbranching assumption, and hence we do not know if MCP(K, N ) spaces are Lipschitz differentiability spaces. Even more, it is known that for example the Heisenberg group satisfies the MCP(K, N ) condition, see [17]. Thus the tangent lines cannot biLipschitz span a part of the tangent.
The paper is organized as follows. In Section 2 we recall the notions of pointed measured Gromov-Hausdorff convergence, tangent functions and Lipschitz differentiability spaces. In Section 3 we define the notion of metric differentiability that we will use in this paper and show, using an identity proved by Kirchheim in [20], that it agrees almost everywhere with the metric speed. We also show that at almost every point of a biLipschitz curve the blow-up will be a tangent line. In Section 4 we consider the blow-ups in Lipschitz differentiability space showing that we have n independent tangent lines at almost every point. In the final section, Section 5, following the ideas of David and Schioppa [12,32], we prove that if tangents split off tangent lines then the n independent tangent lines in a Lipschitz differentiability space span a Euclidean R n in the tangent.
After the completion of this note, we learnt from Schioppa and Preiss that our approach can be used together with the very recent work by Cheeger, Kleiner and Schioppa [11] to improve Theorem 1.2 and show that at almost every point X d ∞ × R d = R n , where n is the dimension of the chart.

Preliminaries
A metric measure space is a triple (X, d, m) where (X, d) is a complete and separable metric space and m a positive Borel measure that is also finite on bounded sets. As the main object of our study will be proper spaces, i.e. metric spaces such that each bounded closed set is also compact, we directly incorporate in the definition of metric measure space also the properness assumption. Consequently m will be a positive Radon measure.
We list here two general properties of metric measure space that we will consider during the paper. The metric measure space (X, d, m) is doubling if for each R > 0 there exists C(R) > 0 such that for every x ∈ X, r ≤ R.
With no loss in generality, the function C can be taken non-decreasing. Morevover a metric measure space (X, d, m) supports a local p-Poincaré inequality for some p ≥ 1 if every ball in X has positive and finite measure and for every g ∈ Lip(X, d) for some positive constant L, where B = B r (x 0 ) and g B = ffl B g(x) dm(x). Here for g ∈ Lip(X, d) we also adopt the following notation: |Dg|(x) := sup y =x d(g(y), g(x)) d(y, x) .

2.1.
Convergence of metric measure spaces. The standard notion of topology on equivalence classes of pointed, proper, separable metric spaces is the one induced by the pointed Gromov-Hausdorff convergence, pGH-convergence in brief. This convergence can be characterized in many equivalent ways. We will adopt the one with ε-isometries.
between compact metric spaces is called an ε-isometry provided (i) it almost preserves distances: for all z, w ∈ X, (ii) it is almost surjective: In order to deal with possibly non-compact spaces, it is customary to fix a distinguished pointx ∈ X and to consider ε-isometries defined on an increasing family of balls centered inx. When a distinguished point is fixed, we use (X, d,x) to denote the pointed metric space.
proper, complete metric spaces converges to a pointed, proper, complete metric space (X ∞ , d ∞ ,x ∞ ), if and only if there exist sequences of positive real numbers {ε i } i∈N , {R i } i∈N with ε i → 0, R i → ∞ and a sequence of ε i -isometries, where B Xi Ri (x i ) is the ball in X i , centered inx and of radius R i . We also consider pointed metric measure spaces: a quadruple (X, d, m,x) where (X, d, m) is a metric measure space andx ∈ X a distinguished point.
of pointed metric measure spaces converges in the pointed measured Gromov-Hausdorff convergence to a pointed metric measure space ( if and only if there exist sequences of positive real numbers {ε i } i∈N , {R i } i∈N with ε i → 0, R i → ∞ and a sequence of ε i -isometries, where C b (X ∞ ) stands for the space of continuous and bounded functions with compact support in X ∞ .
Both, the pGH-convergence and the pmGH-convergence can be used to define and study (measured) tangent spaces.
If (X, d) is a metric space andx ∈ X is a distinguished point, then any limit point in the pGHconvergence of any sequence of the form We use Tan(X, d,x) to denote the set of all possible tangent spaces of (X, d) atx.
