On the notion of parallel transport on $\sf RCD$ spaces

We propose a general notion of parallel transport on $\sf RCD$ spaces, prove an unconditioned uniqueness result and existence under suitable assumptions on the space.


Introduction
More than ten years ago Sturm [21,22] and Lott-Villani [18] introduced the concept of lower Ricci curvature bounds for metric measure spaces. Their approach has been refined in [3] and [10] with the introduction of the class of metric measure spaces with Riemannian Ricci curvature bounded from below, RCD spaces in short, which is currently a very active research area. We refer to the surveys [23,24] for an overview of the topic and detailed references.
Among the various recent contributions, of particular relevance for the current manuscript is the paper [8] by the first author, where a second order calculus has been built. In particular, on RCD(K, ∞) spaces the covariant derivative of vector fields is well defined. Let us mention that in [8] 'vector field' is intended in the sense of abstract L 2 -normed L ∞ -modules and that in our previous paper [12] we showed that on RCD(K, N ) spaces this abstract notion has a canonically more concrete counterpart described in terms of pointed-measured-Gromov-Hausdorff limits of rescaled spaces (this uses the rectifiability results obtained in [19] and in [17], [11]).
In the classical smooth Riemannian framework, covariant derivative and parallel transport are two closely related concepts, thus given the existence of covariant derivative on RCD spaces it is natural to ask: is there a notion of parallel transport in the same setting? In this paper we address this question, our main results being: i) We provide a precise framework and give a rigorous meaning to the 'PDE' defining the parallel transport (see Definitions 3.19, 3.22 and 4.1).
ii) By the nature of our definition, norm-preservation and linearity of the parallel transport are easy to derive, and these in turn will give uniqueness (see Corollary 4.3).
iii) On RCD spaces satisfying a certain regularity property, we are able to show existence of the parallel transport (see Section 4.2). The regularity condition that we use is closely related to the existence of Sobolev vector fields with bounded covariant derivative (see Definition 4.9 for the precise assumption).
We believe that in fact the parallel transport exists on any RCD space, but we are currently unable to get the full proof. An insight on why this should not be too easy to prove is the following: on a space where the parallel transport exists, the dimension of the tangent module must be constant (see Theorem 4.8) and thanks to the aforementioned paper [12] this would in turn imply that the dimension of the pmGH-limits of rescaled spaces is constant. This very same result has been extremely elusive in the context of Ricci-limit spaces and has been obtained only relatively recently by Colding-Naber in [4]. Therefore it would be perhaps too optimistic to hope that the language proposed in [8] and here allows for an easy generalization of such 'constant dimension' result to the RCD setting. In this direction we remark that the assumptions that we put in order to obtain existence of the parallel transport are rather ad hoc and not really interesting from the geometric perspective: the intent with our existence result is just to show that the approach we propose is non-void. In our forthcoming paper [13] we shall study the same problem for more interesting geometric objects like Alexandrov spaces. Let us also mention that in the appendix (see Theorem A.2) we prove that iv) Any RCD(K, ∞) space admits a base of the tangent module made of vectors in W 1,2 C (T X).
That is: if we relax the condition of 'bounded covariant derivative' present in (iii) above into 'covariant derivative in L 2 ', then every RCD space meets the requirement.
Let us now briefly describe our approach. The crucial idea is that we don't study the problem of parallel transport along a single Lipschitz curve, but rather we study the problem along π-a.e. curve, where π is a Borel probability measure on the space Γ(X) of continuous curves such that: π is concentrated on equi-Lipschitz curves, for some C > 0 we have (e t ) * π ≤ Cm for every t ∈ [0, 1], where m is the given reference measure on our RCD space X and e t : Γ(X) → X is the evaluation map defined by e t (γ) := γ t . Measures π of this sort are a special case of so-called test plans introduced in [2]; these can be used to define Sobolev functions on metric measure spaces by a kind of duality argument. The advantage of working with these plans rather than with single curves is that they are naturally linked to Sobolev calculus and thus also to all the functional-analytic machinery built in [8].
Let us now pretend, for the sake of this introduction, that our space X is in fact a smooth Riemannian manifold. In this case a time dependent vector field (v t ) along π is, roughly said, given by a choice, for π-a.e. γ, of time dependent vector fields (v γ t ) on X. Then we say that (v t ) is a parallel transport along π provided for π-a.e. γ the vector field t → v γ t (γ t ) is a parallel transport along γ. This happens if and only if for π-a.e. γ we have ∂ t v γ t + ∇ γ ′ t v γ t = 0 a.e. t.
A relevant part of our paper (the whole Chapter 3) is devoted to showing that the above PDE can be stated even in the non-smooth setting, the key point being that it is possible to define a closed operator acting on L 2 vector fields along π which plays the role of (∂ t + ∇ γ ′ t ), see Definitions 3.17,3.19 and Proposition 3.20.
We conclude this introduction recalling that Petrunin proved in [20] that a certain notion of parallel transport exists along geodesic on Alexandrov spaces. Uniqueness for his construction is still an open problem. The question of comparing our notion and his one is certainly interesting, but outside the scope of this manuscript.
Acknowledgments This research has been supported by the MIUR SIR-grant 'Nonsmooth Differential Geometry' (RBSI147UG4).

Some basic notions
To keep the presentation short, we shall assume the reader familiar with the language proposed in [8] (see also [6]).

