Tangent Lines and Lipschitz Differentiability Spaces

This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Title: Year: Version: Please cite the original version: All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Tangent Lines and Lipschitz Differentiability Spaces Cavalletti, Fabio; Rajala, Tapio Cavalletti, F., & Rajala, T. (2016). Tangent Lines and Lipschitz Differentiability Spaces. Analysis and Geometry in Metric Spaces, 4(1). https://doi.org/10.1515/agms-2016- 0004 2016 ©2016 Fabio Cavalletti and Tapio Rajala, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. Anal. Geom. Metr. Spaces 2016; 4:85–103 Research Article Open Access Fabio Cavalletti* and Tapio Rajala Tangent Lines and Lipschitz Differentiability Spaces DOI 10.1515/agms-2016-0004 Received August 21, 2015; accepted April 15, 2016 Abstract: 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 ndistinct tangent lines, obtained as the blow-up of nLipschitz curves, where nis the dimension of the local measurable chart. Under additional assumptions on the space, such as curvature lower bounds, these ndistinct tangent lines span an n-dimensional part of the tangent space. Keywords: metric geometry; Lipschitz differentiability spaces; tangent of metric spaces; Ricci curvature MSC: 51F99, 53B99 1Introduction 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, 24, 36], these assumptions still have strong geometric implications, [10, 22, 37]. 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 curve besides the trivial one. 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 bi-Lipschitz curve γ:Dom (γ)→Xof the metric space Xat 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 [21], we observe that *Corresponding Author: Fabio Cavalletti: University of Pavia, Dipartimento di Matematica, Pavia, Italy, E-mail: [email protected] Tapio Rajala: University of Jyvaskyla, Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyvaskyla, Finland, E-mail: [email protected] - 10.1515/agms-2016-0004 Downloaded from De Gruyter Online at 09/15/2016 03:08:05PM via The Helsinki University Library, Jyväskylän yliopiston kirjasto / Jyväskylä University Library and Jyväskylän Yliopisto University 86 |Fabio Cavalletti and Tapio Rajala metric derivative coincides with the metric speed at almost every point of Dom (γ). Therefore we deduce that a Lipschitz curve γis metrically differentiable at almost every point (see also Proposition 3.8 for an alternative proof of this fact). Thus broken bi-Lipschitz 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 nindependent Alberti representations. Using the measurability of the metric differential, Lemma 4.1, one can 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 ndistinct tangent lines at every point of the tangent space. Note that Tan(X,d,¯ x)will denote the collection of all the tangent cones obtained by blow-up of the space (X,d)at the point ¯ x. Theorem 1.1 (Theorem 4.5).Let (X,d,m)be a doubling Lipschitz differentiability space and (U,φ)be an ndimensional chart. Then for m-almost every ¯ x∈U, there exist v1,. . . ,vn∈Rnlinearly independent such that for any element (X∞,d∞,¯ x∞)∈Tan(X,d,¯ x)and for each z∈X∞there exist ιz 1,. . . ,ιz n:R→X∞so that i) ιz j(0) = z, for any j= 1,. . . ,n; ii) d∞(ιz j(t),ιz j(s)) = |t−s|, for any j= 1,. . . ,n, for all s,t∈R; iii) d∞(ιz j(t),ιz k(t)) ≥C|t|·|vj−vk|, for any j,k= 1,. . . ,n, for all t∈R; for some positive constant C=C(z). For each z∈X∞, each line ιz iis 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 nbeing the dimension of the chart, at almost every point there is a tangent space bi-Lipschitz equivalent to Rn, 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 [1, 3, 5, 14]), it is known that they infinitesimally look like Euclidean spaces, [16, 27]. 