If (X, d, m) is a metric measure space andx ∈ supp(m) is a distinguished point, for any r > 0, the rescaled and normalized pointed metric measure space is defined as follows: Then a limit point in the pmGH-convergence of the sequence {(X, d/r i , mx ri ,x)} i∈N is a measured tangent space of (X, d, m) atx and to denote the set of all possible measured tangent spaces of (X, d, m) atx we use Tan(X, d, m,x). It is worth noticing that, thanks to compactness properties of the collection of uniformly doubling metric measure space, Tan(X, d, m,x) is always non empty, provided (X, d, m) is doubling.

Tangent functions.
Here we recall few objects and related results presented in [10] and in [18].
If (X, d X ) and (Y, d Y ) are metric spaces and f : X → Y is an ε-isometry, then there exists a (4ε)isometry f ′ : Y → X so that for all x ∈ X and y ∈ Y it holds Such a map is usually called an ε-inverse of f and accordingly we will often adopt the notation f −1 to denote it.
Then to any Lipschitz function g : X → R we can associate a sequence of rescaled functions centered in x: With this in mind, we say that u g : is any ε i -inverse of the approximate isometry f i given by the pmGH convergence of (X, d/r i , mx r ,x) to (X ∞ , d ∞ , m ∞ ,x ∞ ). The term compatible is used to underline that we used the same scaling for the distance and the function g. Remark 2.3. The definition of u g does not depend on the choice of the sequence of the ε i -inverses. Since f i is almost surjective, for any z ∈ X ∞ and i ∈ N sufficiently large, there exists x i ∈ X such that One then easily observes that and since g is Lipschitz, it follows that g i (f −1 i (z)) and g i (f −1 i (z)) have the same limit.
Concerning the existence of compatible tangent functions, the following compactness result holds.
Lemma 2.4. Let (X, d, m) be a doubling metric measure space and a sequence r i → 0 such that where the convergence is in the pmGH sense. Fix also a countable collection F of uniformly Lipschitz functions defined on X. Then possibly choosing a subsequence of {r i } i∈N , for each g ∈ F there exists u g compatible tangent function of g atx.
As one might expect, tangent functions of Lipschitz functions enjoy a generalized notion of linearity. It has different names according to different authors. Here we follow [10] and say that tangent functions to Lipschitz functions, wherever they exists, are generalized linear, see Definition 8.1 of [10]. The terminology used is justified by the fact that being generalized linear on a Euclidean space is the same as being linear in the usual sense, see again [10], Theorem 8.11.

Lipschitz differentiability spaces.
Under fairly general assumption on the structure of the metric measure space, it is proved in [10] that the space of Lipschitz functions has finite dimension in the following sense.
Definition 2.5. Let (X, d) be a metric space and n ∈ N. A Borel set U ⊂ X and a Lipschitz function ϕ : X → R n form a chart of dimension n, (U, ϕ), and a function g : Furthermore a metric measure space (X, d, m) is called a Lipschitz differentiability space if there exists a countable decomposition of X into charts such that any Lipschitz function g : X → R is differentiable at m-almost every point of every chart.
A celebrated result by Cheeger [10] on Lipschitz differentiability spaces can be summarized by the following Theorem 2.6. Let (X, d, m) be a doubling metric measure space supporting a p-Poincaré inequality with constant L ≥ 1 for some p ≥ 1. Then (X, d, m) is a Lipschitz differentiability space.
Subsequently in [7] a finer analysis on curves, and their possible directions with respect to a given chart, was carried on. Here we report only the main statement. We use Γ(X) to denote the set of biLipschitz maps γ : Dom (γ) → X, with Dom (γ) ⊂ R non-empty and compact.
Theorem 2.7 ([7], Theorem 6.6, Corollary 6.7). Let (X, d, m) be a Lipschitz differentiability space and (U, ϕ) an n-dimensional chart. Then for m-a.e. x ∈ U , there exist γ x 1 , . . . , γ x n ∈ Γ(X) such that each Moreover, for any such γ x i , for any Lipschitz g : X → R and m-a.e. x ∈ U , the gradient of g at x with respect to ϕ and γ x 1 , . . . , γ x n equals Dg(x), that is Hence not only the space of Lipschitz functions has locally finite dimension but also each Lipschitz function is locally described in terms of directional derivative with respect to a family of biLipschitz curves.
Here also Keith's results on coordinate functions is worth mentioning: in [18] it is proved that the role of coordinate map ϕ in chart (U, ϕ) can be played by distance functions from a well prepared set. We report here Theorem 2.7 of [18].