Curves in Banach spaces
We recall here some basic results about measurability and integration of Banach-valued maps of a single variable t ∈ [0, 1]. A detailed discussion about this topic can be found e.g. in [5].
Let B be a fixed Banach space. We will denote by B ′ its dual space. A simple function is any map y : [0, 1] → B that can be written in the form where for any topological space X we denote by B(X) the set of Borel subsets of X. A map y : [0, 1] → B is said to be strongly measurable provided there exists a sequence (y n ) n of simple functions y n : [0, 1] → B such that lim n y n (t) − y(t) B = 0 for L 1 -a.e. t ∈ [0, 1], while it is said to be weakly measurable provided [0, 1] ∋ t → ω y(t) ∈ R is a Borel map for every ω ∈ B ′ . It directly follows from the very definition that linear combinations of strongly (resp. weakly) measurable functions are strongly (resp. weakly) measurable. Moreover, if a map y : [0, 1] → B is strongly measurable, then the function [0, 1] ∋ t → y(t) B ∈ R is Borel.
The relation between the strongly measurable functions and the weakly measurable ones is fully described by a theorem of Pettis, which states that a function y : [0, 1] → B is strongly measurable if and only if it is weakly measurable and there exists a Borel set N ⊆ [0, 1] of null L 1 -measure such that y [0, 1] \ N is a separable subset of B.
We now describe how to define B-valued integrals, the so-called Bochner integrals. First of all, given a simple function y : It can be readily checked that this definition does not depend on the particular way of expressing the function y. Further, we say that any strongly measurable function y : [0, 1] → B is Bochner integrable provided there exists a sequence (y n ) n of simple functions y n : [0, 1] → B such that lim n 1 0 y n (t) − y(t) B dt = 0. In particular, the sequence 1 0 y n (t) dt n ⊆ B is Cauchy, so that it makes sense to define It turns out that the value of 1 0 y(t) dt just defined is independent of the approximating simple functions (y n ) n and that it satisfies the fundamental inequality An alternative characterisation of the B-valued integrable maps is given by the following theorem, which is due to Bochner: a strongly measurable function y : It is equipped with the norm 1], B) and thus in particular the curve t → y(t) − z(t) belongs to W 1,p ([0, 1], B) and, obviously, has derivative a.e. equal to 0. Hence to conclude it is sufficient to show that any such curve is a.e. constant. This follows noticing that for any ℓ ∈ B ′ the map t → ℓ(y(t) − z(t)) is in W 1,p ([0, 1]) (by direct verification) and has derivative a.e. equal to 0. Then y ∈ W 1,p ([0, 1], B) and z = y ′ if and only if for some dense set D ⊂ B ′ we have that ℓ • y ∈ W 1,1 ([0, 1]) with (ℓ • y) ′ = ℓ • z a.e. for every ℓ ∈ D.
In particular, if B = Lp(µ) for some Radon measure µ, then y ∈ W 1,p ([0, 1], Lp(µ)) and z = y ′ if and only if for every Borel set E we have that t → E y(t) dµ is in W 1,1 (0, 1) with derivative given by t → E z(t) dµ.
Proof. By assumption and using the fact that the Bochner integral commutes with the application of ℓ we have that for every ℓ ∈ D. The conclusion follows by the density of D in B ′ . For the second claim just observe that the linear span of the set of characteristic functions of Borel sets is dense in Lq(µ) ∼ (Lp(µ)) ′ .
It is important to underline that in general absolute continuity does not imply a.e. differentiability: this has to do with the so-called Radon-Nikodym property of the target Banach space. A sufficient condition for this implication to hold is given in the next theorem: Given any Banach space B, we shall denote by End(B) the space of all linear and continuous maps of B to itself, which is a Banach space if endowed with the operator norm.
Then there exists a unique curve y ∈ Γ(B) such that Moreover, the solution y satisfies y Γ(B) ≤ e c z Γ(B) .

Proof. Given any simple function
is strongly measurable by hypothesis on λ. Now fix y ∈ Γ(B). In particular, y : [0, 1] → B is strongly measurable, hence there exists a sequence (y k ) k of simple functions y k : [0, 1] → B such that lim k y k t − y t B = 0 holds for L 1 -a.e. t ∈ [0, 1]. This grants that λ(t)y k t − λ(t)y t B ≤ c y k t − y t B k → 0 is satisfied for L 1 -a.e. t ∈ [0, 1], thus accordingly the map t → λ(t)y t is strongly measurable as pointwise limit of strongly measurable functions. Moreover, since λ(t)y t B ≤ c y Γ(B) for all t ∈ [0, 1], one has that t → λ(t)y t actually belongs to L ∞ [0, 1], B . Therefore it makes sense to define the function Λy : [0, 1] → B as (Λy)(t) := t 0 λ(s)y s ds for every t ∈ [0, 1]. Note that   We will actually make use of is the following consequence of Theorem 2.5. Proof. Define z(t) := y for every t ∈ [0, 1] and consider the curve y ∈ Γ(B), whose existence is granted by Theorem 2.5. For every t, s ∈ [0, 1], with s < t, we have that Since y(·) B ∈ L ∞ (0, 1), we deduce that the function y is Lipschitz, so that by Theorem 2.4 y is a.e. differentiable. Then (2.5) trivially follows from (2.2). Conversely, let y ∈ LIP [0, 1], B be any curve such that (2.5) holds true. By integration we conclude that y satisfies also property (2.2), proving uniqueness.

Pullback of an L 0 -normed module
The aim of this subsection is to introduce the concept of pullback of an L 0 -normed module and to study its main properties. The following definitions and results mimic the analogous ones for L p -normed modules, which are treated in [8, Subsection 1.6]; a digression similar to the one below has been done in [15].
be σ-finite measured spaces. Let ϕ : X → Y be a map of bounded compression (i.e. ϕ * m X ≤ Cm Y for some C > 0). Let M 0 be an L 0 (m Y )normed module. Then there exists (up to unique isomorphism) a unique couple (N 0 , T), where N 0 is an L 0 (m X )-normed module and T : M 0 → N 0 is a linear map, such that Namely, if two couples (N 0 1 , T 1 ) and (N 0 2 , T 2 ) as above fulfill both (i) and (ii), then there exists a unique module isomorphism Φ : N 0 1 → N 0 2 such that the diagram We construct the map T in the following way: since the space M is dense in M 0 and ϕ * M is continuously embedded into N 0 , the map ϕ * : M → ϕ * M can be uniquely extended to a linear continuous function T : M 0 → N 0 . Such function T also satisfies (i). Moreover, it is clear that the diagram commutes. Therefore, since the set {ϕ * v : v ∈ M } generates N 0 as L 0 (m X )-normed module, we have in particular that the set {Tv : v ∈ M 0 } generates N 0 as L 0 (m X )-normed module, proving (ii).
Uniqueness. Let us choose (N 0 1 , T 1 ), (N 0 2 , T 2 ) satisfying (i) and (ii). For j = 1, 2, denote by V j the set of n i=1 χ A i T j v i as in (ii). Then the unique L 0 (m X )-linear map Ψ : By requiring the condition (i), we force the m X -a.e. equality which shows that the map Ψ is actually well-defined and continuous. There exists a unique module morphism Φ : N 0 1 → N 0 2 that extends Ψ, by density of V 1 in N 0 1 . Such map Φ clearly satisfies Φ • T 1 = T 2 . Finally, by interchanging the roles of N 0 1 and N 0 2 , one can easily conclude that Φ is an isomorphism, getting the thesis. Definition 2.8 (Pullback module) Any couple (N 0 , T) that satisfies Theorem 2.7 will be unambiguously denoted by ϕ * M 0 , ϕ * . Moreover, we shall call ϕ * M 0 the pullback module of M 0 and ϕ * the pullback map. Proposition 2.9 (Universal property of the pullback) Let (X, A X , m X ), (Y, A Y , m Y ) be σ-finite measured spaces. Let ϕ : X → Y be a map of bounded compression. Let M 0 be an L 0 (m Y )-normed module and let N 0 be an L 0 (m X )-normed module. Consider a linear operator T : for a suitable map ℓ ∈ L 0 (m X ). Then there exists a unique L 0 (m X )-linear and continuous operator T : Proof. Denote by V the set of all elements of the form n i=1 χ A i ϕ * v i , with A 1 , . . . , A n ∈ A X partition of X and v 1 , . . . , v n ∈ M 0 , so that V is a dense linear subspace of ϕ * M 0 by property (ii) of Theorem 2.7. Any L 0 (m X )-linear map T : (2.8) Consider T : V → N 0 defined as in (2.8), then (2.6) grants that which shows that T : V → N 0 is well-defined (in the sense that Tw depends only on w and not on the way of representing it) and continuous. Therefore T can be uniquely extended to a linear continuous operator T : ϕ * M 0 → N 0 . We readily deduce from the definition (2.8) that the equality f Tw = T(f w) holds for f : X → R simple function, so that T can be shown to be L 0 (m X )-linear by an approximation argument. Finally, it follows from (2.9) that the inequality (2.7) is satisfied for w ∈ V , whence (2.7) holds by density of V in ϕ * M 0 .