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 a 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 [30] (and its adaptation to metric spaces by Le Donne [25]) 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 splitting of tangents property (defined in Section 5) implies the existence of Rnin each of the tangents at almost every point, where nis 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. ¯ x∈Uany (X∞,d∞,¯ x∞)∈Tan(X,d,¯ x)is of the form (Xd ∞×Rd,dd ∞×|·|,(¯ xd ∞,0)), with d≥n. - 10.1515/agms-2016-0004 Downloaded from De Gruyter Online at 09/15/2016 03:08:05PM via The Helsinki University Library, Jyväskylän yliopiston kirjasto / Jyväskylä University Library and Jyväskylän Yliopisto University Tangent Lines and Lipschitz Differentiability Spaces |87 For the more general CD(K,N)spaces (see [26, 38, 39] for the definitions) isometric splitting of tangents is impossible since already Rnwith 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 [29]. 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 bi-Lipschitz equivalent to a product R×Yfor some metric space Y, the ndimensional Lipschitz chart could result in a piece of the tangent bi-Lipschitz equivalent to Rn. Let us note that for the even more general notion MCP(K,N)of Ricci curvature lower bound (see [28, 39] for the definitions) the above splitting result does not hold even in a topological sense [20]. Moreover, it is not known if a local Poincaré inequality holds in MCP(K,N)spaces without the non-branching 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 [18]. Thus the tangent lines cannot bi-Lipschitz 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 [21], that the metric derivative agrees almost everywhere with the metric speed. We also show that at almost every point of a bi-Lipschitz curve the blow-up will be a tangent line. In Section 4 we consider the blow-ups in a Lipschitz differentiability space showing that we have nindependent tangent lines at almost every point. In the final section, Section 5, following the ideas of David and Schioppa [12, 34], we prove that if tangents split off tangent lines then the nindependent tangent lines in a Lipschitz differentiability space span a Euclidean Rnin the tangent. 2Preliminaries A metric measure space is a triple (X,d,m)where (X,d)is a complete and separable metric space and ma 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 mwill be a positive Radon measure. We list here two general properties of metric measure spaces that we will consider during the paper. The metric measure space (X,d,m)is (uniformly locally) doubling if for each R>0there exists C(R)>0such that 0<m(B2r(x)) ≤C(R)m(Br(x)),for every x∈X,r≤R. With no loss in generality, the function Ccan be taken non-decreasing. Moreover a metric measure space (X,d,m)supports a local p-Poincaré inequality for some p≥1if every ball in Xhas positive and finite measure and for every g∈Lip(X,d) := {l:X→R|lis Lipschitz}, B |g(x)−gB|dm(x)≤Lr  BrL(x0) |Dg|p(x)dm(x)1/p , for some positive constant L, where B=Br(x0)and gB=fflBg(x)dm(x). Here for g∈Lip(X,d)we also adopt the following notation: |Dg|(x) := lim sup y→x, y=x d(g(y),g(x)) d(y,x). - 10.1515/agms-2016-0004 Downloaded from De Gruyter Online at 09/15/2016 03:08:05PM via The Helsinki University Library, Jyväskylän yliopiston kirjasto / Jyväskylä University Library and Jyväskylän Yliopisto University 88 |Fabio Cavalletti and Tapio Rajala 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. A map f: (X,dX)→(Y,dY)between compact metric spaces is called an ε-isometry provided (i) it almost preserves distances: for all z,w∈X, |dX(z,w)−dY(f(z),f(w))|≤ε; (ii) it is almost surjective: ∀y∈Y,∃x∈X:dY(f(x),y)≤ε. In order to deal with possibly non-compact spaces, it is customary to fix a distinguished point ¯ x∈Xand to consider ε-isometries defined on an increasing family of balls centered in ¯ x. When a distinguished point is fixed, we use (X,d,¯ x)to denote the pointed metric space. Definition 2.1. A sequence {(Xi,di,¯ xi)}i∈Nof pointed, proper, complete metric spaces converges to a pointed, proper, complete metric space (X∞,d∞,¯ x∞)in pointed Gromov-Hausdorff, and write (Xi,di,¯ xi)−→ (X∞,d∞,¯ x∞),pGH, if and only if there exist sequences of positive real numbers {εi}i∈N,{Ri}i∈Nwith εi→0,Ri→∞and a sequence of εi-isometries, fi:BXi Ri(¯ xi)−→ BX∞ Ri(¯ x∞),fi(¯ xi) = ¯ x∞, where BXi Ri(¯ xi)is the ball in Xi, centered in ¯ xand of radius Ri. We also consider pointed metric measure spaces: a quadruple (X,d,m,¯ x)where (X,d,m)is a metric measure space and ¯ x∈Xa distinguished point. Definition 2.2. A sequence {(Xi,di,mi,¯ xi)}i∈Nof pointed metric measure spaces converges in the pointed measured Gromov-Hausdorff sense to a pointed metric measure space (X∞,d∞,m∞,¯ x∞) (Xi,di,mi,¯ xi)−→ (X∞,d∞,m∞,¯ x∞),pmGH, if and only if there exist sequences of positive real numbers {εi}i∈N,{Ri}i∈Nwith εi→0,Ri→∞and a sequence of εi-isometries, fi:BXi Ri(¯ xi)−→ BX∞ Ri(¯ x∞),fi(¯ xi) = ¯ x∞, such that lim i→∞ˆX∞ φ(z)d(fi]mi)(z) = ˆX∞ φ(z)dm∞(z),∀φ∈Cb(X∞), where Cb(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 and ¯ x∈Xis a distinguished point, then any limit point in the pGH-convergence of any sequence of the form {(X,d/ri,¯ x)}i∈N, with ri→0, is a tangent space of (X,d)at ¯ x. We use Tan(X,d,¯ x) to denote the set of all possible tangent spaces of (X,d)at ¯ x. If (X,d,m)is a metric measure space and ¯ x∈supp(m)is a distinguished point, for any r>0, the rescaled and normalized pointed metric measure space is defined as follows: X,1 rd,m¯ x r,¯ x,m¯ x r:= ˆBr(¯ x) 1−1 rd(¯ x,z)dm(z)−1 m. - 10.1515/agms-2016-0004 Downloaded from De Gruyter Online at 09/15/2016 03:08:05PM via The Helsinki University Library, Jyväskylän yliopiston kirjasto / Jyväskylä University Library and Jyväskylän Yliopisto University Tangent Lines and Lipschitz Differentiability Spaces |89 Then a limit point in the pmGH-convergence of the sequence {(X,d/ri,m¯ x ri,¯ x)}i∈Nis a measured tangent space of (X,d,m)at ¯ xand to denote the set of all possible measured tangent spaces of (X,d,m)at ¯ xwe use Tan(X,d,m,¯ x). It is worth noticing that, thanks to compactness properties of the collection of uniformly doubling metric measure spaces (see [40], Theorem 27.32 and [17], Lemma 3.32), Tan(X,d,m,¯ x)is always non empty, provided (X,d,m)is doubling. 2.2 Tangent functions Here we recall a few objects and related results presented in [10] and in [19]. If (X,dX)and (Y,dY)are metric spaces and f:X→Yis an ε-isometry, then there exists a (4ε)-isometry f0:Y→Xso that for all x∈Xand y∈Yit holds dX(f0◦f(x),x)≤3ε,dY(f◦f0(y),y)≤ε. Such a map is usually called an ε-inverse of fand accordingly we will often adopt the notation f−1to denote it. Consider now any element (X∞,d∞,m∞,¯ x∞)∈Tan(X,d,m,¯ x)and a sequence of ri→0such that X,1 ri d,m¯ x r,¯ x−→ (X∞,d∞,m∞,¯ x∞),pmGH. Then to any Lipschitz function g:X→Rwe can associate a sequence of rescaled functions centered at ¯ x: gi(x) := g(x)−g(¯ x) ri . If gis L-Lipschitz in (X,d), then so is giin (X,d/ri). With this in mind, we say that ug:X∞→Ris a compatible tangent function of gat ¯ xif lim i→∞gi(f−1 i(z)) = lim i→∞ g(f−1 i(z)) −g(¯ x) ri =ug(z),∀z∈X∞, where f−1 iis any εi-inverse of the approximate isometry figiven by the pmGH convergence of (X,d/ri,m¯ x 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 ugdoes not depend on the choice of the sequence of the εi-inverses. Since fi is almost surjective, for any z∈X∞and i∈Nsufficiently large, there exists xi∈Xsuch that d∞(fi(xi),z)≤εi. One then easily observes that |gi(f−1 i(z)) −gi(f−1 i◦fi(xi))| → 0. If f−1 iand ˆ f−1 iare two distinct εi-inverses of fi, it follows, by the triangle inequality that lim i→∞ 1 ri d(f−1 i◦fi(xi),ˆ f−1 i◦fi(xi)) = 0, and since gis Lipschitz, it follows that gi(f−1 i(z)) and gi(ˆ 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 ri→0such that X,1 ri d,m¯ x r,¯ x−→ (X∞,d∞,m∞,¯ x∞)∈Tan(X,d,m,¯ x), where the convergence is in the pmGH sense. Fix also a countable collection Fof uniformly Lipschitz functions defined on X. Then possibly choosing a subsequence of {ri}i∈N, for each g∈Fthere exists uga compatible tangent function of gat ¯ x. - 10.1515/agms-2016-0004 Downloaded from De Gruyter Online at 09/15/2016 03:08:05PM via The Helsinki University Library, Jyväskylän yliopiston kirjasto / Jyväskylä University Library and Jyväskylän Yliopisto University 90 |Fabio Cavalletti and Tapio Rajala The proof of Lemma 2.4 follows from a standard use of Ascoli-Arzela Theorem. See [23] for details. 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. 