2.4. Geodesics in product spaces. If (X, d X ) and (Y, d Y ) are two metric spaces, we can consider the product distance d XY defined by is again a metric space. We recall an easy lemma on geodesics in product spaces.
Proof. We start with the easy inequality: for a, b, c, d positive real numbers, Then let [0, 1] ∋ t → (γ 1 t , γ 2 t ) ∈ X × Y be a geodesic and suppose by contradiction that γ 1 is not. For ease of notation, we can assume that Expanding the squares and using that γ 1 is not a geodesic, we obtain that We can now use the first inequality we wrote and get , violating the triangle inequality. The claim follows.

Tangent lines
Let us start this section by recalling a result from [31], Theorem 7.10: a more general version of Lebesgue Differentiation Theorem. Here and in the sequel L d denotes the Lebesgue measure on R d . Definition 3.1. Fix x ∈ R d and a sequence of Borel sets {E i } i∈N ⊂ R d . We say that {E i } i∈N shrinks nicely to x provided there exist r i > 0 and α > 0 such that for each i ∈ N we have and For the nicely shrinking sets we have the following general version of Lebesgue Differentiation Theorem.
Theorem 3.2. Let f ∈ L 1 (R d , R) be any function. Associate to each x ∈ R d a sequence {E i (x)} i∈N of sets nicely shrinking to x. Then for every Lebesgue point x of f . In particular it holds for L d -almost every x.
Consider now (X, d) a complete, and separable metric space and note that for the next statement we do not need to assume (X, d) to be proper. Definition 3.3. Let γ : Dom (γ) → X be any curve. We say that γ is metric differentiable at t ∈ Dom (γ) provided the following limit lim s,τ →0 nicely d(γ t+s , γ t+τ ) |s − τ | exists for any sequence of s and τ , where with nicely we ask for the interval with boundary formed by t + s and t + τ to shrink nicely to t. In case the limit exists, we denote it with |dγ|(t).
Remark 3.4. By definition, the existence of |dγ| is a priori a more demanding property compared to existence of metric speed |γ|, for its definition see [6].  for all x, u ∈ R n , whenever the limit exists. In Theorem 2 of [20] it is proved that for Lipschitz functions g, M D exists almost everywhere, with respect to Lebesgue measure, and at almost every point where it exists, it is a seminorm.
Theorem 3.6 (Kirchheim). Let g : R n → X be Lipschitz. Then, for almost every x ∈ R n , M D(g, x)(·) is a seminorm on R n and In the case of Lipschitz curves (n = 1) the quantity M D coincides with the metric speed and at any point where it exists it is also a seminorm. As the objective of this paper is the study of tangent lines, (3.1) is the relevant identity. It is straightforward to observe that if (3.1) holds at t ∈ Dom (γ) then t is a point of metric differentiability and |dγ|(t) = M D(γ, t). Also the converse implication holds. We include here a short proof for reader's convenience.
Proof. Denote the Lipschitz constant of γ by L. Let ǫ > 0 . From the metric differentiability there exists r ǫ > 0 such that if ǫ|t| < |t − s| < |t|r ǫ , we have On the other hand, if 0 < |t − s| < ǫ|t|, we have from the Lipschitz-continuity The claim follows by combining the estimates.
In this paper we prefer to analyze the properties of |dγ| rather than (3.1).
Taking advantage of Theorem 3.2, it is fairly easy to obtain the almost everywhere existence of |dγ|. Proposition 3.8. Let γ : Dom (γ) → X be a Lipschitz curve. Then metric differentiability holds L 1 -a.e. in Dom (γ).
The proof can be obtained already from what was said in Remark 3.5. However, we present here an alternative proof obtained following the ideas of the proof of existence of the metric speed for L 1 -a.e. t ∈ [0, 1], see [6], Theorem 4.1.6.
Proof. Step 1. Consider Λ := γ(Dom (γ)). By continuity of γ, the set Λ is compact and we can consider a dense sequence {x n } ⊂ Λ. We define a sequence of Lipschitz function as follows: The Lipschitz constant of ϕ n coincides with that of γ. For each n ∈ N we denote withφ n a Lipschitz extension of ϕ n . We can assumeφ n to be defined on an interval, say on (a, b), containing Dom (γ). By Rademacher theorem, eachφ n is differentiable L 1 -a.e. and therefore we can define the following map p(t) := sup n∈N |φ n (t)|, at least for almost every t ∈ (a, b).