Some properties of test plans
For the sake of brevity, hereafter we shall use the notation L 1 to indicate the 1-dimensional Lebesgue measure restricted to [0, 1], namely Let π ∈ P Γ(X) be any fixed test plan on X, whence Γ(X), d Γ(X) , π is a metric measure space. Given that the map e t is of bounded compression, it makes sense to consider the pullback module e * t L 2 (T X). Observe that e * t L 2 (T X) is a Hilbert module as soon as (X, d, m) is infinitesimally Hilbertian.
Remark 2.10 Let us define the map e : Γ(X) × [0, 1] → X as e(γ, t) := γ t for every γ ∈ Γ(X) and t ∈ [0, 1]. It can be easily proved that the map e is continuous. This grants that, given any Borel map f : X → R, the function f • e is Borel. Moreover, observe that by Fubini theorem, in other words it holds that e * (π × L 1 ) ≤ C(π) m. Therefore one has that the composition f • e ∈ L 0 (π × L 1 ) is well-defined for any f ∈ L 0 (m).

Theorem 2.11
Let (X, d, m) be a metric measure space. Let π be a test plan on X. Then Proof. First of all, we claim that → 0 as s → t by continuity, for every γ ∈ Γ(X) and t ∈ [0, 1]. Hence we obtain (2.11) as a consequence of the dominated convergence theorem. Observe also that Given t ∈ [0, 1] and n ∈ N, we have that (2.11) and (2.12) yield (2.13) By letting n → ∞ in (2.13), we finally conclude that |f • e s − f • e t | dπ → 0 as s → t, which proves (2.10). The last statement is obvious.
Under further assumptions on (X, d, m), we have at disposal also a notion of 'speed' π ′ t of the test plan π at time t, as described in the following result. For the proof of such fact, we refer to [8,Theorem 2.3.18] or [6, Theorem/Definition 1.32].
Theorem 2.12 (Speed of a test plan) Let (X, d, m) be a metric measure space such that L 2 (T X) is separable. Let π be a test plan on X. Then there exists a unique (up to L 1 -a.e. equality) family π ′ t ∈ e * t L 2 (T X) such that 14) π be a test plan on X and f ∈ W 1,2 (X). Then the a.e. defined map [0, 1] ∋ t → e * t df (π ′ t ) ∈ L 1 (π) is a.e. equal to a Borel map.
Thus by classical arguments the set of t's for which the limit as h → 0 exists is Borel and the limit function, set, say, to 0 when the limit does not exist, is Borel.
In this paper we shall work only on RCD spaces, which in particular are so that W 1,2 (X) is reflexive. In turn this implies, by the arguments in [1], that the tangent module is separable, so that the assumptions of Theorem 2.12 and Proposition 2.13 above are fulfilled.
In the sequel, we will mainly focus our attention on those test plans π that are concentrated on an equiLipschitz family of curves. As illustrated in the next definition, we will refer to them as 'Lipschitz test plans'. Definition 2.14 (Lipschitz test plan) Let (X, d, m) be a metric measure space such that L 2 (T X) is separable. Then a test plan π on X is said to be a Lipschitz test plan provided there exists a constant L ≥ 0 such that 16) or, equivalently, such that π is concentrated on the family of all the L-Lipschitz curves in X.
The smallest constant L ≥ 0 for which (2.16) is satisfied will be denoted by L(π).
Whenever the test plan π is Lipschitz, we have (e * t df )(π ′ t ) ∈ L 2 (π) for every f ∈ W 1,2 (X). One is then led to wonder whether in this case the limit in (2.14) takes place not only in L 1 (π), but also in L 2 (π). The answer is affirmative, as shown in the following simple result: is Lipschitz and Proof. Given any t, s ∈ [0, 1] with s < t, one has that which shows that t → f • e t ∈ L 2 (π) is a Lipschitz map. In particular, it is differentiable at almost every t ∈ [0, 1] by Theorem 2.4, so that (2.17) follows from (2.14).

Introduction of appropriate functional spaces
Throughout all this chapter, (X, d, m) is a given RCD(K, ∞) space and π a test plan on it.
Recall that the space TestF(X) of test functions on X is defined as and that the space of test vector fields on X is defined as

Test vector fields along π
We define the space of vector fields along π as: Thus VF(π) is the collection of maps assigning to each t ∈ [0, 1] an element of e * t L 2 (T X); it is a vector space w.r.t. pointwise operation.
To each V ∈ VF(π) we associate the function The subspace TestVF(π) ⊂ VF(π) of test vector fields along π is defined as: and v i ∈ TestV(X) for every i = 1, . . . , n . Since Proposition 3.1 (Continuity of the test vector fields along π) For any V, W ∈ TestVF(π) we have that In particular, the function Proof. By linearity, it is clear that it is sufficient to prove the claim for V, W of the form and Theorem 2.11. The last statement follows by choosing W = V .
We now want to show that (TestVF(π), · C (π) ) is separable by exhibiting a countable dense subset. To this aim, we first choose three countable families F 1 ⊆ open sets of Γ(X) , F 2 ⊆ LIP [0, 1] and F 3 ⊆ TestV(X) such that given A ⊆ Γ(X) Borel and ε > 0, there exists U ∈ F 1 with π(A∆U ) < ε, 1]) and stable by product and Q-linear combinations, F 3 is a Q-vector space of functions in W 1,2 (X) whose gradients generate L 2 (T X).
We proceed in the following way: is a neighbourhood basis for each point γ ∈ Γ(X). Let us denote by F 1 the set of finite unions of elements of F 1 , so that F 1 is countable. Fix A ∈ B Γ(X) and ε > 0. The measure π is regular, since Γ(X), d Γ(X) is complete and separable. By inner regularity of π, there exists a compact subset K ⊆ A such that π(A \ K) < ε/2. By outer regularity of π, there exists V ⊆ Γ(X) open such that K ⊆ V and π(V \ K) < ε/2. We can then associate to any is infinitesimally Hilbertian, we have that W 1,2 (X) is reflexive and therefore, by [1], separable. Thus let F 3 be any countable dense Q-vector subspace of W 1,2 (X) and notice that since gradients of functions in W 1,2 (X) generate L 2 (T X), the same holds for functions in F 3 .
We now define the class of test vector fields TestVF N (π) as: Clearly TestVF N (π) is a countable subset of TestVF(π). Also, notice that the inequalities valid for any t ∈ [0, 1], A, U ⊂ Γ(X) Borel and v, w ∈ L 2 (T X) and the very definition of pullback module, show that Lemma 3.2 (Separability of TestVF(π)) The family TestVF N (π) is dense in TestVF(π) with respect to the norm · C (π) (and thus also w.r.t. · L 2 (π) ).
By multiplying W i by an appropriate function in F 2 we can also assume that Now let (φ i ) be a Lipschitz partition of the unity subordinate to the cover made with the I i 's and for any i let ψ i ∈ F 2 be such that |φ for any t ∈ [0, 1]. The conclusion follows by the arbitrariness of ε > 0.