2.3 Lipschitz differentiability spaces Under fairly general assumptions on the structure of the metric measure space, it is proved in [10] that the space of germs 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⊂Xand a Lipschitz function φ:X→Rn form a chart of dimension n,(U,φ), and a function g:X→Ris differentiable at x0∈Uwith respect to (U,φ) if there exists a unique Dg(x0)∈Rnsuch that lim sup x→x0 |g(x)−g(x0)−Dg(x0)·(φ(x)−φ(x0))| d(x,x0)= 0. Furthermore a metric measure space (X,d,m)is called a Lipschitz differentiability space if there exists a countable decomposition of Xinto charts such that any Lipschitz function g:X→Ris 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≥1for 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 out. Here we report only the main statement. We use Γ(X)to denote the set of bi-Lipschitz (onto their image) maps γ:Dom (γ)→X, with Dom (γ)⊂Rnon-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 γx 1,. . . ,γx n∈Γ(X)such that: i) (γx i)−1(x) = 0 is a point of density one of (γx i)−1(U)for each i= 1,. . . ,n; ii) {(φ◦γx i)0(0)}i=1,...,nare linearly independent. Moreover, for any such γx i, for any Lipschitz g:X→Rand m-a.e. x∈U, the gradient of gat xwith respect to φand γx 1,. . . ,γx nequals Dg(x), that is g◦γx i0(0) = Dg(x)·φ◦γx i0(0),m−a.e.x∈U, for i= 1,. . . ,n. Hence not only the space of germs 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 bi-Lipschitz curves. Here it is worth mentioning Keith’s results on coordinate functions: in [19] it is proved that the role of the coordinate map φin chart (U,φ)can be played by distance functions from a suitable set. We report here Theorem 2.7 of [19]. - 10.1515/agms-2016-0004 Downloaded from De Gruyter Online at 09/15/2016 03:08:05PM via The Helsinki University Library, Jyväskylän yliopiston kirjasto / Jyväskylä University Library and Jyväskylän Yliopisto University Tangent Lines and Lipschitz Differentiability Spaces |91 Theorem 2.8. Let (X,d,m)be a complete and separable metric measure space admitting a p-Poincaré inequality with mdoubling. Then there exists a measurable differentiable structure {(Ui,φi)}i∈Nsuch that each φi:Ui→Rd(i)is of the form φi(z) = d(z,x1),. . . ,d(z,xd(i)), for some x1,. . . ,xd(i)∈X. For a generalization of the previous result to doubling differentiability spaces, see Corollary 6.31 of [35]. 2.4 Geodesics in product spaces If (X,dX)and (Y,dY)are two metric spaces, we can consider the product distance dXY defined by dXY := qd2 X+d2 Y. Then (X×Y,dXY )is again a metric space. We recall an easy lemma on geodesics in product spaces. By a geodesic in a metric space (X,dX)we mean a map γ: [0,1] →Xsatisfying dX(γs,γt) = |t−s|dX(γ0,γ1)for all s,t∈[0,1], where we use the usual abbreviation γt=γ(t). Lemma 2.9. A curve [0,1] 3t7→ (γ1 t,γ2 t)∈(X×Y,dXY )is a geodesic if and only if γ1is a geodesic in (X,dX) and γ2is a geodesic in (Y,dY). Proof. It is immediate that if γ1and γ2are geodesics, then also (γ1 t,γ2 t)is a geodesic. So, let us show the other direction. We start with the easy inequality: for a,b,c,dpositive real numbers, (a2+b2)(c2+d2)≥(bd +ac)2.(2.1) Then let [0,1] 3t7→ (γ1 t,γ2 t)∈X×Ybe a geodesic and suppose by contradiction that γ1is not. For ease of notation, we can assume that dX(γ1 −s,γ1 s)<dX(γ1 0,γ1 −s) + dX(γ1 0,γ1 s),(2.2) for some s>0. By the fact that (γ1 t,γ2 t)is a geodesic we have d2 X(γ1 −s,γ1 s) + d2 Y(γ2 −s,γ2 s) = qd2 X(γ1 −s,γ1 0) + d2 Y(γ2 −s,γ2 0) + qd2 X(γ1 0,γ1 s) + d2 Y(γ2 0,γ2 s)2 . Expanding the squares and using (2.2), we obtain that d2 Y(γ2 −s,γ2 s)>d2 Y(γ2 −s,γ2 0) + d2 Y(γ2 0,γ2 s) + 2qd2 X(γ1 −s,γ1 0) + d2 Y(γ2 −s,γ2 0)·qd2 X(γ1 0,γ1 s) + d2 Y(γ2 0,γ2 s) −2dX(γ1 −s,γ1 0)dX(γ1 0,γ1 s). We can now use the inequality (2.1) to get d2 Y(γ2 −s,γ2 s)>d2 Y(γ2 −s,γ2 0) + d2 Y(γ2 0,γ2 s) + 2dY(γ2 −s,γ2 0)dY(γ2 0,γ2 s), violating the triangle inequality. The claim follows. 