For the rest of the proof we fix t ∈ Dom (γ) which is a point of differentiability of all ϕ n and a Lebesguepoint of p. We also fix two sequences s m , τ m → 0 with 0 ≤ s m ≤ τ m so that This last condition is equivalent to ask that the interval (t + s m , t + τ m ) shrinks nicely to t. For ease of notation, s = s m , τ = τ m . Then for any n ∈ N we have d(γ t+s , γ t+τ ) τ − s ≥ |ϕ n (γ t+s ) − ϕ n (γ t+τ )| τ − s , and therefore lim inf We can take the supremum over all n without changing the left hand side of the previous inequality and obtaining on the right hand side p(t).
Step 3. Since {x n } n∈N is a dense sequence By assumption t is a Lebesgue-point of p, then lim sup where the last identity follows from Theorem 3.2. For L 1 -a.e. t ∈ Dom (γ): and the claim follows.
3.1. Existence of Tangent lines. From Proposition 3.8 one can prove that at each point of metric differentiability the blow up of the Lipschitz curve is a tangent line. Note that we use the properness assumption of the base space (X, d) in the proof.
Lemma 3.9. Let (X, d) be a complete, proper and separable metric space. Fixx ∈ X and assume the existence of a pointed, proper, complete and separable metric space Then (X ∞ , d ∞ ,x ∞ ) contains an isometric copy of R, in brief a line, limit of γ.
Proof. By assumption there exists a sequence of positive real numbers {r i } i∈N with r i → 0 such that Consider the sequence of approximate isometries f i : X → X ∞ associated to the convergence. For any two real numbers δ, η and i ∈ N sufficiently large it holds: Thanks to metric differentiability Since the limit space is proper, using a diagonal argument, we have convergence of {f i (γ riδ )} i∈N for all rational numbers δ. By density and there exists a curve z : R → X ∞ such that d ∞ (z η , z δ ) = |η − δ| · |dγ|(0).

It follows that z(R) is isometric to R.
In Lemma 3.9 we have not assumed any length structure on the metric space (X, d). Hence the assumption Dom (γ) = [−c, c] could sound a bit restrictive. In what follows we consider γ ∈ Γ(X) with a more general domain. Proposition 3.10. Let (X, d) be a complete, proper and separable metric space. Fixx ∈ X and assume the existence of a pointed, proper, complete and separable metric space (X ∞ , d ∞ ,x ∞ ) ∈ Tan(X, d,x). Let γ ∈ Γ(X) be such that γ 0 =x, 0 is a point of density one of Dom (γ), |dγ|(0) > 0.
Proof. Let us consider fixed sequences of positive numbers ε i → 0, R i → ∞ and where B Xi Ri (x) is the ball in (X, d/r i ), centered inx and of radius R i . Step 1. Denote with I := Dom (γ) and for any positive r we consider I r := {x ∈ R : xr ∈ I}. Consider any sequence ̺ i → 0, then for each n define The set ∩ n∈N I(n) is formed by all real numbers δ such that there exists a subsequence ̺ i k so that ̺ i k δ ∈ Dom (γ) for all k ∈ N. To underline its dependence on {̺ i } i∈N , we also use the following notation We observe that for each n ∈ N L 1 (R \ I(n)) = 0.
Indeed for any M > 0, and j ∈ N, j ≥ n Then since 0 has density one in I, Therefore for any M ∈ R L 1 ((−M, M ) \ I(n)) = 0, consequently L 1 (R \ I(n)) = 0 for all n ∈ N, and finally also holds.
Step 2. Consider now the sequence of radii {r i } for which the pointed Gromov-Hausdorff convergence holds. Consider also a sequence η n → 0 and an enumeration of all rational numbers {q m } m∈N . Consider now a bijection N ∋ h → (n(h), m(h)) ∈ N × N and the associated family of open balls in R: In particular we can consider the sequence where f i is an ε i -isometry from pointed Gromov-Hausdorff convergence. Then since the aforementioned sequence stays in a bounded neighborhood ofx ∞ and (X ∞ , d ∞ ) is proper, there exists a subsequence of {t 1 r i1(k) } k∈N still denoted by {t 1 r i1(k) } k∈N , such that We repeat the construction now with h = 2. Again from (3.2), B 2 ∩ I({r i1(k) }) = ∅ and therefore there exist t 2 ∈ B 2 and a subsequence of {r i1(k) } k∈N , call it {r i2(k) } k∈N for which t 2 r i2(k) ∈ I and Thanks to (3.2), we can repeat the same argument for any h and with a diagonal argument we infer the existence of sequences {t h } h∈N and {r i k } k∈N , such that for any h, for all sufficiently large k we have Step 3. For n, m ∈ N we have: Since |dγ|(0) > 0, we have d ∞ (z n , z m ) = |t n − t m ||dγ|(0).