The space L 2 (π)
We start defining the class of Borel vector fields along π: for every W ∈ TestVF N (π).
Notice that thanks to Lemma 3.2 this notion would be unaltered if we require (3.5) to hold for any W ∈ TestVF(π). Also, Proposition 3.1 ensures that test vector fields are Borel. We have the following basic result: for every t ∈ [0, 1]. and the thesis follows.
We can now define the space L 2 (π): Clearly L 2 (π), · L 2 (π) is a normed space, wherein TestVF(π) is embedded. Adapting the classical arguments concerning the standard L 2 spaces we have the following: Proposition 3.6 (Basic properties of L 2 (π)) The space L 2 (π) is a Hilbert space and Proof. It is clear that the L 2 (π) norm comes from the scalar product To conclude the proof we shall show that if (V n ) is a sequence of Borel vector fields in L 2 (π) such that n V n+1 − V n L 2 (π) < ∞, then such sequence has a limit V ∈ L 2 (π) and for a.e.
we have that g ∈ L 2 (0, 1). Let N := {t : g(t) = +∞} and notice that for every t ∈ proving that (V n,t ) is a Cauchy sequence in e * t L 2 (T X). Then define Notice that for every W ∈ TestVF(π) we have V t , W t dπ = lim n V n,t , W t dπ for all t ∈ [0, 1] \ N , hence the map [0, 1] ∋ t → V t , W t dπ is Borel and, by arbitrariness of W , this shows that V is Borel. Since trivially we have ] t , by (3.6) we see that V ∈ L 2 (π). Now to check that V n → V in L 2 (π) notice that, again by (3.6), the sequence [[V − V n ]] t is dominated in L 2 (0, 1) and that for every t ∈ [0, 1] \ N we have = 0, so that the conclusion follows by the dominate convergence theorem. Proposition 3.7 (Density of TestVF N (π) in L 2 (π)) The space TestVF N (π) is dense in L 2 (π). In particular, L 2 (π) is separable.
Proof. Let (Z k ) be an enumeration of the elements in TestVF N (π), pick a Borel vector field V ∈ L 2 (π) and choose ε > 0. Then for every k ∈ N let G k := t ∈ [0, 1] : [[V − Z k ]] t < ε and put G 1 :=G 1 and G k : Observe that W ∞ − V L 2 (π) < ε by definition of G k . Moreover, for each m ≥ 1 one has that Hence to conclude it is sufficient to show that each W m belongs to the L 2 (π)-closure of TestVF N (π) and in turn this will follow if we prove that for Z ∈ TestVF N (π) and G ⊂ [0, 1] Borel the vector field χ G Z belongs to the L 2 (π)-closure of TestVF N (π). To see this, simply let (ϕ n ) ⊂ LIP([0, 1]) be uniformly bounded and a.e. converging to χ G , notice that ϕ n Z ∈ TestVF(π) and that an application of the dominate convergence theorem shows that ϕ n Z → χ G Z in L 2 (π). Now consider the speed π ′ t , associated to any test plan π by Theorem 2.12.
Proposition 3.8 The (equivalence class up to a.e. equality of the) map t → π ′ t is an element of the space L 2 (π).
Proof. We have π ′ t ∈ e * t L 2 (T X) for a.e. t ∈ [0, 1] and by the very definition of test plan. Hence we need only to show that t → π ′ t has a Borel representative in the sense of Definition 3.3.
Notice that for any f ∈ W 1,2 (X) by Proposition 2.13 we have that the map t → (e * t df )(π ′ t ) has a Borel representative. Hence the same holds for t → ψ(t) χ U e * t ∇f, π ′ t for every ψ ∈ LIP([0, 1]) and U ⊂ Γ(X) Borel. Therefore there exists a Borel negligible set N ⊂ [0, 1] such that for every V ∈ TestVF N (π) the map t → V t , π ′ t dπ, set to 0 on N , is Borel. This is sufficient to conclude.
Notice that the notion of pullback module e * L 2 (T X), e * makes no (explicit) reference to the concept of 'test vector field' as defined it in Section 3.1. Thus this last proposition is also telling that the choice of using these test object to check Borel regularity, which a priori might seem arbitrary, leads in fact to a canonical interpretation of L 2 (π).
Remark 3.10 (L 2 (π) as direct integral) The construction of L 2 (π) can be summarized by saying that such space is the direct integral of the e * t L 2 (T X), the space of Borel vector fields being the so-called 'measurable sections' and the set TestVF N (π) being the one used to check measurability.

The space C (π)
Here we introduce and briefly study those vector fields in VF(π) which are 'continuous in time'. We start with the following definition: Definition 3.11 (The space C (π)) Let V ∈ VF(π). Then we say that V is a continuous vector field provided for every W ∈ TestVF N (π) and We denote the family of all continuous vector fields by C (π) and, for every V ∈ C (π), we put V C (π) := max Lemma 3.2 ensures that this definition would be unaltered if we require (3.9) to hold for any W ∈ TestVF(π). Also, Proposition 3.1 gives that TestVF(π) ⊂ C (π). It is not obvious that C (π) is a vector space, the problem being in checking that (3.10) holds for linear combinations. This will be a consequence of the density of TestVF N (π) in C (π), which is part of the content of the next result: Proposition 3.12 (C (π), · C (π) ) is a separable Banach space, with TestVF N (π) being dense.
Proof. Let V 1 , V 2 ∈ C (π) and notice that using (3.9), (3.10) and arguing exactly as in the proof of Lemma 3.2 we can find (W 1,n ), (W 2,n ) ⊂ TestVF N (π) such that the functions t → [[V i − W i,n ]] t uniformly converge to 0 as n → ∞, i = 1, 2. Now observe that since W 1,n + W 2,n ∈ TestVF N (π) the map t → [[W 1,n + W 2,n ]] t is continuous and that for every t ∈ [0, 1] we have Hence t → [[V 1 + V 2 ]] t is the uniform limit of continuous functions and thus continuous itself. Since trivially C (π) is closed by multiplication by scalars we proved that it is a vector space. That · C (π) is a complete norm on it is trivial and the density of TestVF N (π) has already been shown, hence the proof is finished.
A useful consequence of the density of test vector fields is the following strengthening of the continuity property: Corollary 3.13 Let V ∈ C (π). Then the map t → |V t | 2 ∈ L 1 (π) is continuous.
Proof. For V ∈ TestVF(π) the claim has been proved in Proposition 3.1. Now notice that for V, W ∈ C (π) we have thus showing that if V n → V in C (π) then t → |V n,t | 2 ∈ L 1 (π) uniformly converge to t → |V t | 2 ∈ L 1 (π). The conclusion then follows from the density of TestVF(π) in C (π).