3Tangent lines Let us start this section by recalling a result from [33], Theorem 7.10: a more general version of Lebesgue Differentiation Theorem. Here and in the sequel Lddenotes the Lebesgue measure on Rd. - 10.1515/agms-2016-0004 Downloaded from De Gruyter Online at 09/15/2016 03:08:05PM via The Helsinki University Library, Jyväskylän yliopiston kirjasto / Jyväskylä University Library and Jyväskylän Yliopisto University 92 |Fabio Cavalletti and Tapio Rajala Definition 3.1. Fix x∈Rdand a sequence of Borel sets {Ei}i∈N⊂Rd. We say that {Ei}i∈Nshrinks nicely to x provided there exist ri>0and α>0such that for each i∈Nwe have Ei⊂Bri(x)and Ld(Ei)≥αLd(Bri(x)). For the nicely shrinking sets we have the following general version of Lebesgue Differentiation Theorem. Theorem 3.2. Let f∈L1(Rd,R)be any function. Associate to each x∈Rda sequence {Ei(x)}i∈Nof sets nicely shrinking to x. Then f(x) = lim i→∞ 1 Ld(Ei(x)) ˆEi(x) f(y)dy, for every Lebesgue point xof f. In particular it holds for Ld-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)→Xbe any curve. We say that γis metric differentiable at t∈Dom (γ)provided the following limit lim s,τ→0 nicely d(γt+s,γt+τ) |s−τ| exists for any sequence of sand τ, where with nicely we ask for the interval with boundary formed by t+s and t+τto 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]. Actually the two notions are different. Consider for instance the curve γ: [−1,1] →R2defined by γ(t) := (t,t)for t≥0and γ(t) := (t,−t)for t≤0. Then the metric speed always exists and is 1, while |dγ|does not exists for t= 0. The converse trivially holds. For curves with values in a Euclidean space, at any point of differentiability, |dγ|(t0)coincides with the modulus of the derivative. Remark 3.5. Another notion of differentiability for maps with values in metric spaces was introduced by Kirchheim in [21]: for any g:Rn→(X,d)consider the following quantity MD(g,x)(u) := lim r&0 1 rd(g(x+ru),g(x)) for all x,u∈Rn, whenever the limit exists. In Theorem 2 of [21] it is proved that for Lipschitz functions g,MD exists almost everywhere, with respect to Lebesgue measure, and at almost every point where it exists, it is a seminorm. Theorem 3.6 ([21]).Let g:Rn→Xbe Lipschitz. Then, for almost every x∈Rn,MD(g,x)(·)is a seminorm on Rnand d(g(z),g(y)) −MD(g,x)(z−y) = o(|z−x|+|z−y|).(3.1) In the case of Lipschitz curves (n= 1) the quantity MD 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 tis a point of metric differentiability and |dγ|(t) = MD(γ,t)(1). Also the converse implication holds. We include here a short proof for the reader’s convenience. Lemma 3.7. Suppose a Lipschitz curve γ: [−c,c]→Xis metric differentiable at 0. Then d(γt,γs)−|dγ|(0) ·|t−s|=o(|t|+|t−s|). - 10.1515/agms-2016-0004 Downloaded from De Gruyter Online at 09/15/2016 03:08:05PM via The Helsinki University Library, Jyväskylän yliopiston kirjasto / Jyväskylä University Library and Jyväskylän Yliopisto University Tangent Lines and Lipschitz Differentiability Spaces |99 Now we just observe that for j,l= 1,. . . ,n d∞(zj h,zl h) = lim k→∞d∞(fik◦γj thrik ,fik◦γl thrik) = lim k→∞ 1 rik d(γj thrik ,γl thrik) ≥1 Llim k→∞ 1 rik |φ◦γj thrik −φ◦γl thrik| ≥th L|φ◦γj0(0) −φ◦γl0(0)|, where Lis the Lipschitz constant of φ. Therefore we have proved that d∞(zj h,zl h)≥th L|φ◦γj0(0) −φ◦γl0(0)|,(4.2) that implies, by linear independence, that d∞(zj h,zl h)>0, for all h∈N. Since intersection for different times is not possible (at time 0 they start from the same point, with the same speed), the claim follows. We summarize the disjointness property of the isometric embeddings of R. Corollary 4.4. Let (X,d,m)be a Lipschitz differentiability space and (U,φ)be an n-dimensional chart. Then for m-almost every ¯ x∈U, there exist v1,. . . ,vn∈Rnlinearly independent such that for any element (X∞,d∞,¯ x∞)∈Tan(X,d,¯ x)there exist ι1,. . . ,ιn:R→X∞so that i) ιj(0) = ¯ x∞, for any j= 1,. . . ,n; ii) d∞(ιj(t),ιj(s)) = |t−s|, for any j= 1,. . . ,n, for all s,t∈R; iii) d∞(ιj(t),ιk(t)) ≥C|t|·|vj−vk|, for any j,k= 1,. . . ,n, for all t∈R; for some positive constant C. Each of the ιiis obtained as the limit of a Lipschitz curve. 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. 