Define therefore the curve: . Now observe that the set of points {t h } h∈N is dense in R, indeed for each h ∈ N the inclusion t h ∈ B h holds. It follows that γ ∞ can be extended by continuity to any s ∈ R. So we have proved the existence of

The claim follows.
Remark 3.11. The constructions done in the previous proof can be done simultaneously for finitely many curves. In particular suppose to have γ 1 , . . . , γ n ∈ Γ(X) such that γ j 0 =x, 0 is a point of density one of Dom (γ j ) and |dγ j |(0) > 0, for j = 1, . . . , n. Then there exists a dense countable set of times {t h } and a subsequence i k such that: for any h, η ∈ N and j = 1, . . . , n.

3.2.
Change of base point. It is also possible to extend Proposition 3.10 to almost any other point of the tangent space that is, if (X ∞ , d ∞ ,x ∞ ) is a pointed tangent space of (X, d,x) and z ∞ ∈ X ∞ , then one can find a tangent line passing through z ∞ , obtained as the blow-up of the same curve. This can be obtained using the fact that for almost everyx ∞ and for every z ∞ ∈ X ∞ also (X ∞ , d ∞ , z ∞ ) is a pointed tangent space, provided the ambient measure m is doubling. This has been proved by Preiss in [28] in the Euclidean framework and adapted to the metric space case by Le Donne in [23]. x ∈ X, for all (X ∞ , d ∞ ,x ∞ ) ∈ Tan(X, d,x), and for all z ∞ ∈ X ∞ we have Combining Proposition 3.10 and Theorem 3.12 we have Corollary 3.13. Let (X, d, m) be a doubling metric measure space,x ∈ X outside the exceptional set of Theorem 3.12 and γ ∈ Γ(X) such that γ 0 =x, 0 is a point of density one of Dom (γ), |dγ|(0) > 0.
Then for any (X ∞ , d ∞ ,x ∞ ) ∈ Tan(X, d,x) and any z ∞ ∈ X ∞ there exists a line, limit of γ, passing through z ∞ .

Tangent lines in Lipschitz differentiability spaces
In order to apply metric differentiability to Lipschitz differentiability spaces, a Borel regularity with respect to a precise Polish space is needed. We therefore recall few definitions from [7] that will be needed only in this section.
For a metric space (X, d) define H(X) to be the collection of non-empty compact subsets of R × X with the Hausdorff metric, so that H(X) is complete and separable. Moreover identify Γ(X) with its isometric image in H(X) via the map γ → graph(γ) and consider One can show (Lemma 2.7, [7]) that Γ(X) is a Borel subset of H(X) and A(X) is a Borel subset of X × H(X). Modifying Lemma 2.8 of [7] we obtain Lemma 4.1. Let (X, d) be a complete and separable metric space. The map F : A(X) → R∪{∞} defined as Proof. The proof is a slight modification of the proof of Lemma 2.8 of [7]. Let q, δ, ǫ > 0 and α ∈ (0, 1]. The set of (γ t0 , γ) ∈ A(X) with for all t, s with t 0 + t, t 0 + s ∈ Dom (γ) and |t|, |s| ≤ δ and |t − s| ≥ α max{t, s}, is closed. After taking suitable countable intersection and unions as in [7], the set where F belongs to some open set of R is Borel and the claim follows.
Now we obtain the following improved version of Theorem 2.7, stated in Section 2.3.
Proposition 4.2. Let (U, ϕ) be an n-dimensional chart in a Lipschitz differentiability space (X, d, m). Then for almost every x ∈ U , there exists γ x 1 , . . . , γ x n ∈ Γ(X) such that for each i = 1, . . . , n i) (γ x i ) −1 (x) = 0 is a point of density one of (γ x i ) −1 (U ); ii) the metric differential in 0 exists and |dγ x i |(0) > 0; iii) (ϕ • γ x i ) ′ (0) are linearly independent. Moreover, for any such γ x i , for any Lipschitz g : X → R and almost every x ∈ U , the gradient of g at x with respect to ϕ and γ x 1 , . . . , γ x n equals Dg(x), that is Even though the proof of Proposition 4.2 contains no novelty with respect to Theorem 2.7, we included it here for reader's convenience.