The spaces W 1,2 (π) and H 1,2 (π)
Throughout all this section we shall further assume that the test plan π is Lipschitz in the sense of Definition 2.14.
Let v ∈ W 1,2 C (T X) and notice that the map from L 0 (T X) to e * t L 0 (T X) defined by Hence by the universal property of the pullback given in Proposition 2.9 we know that there exists a unique L 0 (π)-linear continuous operator, which we shall call Cov(v, ·) from e * t L 0 (T X) to e * t L 0 (T X) such that and such operator satisfies the bound We shall be interested in such covariant differentiation along the speed of our test plan: for every t ∈ [0, 1] such that π ′ t exists we define the map Cov t : Notice the following simple proposition: Proposition 3.14 For every t ∈ [0, 1] such that π ′ t exists, the map Cov t is linear and continuous from W 1,2 C (T X) to e * t L 2 (T X). Moreover, for every v ∈ W 1,2 C (T X) the (equivalence class up to a.e. equality of the) a.e. defined map t → Cov t (v) ∈ e * t L 2 (T X) is an element of L 2 (π).
Proof. The continuity of Cov t as map from W 1,2 C (T X) to e * t L 2 (T X) is a direct consequence of the bounds (3.11) and our assumption (2.16): Thanks to this bound, to conclude it is sufficient to show that for any v ∈ W 1,2 is a.e. equal to a Borel element of VF(π). Taking into account that t → π ′ t ∈ L 2 (π) by Proposition 3.8, that TestVF(π) is dense in L 2 (π), the second claim in Proposition 3.6 and the bound (3.11), we see that to conclude it is sufficient to show that t → Cov(v, V t ) is a Borel vector field in VF(π) for any V ∈ TestVF(π).
Thus fix such V , say V t = i φ i (t) χ A i e * t v i , and let W t = j ψ j (t) χ B j e * t w j ∈ TestVF(π) be arbitrary. Notice that since |v i |, |w j | ∈ L 2 ∩ L ∞ (X), we have that v i , ∇ w j v ∈ L 1 (X) and thus by Theorem 2.11 we deduce that t → v i , ∇ w j v • e t ∈ L 1 (π) is continuous for every i, j. Therefore is continuous, thus establishing, by the arbitrariness of W , the Borel regularity of t → Cov(v, V t ).
The 'compatibility with the metric' of the covariant derivative yields the following simple but crucial lemma: Lemma 3.15 Let v, w ∈ TestV(X). Then the map t → v, w • e t ∈ L 2 (π), which is Lipschitz by Proposition 2.15, satisfies Proof. Recall from [8] that it holds and notice that from the defining property of pointwise norm in the pullback and by polar- holds for every Z ∈ e * t L 2 (T X) of the form Z t = e * t z for some z ∈ L 2 (T X). Since both sides of this identity are L ∞ (π)-linear and continuous in Z, we see that (3.13) holds for generic Z ∈ e * t L 2 (T X). The conclusion comes picking Z = π ′ t and recalling Proposition 2.15. We now want to introduce a new differential operator, initially defined only on TestVF(π) and then extended to more general vector fields. To this aim the following lemma will be useful.
Then for a.e. t it holds: Proof. For the first in (3.15) we notice that our assumption (3.14) and Proposition 3.9 1] (·, t) ψ j (t) e * w j as elements in e * L 2 (T X) ∼ L 2 (π), thus we can differentiate in time and conclude using again Proposition 3.9.
For the second in (3.15) start noticing that our assumption (3.14) and the very definition of pullback imply that for any i, j and every t ∈ [0, 1] it holds > 0 . This identity and the locality of the covariant derivative give that χ C ϕ i (t) ∇ z v i = χ C ψ j (t) ∇ z w j for every z ∈ L 2 (T X). Applying the pullback map on both sides and noticing that χ C • e t ≥ χ A i ∩B j we deduce that for every Z of the form Z t = e * t z. From the L ∞ (π)-linearity in Z of both sides and the arbitrariness of i, j the conclusion follows.
We can now define the convective derivative of test vector fields: Definition 3.17 (Convective derivative along a test plan) We define the convective derivative operator D π : TestVF(π) → L 2 (π) as follows: to the element V ∈ TestVF(π), of the form V t = n i=1 ϕ i (t) χ A i e * t v i , we associate the vector field D π V ∈ L 2 (π) given by For the sake of simplicity, we will briefly write D π V t instead of ( D π V ) t .
Notice that Lemma 3.16 ensures that the right hand side of (3.16) depends only on V and not on the way we write it as V t = n i=1 ϕ i (t) χ A i e * t v i . The fact that the right hand side of (3.16) defines a Borel vector field in VF(π) follows directly from Proposition 3.14; to see that it belongs to L 2 (π) notice that (t → e * t v i ), Cov π (v i ) ∈ L 2 (π) for every i and that the ϕ i 's are Lipschitz.
Therefore the definition is well posed and is then clear that D π is a linear operator.
The convective derivative has the following simple and crucial property, which is a direct consequence of Lemma 3.15.
Proposition 3.18 Let V, W ∈ TestVF(π). Then the map t → V t , W t ∈ L 2 (π) is Lipschitz and satisfies Proof. By bilinearity, to prove (3.17) is sufficient to consider the case V t = ϕ(t) χ A e * t v and W t = ψ(t) χ B e * This last proposition allows to 'integrate by parts' and extend the definition of convective derivative to 'Sobolev vector fields along π'.
Let us define the support spt(V ) of a test vector field V ∈ TestVF(π) as the closure of the set of t's for which V t = 0 and let us introduce the space of sections with compact support in (0, 1): TestVF c (π) := V ∈ TestVF(π) : spt(V ) ⊆ (0, 1) .
This choice of terminology is consistent with that of Definition 3.17: Proof. Fix W ∈ TestVF c (π). We know from Proposition 3.18 that [0, 1] ∋ t → V t , W t dπ is an absolutely continuous function, so that (3.17) gives, after integration, that This proves that V satisfies (3.18) with Z = D π V .
(i) Let (V n ) ⊆ W 1,2 (π) be a sequence such that V n → V and D π V n → Z in L 2 (π) for some V, Z ∈ L 2 (π). Then for arbitrary W ∈ TestVF c (π) we have proving that V ∈ W 1,2 (π) with D π V = Z, which was the claim.
We just proved that TestVF(π) is contained in W 1,2 (π), but we don't know if it is dense. Hence the following definition is meaningful: Clearly, H 1,2 (π) is a separable Hilbert space. A key feature of elements of H 1,2 (π) is that they admit a continuous representative (much like Sobolev functions on the interval): Theorem 3.23 The inclusion TestVF(π) ֒→ C (π) uniquely extends to a linear continuous and injective operator ι : H 1,2 (π) → C (π).
Whenever we will consider an element V of H 1,2 (π), we will always implicitly refer to its unique continuous representative ι(V ) ∈ C (π).
Among the several properties of the test sections that can be carried over to the elements of H 1,2 (π), the most important one is the Leibniz formula for the convective derivatives: Proposition 3.24 (Leibniz formula for D π ) Let V ∈ W 1,2 (π) and W ∈ H 1,2 (π). Then the function t → V t , W t is in W 1,1 ([0, 1], L 1 (π)) and its derivative is given by Proof. For W ∈ TestVF(π) the claim is a direct consequence of point (iii) in Proposition 3.21. The general case can be achieved by approximation noticing that the simple inequalities allow to pass to the limit in the distributional formulation of d dt V t , W t as W varies in H 1,2 (π).
(i) If w ∈ TestV(X) we have that W ∈ TestVF(π) by definition and in this case formula (3.21) holds by the definition (3.16) and Proposition 3.20. The general case can then be obtained by approximating w with vector fields in TestV(X) w.r.t. the W 1,2 C (T X) topology, using the bounds and recalling the closure of D π .
(ii) The claim about continuity is obvious, so we concentrate on the other one. Assume at first that a belongs to the space A defined as and that W ∈ TestVF(π). In this case aW belongs to TestVF(π) as well and formula (3.22) is a direct consequence of the definitions. Then using the trivial bounds the W 1,2 (π)-density of TestVF(π) in H 1,2 (π) and the closure of D π , we conclude that aW ∈ H 1,2 (π) for every a ∈ A and W ∈ H 1,2 (π) and that (3.22) holds in this case. Now let W be as in the assumptions and notice that we also have the bounds therefore using again the closure of D π , to conclude it is sufficient to prove that A is dense in W 1,2 ([0, 1], L 2 (π)). To this aim we argue as follows: for every n ∈ N let (E n i ) i∈N be a Borel partition of supp(π) ⊂ Γ(X) made of sets with positive π-measure and diameter ≤ 1 n . Then for every n, N ∈ N let P N n : L 2 (π) → L 2 (π) be defined by It is clear that P N n has operator norm ≤ 1 for every n, N ∈ N and an application of the dominated convergence theorem shows that for every f ∈ C b Γ(X) , the limits being intended in L 2 (π). Therefore (3.23) also holds for every f ∈ L 2 (π). The linearity and continuity of P N n also grants that if t → a t belongs to W 1,2 ([0, 1], L 2 (π)), then also t → P N n (a) t := P N n (a t ) belongs to W 1,2 ([0, 1], L 2 (π)) with All these considerations imply that for every a ∈ W 1,2 ([0, 1], L 2 (π)) and thus to conclude it is sufficient to prove that P N n (a) belongs to the W 1,2 ([0, 1], L 2 (π))-closure of A for every n, N ∈ N and a ∈ W 1,2 ([0, 1], L 2 (π)).
Since N i=1 g i,j χ E n i ∈ A for every j, the proof is finished.
4 Parallel transport on RCD spaces 4.1 Definition and basic properties of parallel transport