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 every ¯ x∈U, there exist v1,. . . ,vn∈Rnlinearly independent such that for any element (X∞,d∞,¯ x∞)∈Tan(X,d,¯ x)and for each z∈X∞there exist ιz 1,. . . ,ιz n:R→X∞so that i) ιz j(0) = z, for any j= 1,. . . ,n; ii) d∞(ιz j(t),ιz j(s)) = |t−s|, for any j= 1,. . . ,n, for all s,t∈R; iii) d∞(ιz j(t),ιz k(t)) ≥C|t|·|vj−vk|, for any j,k= 1,. . . ,n, for all t∈R; for some positive constant C=C(z). For each z∈X∞, each line ιz iis obtained as the blow-up of a Lipschitz curve, with the blow-up depending on z. 5Tangent 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 available. For instance in [10], Theorem 8.5, existence of integral curves for tangent functions was proved. This in turn implies the existence - 10.1515/agms-2016-0004 Downloaded from De Gruyter Online at 09/15/2016 03:08:05PM via The Helsinki University Library, Jyväskylän yliopiston kirjasto / Jyväskylä University Library and Jyväskylän Yliopisto University 100 |Fabio Cavalletti and Tapio Rajala 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 nis the dimension of a chart of the measurable differentiable structure of (X,d,m)seen as a Lipschitz differentiability space and if dis the dimension of a Euclidean tangent space at x, then n≤dat m-a.e. point of X. More precisely, we are interested in a special class of metric measure spaces (X,d,m)having the splitting of tangents property: if (X∞,d∞,¯ x∞)∈Tan(X,d,¯ x) for some ¯ x∈Xand if X∞contains an isometric copy of Rgoing through ¯ x∞, then (X∞,d∞)is isometric to (R×Y,|·|×dY) where (Y,dY)is a metric space, and the same property holds for Y. In particular if ι:R→X∞parametrizes the isometric copy of Rcontained in X∞, then there exists an isometry h: (X∞,d∞)→(R×Y,|·|×dY), such that h◦ι(R)⊂R×{¯ y}, for some ¯ y∈Y. The same property has to hold for the metric space (Y,dY). 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∈Uany (X∞,d∞,¯ x∞)∈ Tan(X,d,¯ x)is of the form (Xd ∞×Rd,dd ∞×|·|,(¯ xd ∞,0)), with d≥n. 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 Xthere 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∈Rand 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 [31, 32], we immediately have that any tangent space contains an Rnpart with dimension at least the dimension of the chart. For a comprehensive treatise on the above mentioned family of spaces we refer to [26, 38, 39] for the defintion of CD(K,N)and to [2, 3] for the infinite dimensional Riemannian version. Finally RCD*(K,N)with N∈Rhas 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 [27] states that for (X,d,m)verifying RCD*(K,N), at m-a.e. x∈Xthere exists a unique tangent space and it is isomorphic, in the sense of metric measure spaces, to (Rd,|·|,Ld), with dvarying 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∈Rwith N>1. Then there exists a countable collection {Rj}j∈Nof m-measurable subsets of X, covering Xup to an m-negligible set, such that each Rjis bi- Lipschitz to a measurable subset of Rkj, for some 1≤kj≤N,kjpossibly depending on j. However, if a complete Lipschitz differentiability space is bi-Lipschitz embeddable into some Euclidean space Rk, then at almost every point all the tangent spaces are bi-Lipschitz equivalent to Rn, where n≤kis - 10.1515/agms-2016-0004 Downloaded from De Gruyter Online at 09/15/2016 03:08:05PM via The Helsinki University Library, Jyväskylän yliopiston kirjasto / Jyväskylä University Library and Jyväskylän Yliopisto University Tangent Lines and Lipschitz Differentiability Spaces |101 the dimension of the chart given by the Lipschitz differentiability, see Corollary 8.1 in [12]. Moreover (see [7]) any positive measure subset of a Lipschitz differentiability space is itself a Lipschitz differentiability space. Since from inner regularity any measurable subsets of positive measure can be approximated, up to a set of measure zero, from inside with an increasing family of compact sets (obtaining therefore also the completeness), combining these two result with Theorem 5.3 we get that for RCD*(K,N)spaces at almost every point the tangent is Rnwhere the nis the dimension of the chart. Let us note that it is still unknown if in this context the dimension nof the tangent (and the chart) depends on the point. We prove Theorem 5.1, which is valid without the bi-Lipschitz embeddability to Rn. Proof of Theorem 5.1. Step 1. By Corollary 4.4 any element (X∞,d∞,¯ x∞)∈Tan(X,d,¯ x)has ndistinct isometric copies of R: ιj:R→X∞,ιj(0) = ¯ x∞,j= 1,. . . ,n, and each ιjis the blow-up of has a corresponding Lipschitz curve γj, see Proposition 4.3. By the splitting property, there exists an isometry h1: (X∞,d∞)−→ (X1 ∞×R,d1 ∞×|·|),h1(¯ x∞) = (¯ x1 ∞,0), with h1(ι1(R)) = {(¯ x1 ∞,t) : t∈R}. Since the ngeodesics are all disjoint, composing isometries and applying Lemma 2.9 we deduce the existence of n−1geodesics, again denoted with ιj: (R,|·|)→(X1 ∞,d1 ∞),j= 2,. . . ,n. By Lemma 2.9 we can also deduce that ι2(R),. . . ,ιn(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∞∼Xn ∞×Rn. It might be the case that for some j= 3,. . . ,n,ιj(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→RnLipschitz and any ¯ x∈Usuch that Corollary 3.13 applies. Fix also (X∞,d∞,¯ x∞)∈Tan(X,d,¯ x)and uφ, the tangent function of φat ¯ x. Note that, possibly passing to subsequences, uφis well-defined. Repeating the argument of Step 1. changing the reference point (see [25], Theorem 1.1), we have the following: for some d∈Nall the possible splittings obtained from the lines of Theorem 4.5 give a decomposition of the following type: X∞=Xd ∞×Rd, where the identity holds in the sense of metric spaces, and Rdis equipped with the Euclidean distance. Step 3. We now show that d≥n. Consider the sequence {ri}i∈Nproducing (X∞,d∞,¯ x∞)as the tangent space and the 3εi-isometries fiand f−1 i. Let z∈X∞be any point, then by definition uφ(z) = lim i→∞ φ(f−1 i(z)) −φ(¯ x) ri . As observed in the proof of Proposition 4.3, after a suitable reparametrization with unit speed, there exists a sequence of times {ti}i∈Nwith ti→1as i→∞such that d∞(ιj(1),fi(γj tiri)) →0, for each j= 1,. . . ,n. We pose z=ιj(1) and observe that 1 ri |φ(γj tiri)−φ(f−1 i(z))|≤L1 ri d(γj tiri,f−1 i(z)) ≤Lεi+d∞(fi(γj tiri),fi(f−1 i(z))) ≤Lεi+d∞(fi(γj tiri),z) + d∞(z,fi(f−1 i(z))) ≤Cεi. - 10.1515/agms-2016-0004 Downloaded from De Gruyter Online at 09/15/2016 03:08:05PM via The Helsinki University Library, Jyväskylän yliopiston kirjasto / Jyväskylä University Library and Jyväskylän Yliopisto University 102 |Fabio Cavalletti and Tapio Rajala It therefore follows that uφ(ιj(1)) = lim i→∞ φ(γj tiri)−φ(¯ x) ri = (φ◦γj)0(0). Using a different ticonverging to some other real number, it is easy to observe that s7→ uφ(ιj(s)) is linear and Spanuφ(ι1(1)),. . . ,uφ(ιn(1))=Rn. Thanks to Proposition 3.1 of [12], the same argument works for any z∈X∞. We can therefore consider the isometries ιz 1,. . . ,ιz nsuch that ιz j(0) = zfor j= 1,. . . ,nsuch that s7→ uφ(ιz j(s)) is linear, for any z∈X∞and j= 1,. . . ,n. Finally, we consider ¯ uφ, the restriction of uφto n¯ xd ∞o×Rd→Rn. The claim can now be proven via showing that ¯ uφis 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 s7→ uφ(ιz j(s)), together with Theorem 4.5. References [1] L. Ambrosio, N. Gigli, A. Mondino, and T. Rajala, Riemannian Ricci curvature lower bounds in metric measure spaces with σ-finite measure, Trans. Amer. Math. Soc. 367 (2015), no. 7, 4661–4701. [2] L. Ambrosio, N Gigli and G. Savaré, Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds, Annals of Probab. 43 (2015), no. 1, 339–404. [3] L. Ambrosio, N Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J. 163 (2014), 1405–1490. [4] L. Ambrosio and B. Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000), 527–555. [5] L. Ambrosio, A. Mondino and G. Savaré, Nonlinear diffusion equations and curvature conditions in metric measure spaces, preprint arXiv:1509.07273. [6] L. Ambrosio and P. Tilli, Topics on Analysis in Metric Spaces. Oxford University press, Oxford Lecture Series in Mathematics and Its Applications, 2004. [7] D. Bate, Structure of measures in Lipschitz differentiability spaces, Journal Amer. Math. Soc. 28 (2015), 421–482. [8] D. Bate and S. Li, Characterizations of rectifiable metric measure spaces, preprint, arXiv:1409.4242. [9] M. Bourdon and H. Pajot, Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings, Proc. Amer. Math. Soc. 127 (1999), no. 8, 2315–2324. [10] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9(1999), 428–517. [11] J. Cheeger and B. Kleiner and A. Schioppa, Infinitesimal