Proof. By Theorem 6.6 of [7] we have the existence of a countable decomposition U = ∪ j U j of U into sets with n ϕ-independent Alberti representations (whose definition can be found in Section 2 of [7]). We consider for k = 1, . . . , n the Borel function F k : where ϕ k is the k-th component of the coordinate map ϕ : U → R n . Moreover, we define F 0 to be the function F considered in Lemma 4.1.
For each k = 0, . . . , n all the assumption of Proposition 2.9 of [7] are satisfied. The case k = 0 follows from Proposition 3.8 and Lemma 4.1, while the case k ≥ 1 from Lemma 2.8 of [7]. Then we can repeat the same argument for the n ϕ-independent Alberti representations on each U j .
Hence for each j ∈ N there exists V j ⊂ U j with m(U j \ V j ) = 0 such that for each x ∈ V j there exist γ x 1 , . . . , γ x n ∈ Γ(X) such that i) (γ x i ) −1 (x) = 0 is a point of density one of (γ x i ) −1 (V j ); ii) the metric differential in 0 exists and |dγ x i |(0) > 0; iii) (ϕ • γ x i ) ′ (0) are linearly independent, for i = 1, . . . , n and for each k = 0, . . . , n the map . This proves the first part of the statement. The second part just follows from Theorem 2.7.
We can now use the previous result to obtain the following Proposition 4.3. Let (X, d, m) be a Lipschitz differentiability space and (U, ϕ) be an n-dimensional chart. Then for m-almost everyx ∈ U , any element (X ∞ , d ∞ ,x ∞ ) ∈ Tan(X, d,x) contains n disjoint (neglectingx ∞ ) isometric copies of R, obtained as limits of Lipschitz curves.
Proof. Take any pointed metric measure space (X ∞ , d ∞ ,x ∞ ) ∈ Tan(X, d,x) and the corresponding sequence of dilations r i > 0, with r i → 0.
The existence of n isometric copies of R follows straightforwardly from Definition 2.2, Proposition 3.10 and Proposition 4.2. It only remains to prove that the copies are disjoint. To this end we consider a chart (U, ϕ) withx ∈ U and use Remark 3.11: there exists a dense sequence {t h } h∈N ⊂ R and a subsequence i k such that: for j = 1, . . . , n, where f i k is the sequence of approximate isometries and γ j are given by Proposition 4.2. Recall that the closure in d ∞ of each {z j h : h ∈ N} forms the isometric copies of R in X ∞ . Via a reparametrization, without loss of generality, we may also assume that |dγ j |(0) = 1 for all j = 1, . . . , n.
Now we just observe that for j, l = 1, . . . , n where L is the Lipschitz constant of ϕ. Therefore we have proved that Since intersection for different times is not possible (at time 0 they start from the same point, with the same speed), the claim follows.
If the Lipschitz differentiability space is also doubling, one can argue as in Corollary 3.13 to obtain information on lines through any point of the tangent space.

Tangent lines in spaces with splitting tangents
As stated in Theorem 2.6, doubling metric measure spaces supporting a local p-Poincaré inequality are Lipschitz differentiability spaces and in particular Corollary 4.4 applies. For this class of more regular metric measure spaces, results on the structure of tangent spaces were already at disposal. For instance in [10], Theorem 8.5, existence of integral curves for tangent functions was proved. This in turn implies the existence of sufficiently many geodesic lines in the tangent space. But no explicit relation between geodesic lines in the tangent space and curves on the metric measure space was shown to exist. Therefore Corollary 4.4 brings new information also on the structure of tangent spaces for doubling metric measure space supporting a local p-Poincaré inequality.
In this last section we show that if n is the dimension of a chart of the measurable differentiable structure of (X, d, m) seen as a Lipschitz differentiability space and if d is the dimension of a Euclidean tangent space at x, then n ≤ d at m-a.e. point of X. In particular, we are interested in a special class of metric measure spaces (X, d, m) having the splitting of tangents property: if for somex ∈ X and if X ∞ contains an isometric copy of R going throughx ∞ , then ( We obtain the following result. Theorem 5.1. Suppose that (X, d, m) is a doubling Lipschitz differentiability space with the splitting of tangents property. Let (U, ϕ) be an n-dimensional chart of (X, d, m). Then for m-a.e.x ∈ U any Compare Theorem 5.1 to the result from [16], that can be rephrased as Theorem 5.2. Suppose that (X, d, m) is a geodesic doubling metric measure space with the splitting of tangents property. Then at m-a.e. point in X there exists a Euclidean tangent space.