Definition and uniqueness
We shall frequently use the fact that, since H 1,2 (π) is continuously embedded into C (π) by Theorem 3.23, any vector field V ∈ H 1,2 (π) has pointwise values V t ∈ e * t L 2 (T X) defined at every time t ∈ [0, 1]. Definition 4.1 (Parallel transport) Let K ∈ R, (X, d, m) an RCD(K, ∞) space and π be a Lipschitz test plan on X. A parallel transport along π is an element V ∈ H 1,2 (π) such that D π V = 0.
The linearity of the requirement D π V = 0 ensures that the set of parallel transports along π is a vector space. From Proposition 3.24 we deduce the following simple but crucial result: Proposition 4.2 (Norm preservation) Let V be a parallel transport along the Lipschitz test plan π. Then t → |V t | 2 ∈ L 1 (π) is constant.
Proof. We know from Corollary 3.13 that t → |V t | 2 ∈ L 1 (π) is continuous. Hence the choice W = V in Proposition 3.24 tells that such map is absolutely continuous with derivative given by d dt This is sufficient to conclude.

Remark 4.4
We emphasize that the norm preservation property is a consequence of the Leibniz formula in Proposition 3.24. We don't know if such formula holds for V, W ∈ W 1,2 (π) and this is why we defined the parallel transport as an element of H 1,2 (π) with null convective derivative, as opposed to an element of W 1,2 (π) with the same property.

Some consequences of existence of parallel transport
In this section we assume existence of parallel transport along some/all Lipschitz test plans and see what can be derived from such assumption. It will be convenient to recall the concept of base of a module, referring to [8] for a more detailed discussion. Let µ be a Borel measure on a Polish space Y, M a L 2 (µ)-normed module, v 1 , . . . , v n ∈ M and E ⊂ Y a Borel set. Then the v i 's are said to be independent on E provided for any f i ∈ L ∞ (µ) we have If v 1 , . . . , v n are both independent and generators of M on E we say that they are a base of M on E and in this case we say that the dimension of M on E is n.
Recall that there always exists a (unique up to µ-negligible sets) Borel partition (E i ) i∈N∪{∞} of Y, called dimensional decomposition of M , such that the dimension of M on E i is i for every i ∈ N and for no Borel subset F of E ∞ with positive measure the dimension of M on F is finite.
For a separable Hilbert module H on a space with finite measure µ one can always find an orthonormal base (v n ) n∈N i.e. a sequence whose L ∞ -linear combinations are dense in H and such that for some Borel partition (E n ) n∈N∪{∞} the following hold: and it is easily verified that if these hold, then necessarily the E n 's form the dimensional decomposition of H (the role of the assumption about finiteness of µ is to ensure that the v i 's are elements of H : their pointwise norm is in L ∞ (µ) and thus in general it may be not in L 2 (µ)). Given such a base and v ∈ H there are functions a n ∈ L 2 (µ) such that v = n∈N a n v n (4.1) meaning that the sequence converges absolutely in H . A choice for the a n is a n := v, v n and for any two sequences (a n ), (ã n ) for which (4.1) holds we have a n =ã n µ − a.e. on E m , ∀m ∈ N ∪ {∞}, m > n.
With this said, it is easy to see that parallel transport sends bases into bases: Proposition 4.5 Let π be a Lipschitz test plan such that for every t ∈ [0, 1] andV t ∈ e * t L 2 (T X) there exists a (necessarily unique) parallel transport V along π such that V t =V t . Also, let (E n ) n∈N∪{∞} be the dimensional decomposition of e * 0 L 2 (T X) and (V n ) ⊂ e * 0 L 2 (T X), n ∈ N an orthonormal base of e * 0 L 2 (T X) and denote by t → V n,t the parallel transport ofV n along π.
Then for every t ∈ [0, 1] the partition (E n ) n∈N∪{∞} is also the dimensional decomposition of e * t L 2 (T X) and the set (V n,t ) is an orthonormal base of e * t L 2 (T X). Proof. For every t, s ∈ [0, 1] consider the map sendingV ∈ e * t L 2 (T X) to V s ∈ e * s L 2 (T X) where V ∈ H 1,2 (π) is the parallel transport such that V t =V . Proposition 4.2 ensures that this map preserves the pointwise norm. Since it is clearly linear, it is easily verified that it must be an isomorphism of e * t L 2 (T X) and e * s L 2 (T X). The conclusions follow.
We shall apply this result to show that, under the same assumptions, we have W 1,2 (π) = H 1,2 (π): Proposition 4.6 (H = W ) Let π be a Lipschitz test plan such that for every t ∈ [0, 1] and V t ∈ e * t L 2 (T X) there exists a (necessarily unique) parallel transport V along π such that V t =V t .
Proof. Let V ∈ W 1,2 (π), (V i ) ⊂ e * 0 L 2 (T X), i ∈ N, an orthonormal base of e * 0 L 2 (T X) and t → V i,t the parallel transport ofV i along π. Then by Proposition 4.5 above we see that being intended that the series converges absolutely in e * t L 2 (T X) for a.e. t. By Proposition 3.24 we see that t → a i,t is in W 1,1 ([0, 1], L 1 (π)) with derivative given by In particular, since |V i,t | ≤ 1 we see that a i,t , a ′ i,t ∈ L 2 ([0, 1], L 2 (π)) and in turn this implies -by Proposition 2.3 -that t → a i,t is in W 1,2 ([0, 1], L 2 (π)). This fact and point (ii) in Proposition 3.25 give that (t → a i,t V i,t ) ∈ H 1,2 (π) for every i ∈ N and therefore (t → n i=0 a i,t V i,t ) ∈ H 1,2 (π) for every n ∈ N. Hence to conclude it is sufficient to show that these partial sums are a W 1,2 (π)-Cauchy sequence, as then it is clear from (4.2) that the limit coincides with V . From (4.2) and (4.3) we have that hence the conclusion follows from the identity We shall now prove that if the parallel transport exists along all Lipschitz test plans, then the dimension of X, intended here as the dimension of the tangent module, must be constant. We shall use the following simple lemma (for simplicity we state it for Hilbert modules, but in fact the same holds for general ones): To see that they are independent, let f 0 , . . . , f n−1 ∈ L ∞ (Y) be such that i f i ϕ * v i = 0 on ϕ −1 (E). Then it holds and therefore f i = 0 m Y -a.e. on ϕ −1 (E) for every i = 0, . . . , n − 1.
We now prove that the dimension is constant: Theorem 4.8 (From parallel transport to constant dimension) Let (X, d, m) be a RCD(K, ∞) space such that for any Lipschitz test plan π, any t ∈ [0, 1] and anyV ∈ e * t L 2 (T X) there exists the parallel transport V along π such that V t =V .
Then the tangent module has constant dimension, i.e. in its dimensional decomposition (E i ) i∈N∪{∞} one of the E i 's has full measure and the others are negligible.
Proof. We argue by contradiction, thus we shall assume that for some i, j ∈ N ∪ {∞}, i = j, we have m(E i ), m(E j ) > 0. Let F 0 ⊂ E i and F 1 ⊂ E j be bounded, with positive and finite measure, consider let π be the unique optimal geodesic plan connecting them and recall that it is a test plan (see [14]). Since π(e −1 0 (F 0 )) = µ 0 (F 0 ) = 1 and the dimension of L 2 (T X) on F 0 is i, by Lemma 4.7 above we see that for the dimensional decomposition (Ẽ 0 n ) n∈N∪{∞} of e * 0 L 2 (T X) we have π(Ẽ 0 i ) = 1 and π(Ẽ 0 k ) = 0 for every k = i. Similarly, for the the dimensional decomposition (Ẽ 1 n ) n∈N∪{∞} of e * 1 L 2 (T X) we have π(Ẽ 1 j ) = 1 and π(Ẽ 1 k ) = 0 for every k = j. In particular we have Now notice that from basic considerations about optimal transport we have that π is concentrated on geodesics starting from F 0 and ending in F 1 . The constant speed of any such geodesic is bounded from above by sup x∈F 0 ,y∈F 1 d(x, y) < ∞ so that π is a Lipschitz test plan. Therefore from Proposition 4.5 we know that the dimensional decomposition of e * t L 2 (T X) does not depend on t. This however contradicts (4.4), hence the proof is completed.