structure of differentiability spaces, and metric differentiation, preprint arXiv:1503.07348. [12] G.C. David, Tangents and rectifiability of Ahlfors regular Lipschitz differentiability spaces, Geom. Funct. Anal. 25 (2015), no. 2, 553–579. [13] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1–61. [14] M Erbar, Kuwada and K.T. Sturm, On the Equivalence of the Entropic Curvature-Dimension Condition and Bochner’s Inequality on Metric Measure Space, Invent. Math. 201 (2015), no. 3, 993 – 1071. [15] N. Gigli, The splitting theorem in non-smooth context, preprint, arXiv:1302.5555. [16] N. Gigli, A. Mondino and T. Rajala, Euclidean spaces as weak tangents of infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded below, J. Reine Angew. Math. 705 (2015), 233–244. [17] N. Gigli, A. Mondino and G. Savaré, Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, to appear in Proc. London Math. Soc. doi: 10.1112/plms/pdv047. [18] N. Juillet, Geometric inequalities and generalized Ricci bounds in the Heisenberg group, Int. Math. Res. Notices 2009 (2009), 2347–2373. [19] S. Keith, Measurable Differentiable Structures and the Poincaré Inequality, Indiana Univ. Math. J. 53 (2004), 1127–1150. [20] C. Ketterer and T. Rajala, Failure of topological rigidity results for the measure contraction property, Potential Anal. 42 (2015), no. 3, 645–655. [21] B. Kirchheim, Rectifiable metric space: local structure and regularity of the Hausdorff measure, Proc. Am. Math. Soc. 121 (1994), 113–123. [22] R. Korte, Geometric implications of the Poincaré inequality, Result. Math. 50 (2007), 93–107. [23] B. Kleiner and J. Mackay, Differentiable structures on metric measure spaces: A Primer preprint, arXiv:1108.1324. [24] T. Laakso, Ahlfors Q-regular spaces with arbitrary Q>1admitting weak Poincaré inequality, Geom. Funct. Anal. 10 (2000), 111–123. - 10.1515/agms-2016-0004 Downloaded from De Gruyter Online at 09/15/2016 03:08:05PM via The Helsinki University Library, Jyväskylän yliopiston kirjasto / Jyväskylä University Library and Jyväskylän Yliopisto University Tangent Lines and Lipschitz Differentiability Spaces |103 [25] E. Le Donne, Metric spaces with unique tangents, Ann. Acad. Sci. Fenn. Math. 36 (2011), 683–694. [26] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169 (2009), 903–991. [27] A. Mondino and A. Naber, Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds I, preprint, arXiv:1405.2222. [28] S.-I. Ohta, On the measure contraction property of metric measure spaces, Comment. Math. Helv. 82 (2007), 805–828. [29] S.-I. Ohta, Splitting theorems for Finsler manifolds of nonnegative Ricci curvature, J. Reine Angew. Math. 700 (2015), 155– 174. [30] D. Preiss, Geometry of measures in Rn: distribution, rectifiability, and densities, Ann. of Math. 125 (1987), 537–643. [31] T. Rajala, Local Poincaré inequalities from stable curvature conditions on metric spaces, Calc. Var. Partial Differential Equations 44 (2012), 477–494. [32] T. Rajala, Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm, J. Funct. Anal. 263 (2012), 896–924. [33] W. Rudin, Real and complex analysis, McGraw-Hill Book Company, International Edition 1987. [34] A. Schioppa Derivations and Alberti representations, preprint, arXiv:1311.2439. [35] A. Schioppa On the relationship between derivations and measurable differentiable structures, Ann. Acad. Sci. Fenn. Math., 39 (2014), no. 1, 275–304. [36] S. Semmes, Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities, Selecta Math. 2(1996), 155–295. [37] S. Semmes, Some novel types of fractal geometry, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2001. [38] K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006), 65–131. [39] K.T. Sturm, On the geometry of metric measure spaces. II, Acta Math. 196 (2006), 133–177. [40] C. Villani, Optimal transport. Old and new, Grundlehren der Mathematischen Wissenschaften,338, Springer-Verlag, Berlin, (2009). - 10.1515/agms-2016-0004 Downloaded from De Gruyter Online at 09/15/2016 03:08:05PM via The Helsinki University Library, Jyväskylän yliopiston kirjasto / Jyväskylä University Library and Jyväskylän Yliopisto University