Theorem 5.2 was formulated in [16] for RCD * (K, N ) spaces (metric measure spaces with Riemannian Ricci curvature bounded below by K ∈ R and dimension from above by N ), for which any tangent is an RCD * (0, N ) space having the splitting property, as was shown by Gigli [15]. Theorem 5.1 now shows that taking into account the fact that RCD * (K, N ) spaces are doubling and support a local Poincaré inequality [29,30], we immediately have that any tangent space contains an R n part with dimension at least the dimension of the chart. For a comprehensive treatise on the above mentioned family of spaces we refer to [24,35,36] for the defintion of CD(K, N ) and to [2,3] for the infinite dimensional Riemannian version. Finally RCD * (K, N ) with N ∈ R has been introduced independently in [5] and [14].
For RCD * (K, N ) spaces more can be said on the relation of the charts and the tangent spaces than the conclusion of Theorem 5.1. A recent result by Mondino and Naber in [25] states that for (X, d, m) verifying RCD * (K, N ), at m-a.e. x ∈ X there exists a unique tangent space and it is isomorphic, in the sense of metric measure spaces, to (R d , | · |, L d ), with d varying measurably in x. Moreover, they proved the following theorem. Theorem 5.3. Let (X, d, m) be an RCD * (K, N ) space for some K, N ∈ R with N > 1. Then there exists a countable collection {R j } j∈N of m-measurable subsets of X, covering X up to an m-negligible set, such that each R j is biLipschitz to a measurable subset of R kj , for some 1 ≤ k j ≤ N , k j possibly depending on j.
Combining this result with the fact that if a Lipschitz differentiability space is (locally) biLipschitz embeddable into a Euclidean space, then at almost every point all the tangent spaces are biLipschitz equivalent to R n , where n is the dimension of the chart, see Corollary 8.3 in [12]. Therefore, for RCD * (K, N ) spaces at almost every point the tangent is R n where the n is the dimension of the chart. Let us note that it is still unknown if in this context the dimension n of the tangent (and the chart) depends on the point.
We prove Theorem 5.1, which is valid without the biLipschitz embeddability to R n .
By the splitting property, there exists an isometry with h 1 (ι 1 (R)) = {(x 1 ∞ , t) : t ∈ R} and splitting the measure. Since the n geodesics are all disjoint, composing isometries and applying Lemma 2.9 we deduce the existence of n − 1 geodesics, again denoted with ι j : (R, | · |) → (X 1 ∞ , d 1 ∞ ), j = 2, . . . , n. By Lemma 2.9 we can also deduce that ι 2 (R), . . . , ι n (R) are all disjoint and we can use again the splitting property to rule out another isometric copy of R.
The same reasoning cannot be repeated to obtain a splitting of the form X ∞ ∼ X n ∞ × R n . It might be the case that for some j = 3, . . . , n, ι j (R) is already contained in the Euclidean component of the tangent space, and therefore the projection in the purely metric component of X ∞ could be the constant geodesic, not producing a new component to rule out via the splitting property.
Step 2. Consider the n-dimensional chart (U, ϕ) with ϕ : U → R n Lipschitz and anyx ∈ U such that Corollary 3.13 applies. Fix also (X ∞ , d ∞ , m ∞ ,x ∞ ) ∈ Tan(X, d, m,x) and u ϕ , the tangent function of ϕ atx. Note that, possibly passing to subsequences, u ϕ is well-defined.
Repeating the argument of Step 1. changing the reference point (see [23], Theorem 1.1), we have the following: for some d ∈ N all the possible splittings obtained from the lines of Theorem 4.5 give a decomposition of the following type: X ∞ = X d ∞ × R d , where the identity holds in the sense of metric measure spaces, and R d is equipped with the Euclidean distance and the Lebesgue measure.
Finally, we considerū ϕ , the restriction of u ϕ to x d ∞ × R d → R n . The claim can now be proven via showing thatū ϕ is a quotient map (again we refer to [12] for the relative definition). This can be obtained repeating verbatim the proof of Corollary 5.1 of [12] and using the linearity of s → u ϕ (ι 0 j (s)), together with Theorem 4.5.