Existence of the parallel transport in a special case
It is unclear to us whether on general RCD spaces the parallel transport exists or not. Aim of this section it to show at least that the theory we propose is not empty, i.e. that under suitable assumptions on the space, the parallel transport exists. We won't insist in trying to make such assumptions as general as possible (for instance, the 'good base' defined below could consist in different vector fields on different open sets covering our space) as our main concern is only to show that in some circumstances our notion of parallel transport can be shown to exist.
We shall work with spaces admitting the following sort of base for the tangent module: Definition 4.9 (Good base) Let (X, d, m) be a given RCD(K, N ) space, for some K ∈ R and N ∈ (1, ∞). Let us denote by (A k ) n k=1 the dimensional decomposition of X. Then a family W = {w 1 , . . . , w n } ⊆ H 1,2 C (T X) of Sobolev vector fields on X is said to be a good basis for L 2 (T X) provided there exists M > 0 such that the following properties are satisfied: (i) For any k = 1, . . . , n, we have that w 1 , . . . , w k constitute a basis for L 2 (T X) on A k and m-a.e. in A k , for every i, j = 1, . . . , k with i = j, (4.5) Let us notice that the 'hard' assumption here is given by point (ii) (perhaps coupled with the lower bound in (i)), which imposes an L ∞ bound on covariant derivative, when in our setting L 2 -ones are more natural (compare with Theorem A.2). Let us mention in particular that for spaces admitting a good base it is not hard to prove, regardless of parallel transport, that the dimension is constant (see Proposition 4.11 below). Let us start the technical work with the following simple lemma: Lemma 4.10 Let H be a Hilbert module on X, A ⊂ X be a Borel set and M > 1. Also, for k ∈ N let w 1 , . . . , w k ∈ H be such that hold m-a.e. in A, for every i, j = 1, . . . , k with i = j.
and in particular w ∈ H | A if and only if h i ∈ L 2 (m) | A for every i = 1, . . . , k.
Proof. For the second in (4.7) we notice that w i , w j ≤ M 2 m-a.e. on A for every i, j, thus For the first inequality we recall that |w i | > M −1 and w i , w j ≥ − 1 M 2 k for i = j m-a.e. on A to deduce The constant dimension now easily follows from the lemma and the fact that if a good base exists, then there is another one for which the functions w i , w j are Lipschitz, as shown in the proof of the following proposition.
Proposition 4.11 Let (X, d, m) be a RCD(K, N ) space admitting a good base for L 2 (T X).
Then the tangent module has constant dimension, i.e. in its dimensional decomposition (E k ) n k=1 one of the E k 's has full measure and the others are negligible. Proof. Let k ∈ N be the maximal index such that m(E k ) > 0 (its existence follows from the finiteness results in [16] and [12]). To conclude it is enough to show that on a neighbourhood of E k the tangent module has dimension ≥ k. Let (w i ) k i=1 be a good base, f ∈ C ∞ c (R) be such that f (z) = z for every z ∈ [0, M ] and consider the vector fieldsw i := f (|w i | 2 )w i . Notice that by [8] we know that f (|w i | 2 ) ∈ W 1,2 (X) with ∇f (|w i | 2 ) = 2f ′ (|w i | 2 )∇w i (·, w i ), hence by (4.6) and the choice of f we see that f (|w i | 2 ) is bounded with bounded gradient. It follows (see [8]) thatw i ∈ H 1,2 C (T X) with so that from the expression of ∇f (|w i | 2 ) we see thatw i is bounded with bounded covariant derivative. Hence g i,j := w i ,w j belongs to W 1,2 (X) (see [8]) and is bounded with bounded gradient as well. By the Sobolev-to-Lipschitz property (see [3], [7], [9]) we deduce that g i,j has a Lipschitz, in particular continuous, representative. By construction, the bounds (4.5) hold on E k for thew i 's, hence the continuity of g i,j grants that they hold also on some neighbourhood of E k , and by Lemma 4.10 above this is sufficient to conclude that the vector fieldsw i are independent -by the first in (4.7) -on such neighbourhood, thus concluding the proof.
We now prove existence of the parallel transport for the class of those RCD spaces that admit a good base for their tangent module. In the proof we shall use, for simplicity, the fact just proved that the dimension must be constant, but in fact the same argument works even without knowing a priori this fact (albeit since constant dimension follows so directly from the existence of a good base, this remark is perhaps irrelevant).
Theorem 4.12 (Existence of the parallel transport) Let (X, d, m) be any RCD(K, N ) space, K ∈ R and N ∈ (1, ∞), that admits a good base. Let π be a Lipschitz test plan on X and fix V ∈ e * 0 L 2 (T X). Then there exists the parallel transport V ∈ H 1,2 (π) along π such that V 0 = V . Proof. We know by Proposition 4.11 that the dimension of the tangent module must be constant. From this fact and Definition 4.9 we know that for some n we have w 1 , . . . , w n ∈ H 1,2 C (T X) for which (4.5) and (4.6) hold on X. Put W i,t := e * t w i . By point (i) in Proposition 3.25 we have that W i ∈ H 1,2 (π) with D π W i,t = Cov t (w i ) and therefore from (3.11), (4.6) and the assumption that π is Lipschitz we get (4.8) Also, by the defining property of the pullback map and from (4.5) we have that for every t ∈ [0, 1] it holds for every i, j = 1, . . . , n with i = j, thus Lemma 4.10 grants that there are functions g 1 , . . . , g n ∈ L 2 (π) such that V = n i=1 g i e * 0 w i . A similar argument applied to the pullback of the map e (recall Proposition 3.9 and the definition (3.7)) and based on the bound (4.8) shows that there are functions H i,j ∈ L ∞ (π × L 1 ) such that a.e. t. (4.9) It will be technically convenient to fix once and for all Borel representatives, still denoted by H i,j , of these functions such that sup γ,t |H i,j,t (γ)| = H i,j L ∞ (π×L 1 ) ∀i = 1, . . . , n. (4.10) We shall look for a parallel transport of the form V := i g i W i with g i ∈ AC 2 ([0, 1], L 2 (π)). Notice that Lemma 4.10 grants that any such V belongs to H 1,2 (π) with H j,i,t g j,t W i,t for L 1 -a.e. t ∈ [0, 1].
Hence our V is the desired parallel transport if and only if the functions g 1 , . . . , g n solve the system g i,0 = g i ∀i = 1, . . . , n, g ′ i,t + n j=1 H j,i,t g j,t = 0 ∀i = 1, . . . , n and for L 1 -a.e. t ∈ [0, 1]. (4.11) To solve this system we shall apply Theorem 2.6 to the Banach (in fact Hilbert) space B := [L 2 (π)] n equipped with the norm For every t ∈ [0, 1] define λ t ∈ End(B) as For the latter, notice that since B is separable it is sufficient to prove that for any f ∈ B the map t → λ t f ∈ B is weakly measurable. Since B is also Hilbert we need to show that for every f, g ∈ B the map t → λ t f, g B ∈ R is measurable. Since we have H j,i,t f j g i dπ, the conclusion follows from Fubini's theorem.

A Sobolev base of the tangent module
In this appendix we show that one can always build a base of the tangent module of a RCD space which has at least Sobolev regularity, as opposed to just L 2 regularity. The basic idea used in the construction is based on the observation that 'being a base' is a non-linear requirement. Technically speaking, the crucial argument is contained in the following lemma: Lemma A.1 Let (X, d, m) be a RCD(K, ∞) space and w ∈ L 2 (T X).
We claim that the vector v := i≥1 β i α i w i satisfies the conclusion of the statement and start observing that β i ≤ 3 −i and thus β i α i w i W 1,2 C (T X) ≤ 3 −i α i −1 w i W 1,2 C (T X) ≤ 3 −i by definition of α i . Hence the series converges in W 1,2 C (T X), so that v is well defined and belongs to H 1,2 C (T X). Now notice that by construction and (A.1) we havem(E n ) → m ∞ andm E n \{|w| > 0} = 0, so thatm {|w| > 0} \ ∪ n E n = 0. Hence to conclude it is sufficient to show that for every n ≥ 1 it holds v, w = 0m-a.e. on E n . Fix such n, let m > n and observe that by definition of the α i 's and β i 's we have β m α m w m , w ≤ 3 n−m+1 β n+1 α −1 m w m , w ≤ 3 n−m+1 β n+1 ,m − a.e., so that m>n βm αm w m , w ≤ 3 2 β n+1 ≤ 1 2 γ n+1 . On the other hand by construction we have that n i=1 β i α i w i , w ≥ γ n+1 holdsm-a.e. on E n , granting that | v, w | ≥ 1 2 γ n+1m -a.e. on E n .
By repeatedly applying Lemma A.1, we can find a family of H 1,2 C (T X)-Sobolev generators of the tangent module on any RCD(K, ∞) space X, as follows: Theorem A.2 (Sobolev base of the tangent module) Let (X, d, m) be an RCD(K, ∞) space, for some constant K ∈ R. Suppose that the dimensional decomposition of X is given by (A n ) n∈N . Then there exists a sequence of vector fields (v n ) n≥1 ⊆ H 1,2 C (T X) such that v 1 , . . . , v n is a basis for L 2 (T X) on A n , for every n ∈ N + .
Proof. The thesis can be equivalently rewritten in the following way: We build the sequence (v n ) n by means of a recursive argument. First of all, choose a vector field w ∈ L 2 (T X) such that 0 < |w| ≤ 1 m-a.e. in k≥1 A k , then pick v 1 ∈ H 1,2 C (T X) such that v 1 , w = 0 m-a.e. in k≥1 A k , whose existence is granted by Lemma A.1. Thus in particular we have |v 1 | > 0 m-a.e. in k≥1 A k , proving (A.3) for n = 1. Now suppose to have already found v 1 , . . . , v n satisfying the required property. It can be easily seen that there exists w ∈ L 2 (T X) such that v 1 , w = . . . = v n , w = 0 and 0 < |w| ≤ 1 hold m-a.e. in the set k>n A k . Hence Lemma A.1 ensures the existence of a vector field v n+1 ∈ H 1,2 C (T X) such that v n+1 , w = 0 m-a.e. in k>n A k .
Now take any f 1 , . . . , f n+1 ∈ L ∞ (m) such that n+1 i=1 f i v i = 0 m-a.e. in k>n A k , thus one has f n+1 v n+1 , w = n+1 i=1 f i v i , w = 0 m-a.e. in k>n A k , from which we can deduce that f n+1 = 0 holds m-a.e. in k>n A k . Therefore n i=1 f i v i = 0 m-a.e. in k>n A k and accordingly also f 1 = . . . = f n = 0 m-a.e. in k>n A k , as a consequence of the independence of v 1 , . . . , v n . This grants that the vector fields v 1 , . . . , v n+1 are independent on k>n A k , proving (A.3) for n + 1. The thesis is then achieved.