A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries

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. A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries Le Donne, Enrico Le Donne, E. (2017). A Primer on Carnot Groups: Homogenous Groups, Carnot- Carathéodory Spaces, and Regularity of Their Isometries. Analysis and Geometry in Metric Spaces, 5(1), 116-137. https://doi.org/10.1515/agms-2017-0007 2017 Open Access. ©2017 Enrico Le Donne, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution- Non-Commercial-NoDerivs 4.0 License. Anal. Geom. Metr. Spaces 2017; 5:116–137 Survey Paper Open Access Enrico Le Donne* A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries https://doi.org/10.1515/agms-2017-0007 Received November 18, 2016; revised September 7, 2017; accepted November 8, 2017 Abstract: Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remarks. We consider them as special cases of graded groups and as homogeneous metric spaces. We discuss the regularity of isometries in the general case of Carnot-Carathéodory spaces and of nilpotent metric Lie groups. Keywords: Carnot groups, sub-Riemannian geometry, sub-Finsler geometry, homogeneous spaces, homogeneous groups, nilpotent groups, metric groups MSC: 53C17, 43A80, 22E25, 22F30, 14M17. Carnot groups are special cases of Carnot-Carathéodory spaces associated with a system of bracket-generating vector fields. In particular, they are geodesic metric spaces. Carnot groups are also examples of homogeneous groups and stratified groups. They are simply connected nilpotent groups and their Lie algebras admit special gradings: stratifications. The stratification can be used to define a left-invariant metric on each group in such a way that the group is self-similar with respect to this metric. Namely, there is a natural family of dilations on the group under which the metric behaves like the Euclidean metric under Euclidean dilations. Carnot groups, and more generally homogeneous groups and Carnot-Carathéodory spaces, appear in several mathematical contexts. Such groups appear in harmonic analysis, in the study of hypoelliptic differential operators, as boundaries of strictly pseudo-convex complex domains, see the books [26, 96] as initial references. Carnot groups, with subFinsler distances, appear in geometric group theory as asymptotic cones of nilpotent finitely generated groups, see [41, 85]. SubRiemannian Carnot groups are limits of Riemannian manifolds and are metric tangents of subRiemannian manifolds. SubRiemannian geometries arise in many areas of pure and applied mathematics (such as algebra, geometry, analysis, mechanics, control theory, mathematical physics), as well as in applications (e.g., robotics), for references see the book [75]. The literature on geometry and analysis on Carnot groups is plentiful. In addition to the previous references, we also cite some among the more authoritative ones [3, 16, 36, 51, 56, 70, 80, 87, 90, 98, 99, Hei95]. The setting of Carnot groups has both similarities and differences compared to the Euclidean case. In addition to the geodesic distance and the presence of dilations and translations, on each Carnot group one naturally considers Haar measures, which are unique up to scalar multiplication and are translation invariant. Moreover, in this setting they have the important property of being Ahlfors-regular measures. In fact, the measure of each r-ball is the measure of the unit ball multiplied by rQ, where Qis an integer depending only on the group. With such a structure of metric measure space, it has become highly interesting to study geo- *Corresponding Author: Enrico Le Donne: Department of Mathematics and Statistics, P.O. Box 35, FI-40014, University of Jyväskylä, Finland, E-mail: [email protected] Brought to you by | Jyväskylän Yliopisto University Authenticated Download Date | 1/30/18 12:38 PM A Primer on Carnot Groups |117 metric measure theory, and other aspects of analysis or geometry, in Carnot groups. On the one hand, with so much structure many results in the Euclidean setting generalize to Carnot groups. The most celebrated example is Pansu’s version of Rademacher’s theorem for Lipschitz maps (see [85]). Other results that have been generalized are the Isoperimetric Inequality [82], the Poincaré Inequality [52], the Nash Embedding Theorem [60], the Myers-Steenrod Regularity Theorem [32], and,at least partially, the De-Giorgi Structure Theorem [37]. On the other hand, Carnot groups exhibit fractal behavior, the reason being that on such metric spaces the only curves of finite length are the horizontal ones. In fact, except for the Abelian groups, even if the distance is defined by a smooth subbundle, the squared distance is not smooth and the Hausdorff dimension of the space differs from the topological dimension. Moreover, these spaces contain no subset of positive measure that is biLipschitz equivalent to a subset of a Euclidean space and do not admit biLipschitz embedding into reflexive Banach spaces, nor into L1, see [4, 29–31, 93]. For this reason, nonAbelian Carnot groups, together with the classical fractals and boundaries of hyperbolic groups, are the main examples in an emerging field called ‘non-smooth analysis’, in addition see [14, 15, 17, 18, 24, 28, 42, 43, 47, 48, 53, 54, 59, 66, 73, 92]. A non-exhaustive, but long-enough list of other contribution in geometric measure theory on Carnot groups is [5–9, 12, 22, 33, 34, 47, 49, 56–58, 63, 67–69, 71, 76, 77, 79, 81, 86, 88, 89], and more can be found in [61, 91]. In this essay, we shall distinguish between Carnot groups, homogeneous groups, and stratified groups. In fact, the latter are Lie groups with a particular kind of grading, i.e., a stratification. Hence, they are only an algebraic object. Instead, homogeneous groups¹are Lie groups that are arbitrarily graded but are equipped with distances that are one-homogeneous with respect to the dilations induced by the grading. Finally, for the purpose of this paper, the term Carnot group will be reserved for those stratified groups that are equipped with homogeneous distances making them Carnot-Carathéodory spaces. Namely, they are equipped with a subFinsler distance. It is obvious that up to biLipschitz equivalence every Carnot group has one unique geometric structure. This is the reason why this term sometimes replaces the term stratified group. From the metric viewpoint, Carnot groups are peculiar examples of isometrically homogeneous spaces. In this context we shall differently use the term ‘homogeneous’: it means the presence of a transitive group action. In fact, on Carnot groups left-translations act transitively and by isometries. In some sense, these groups are quite prototypical examples of geodesic homogeneous spaces. An interesting result of Berestovskii gives that every isometrically homogeneous space whose distance is geodesic is indeed a Carnot-Carathéodory space, see Theorem 5.5. Hence, Carnot-Carathéodory spaces and Carnot groups are natural examples in metric geometry. This is the reason why they appear in different contexts. A purpose of this essay is to present the notion of a Carnot group from different viewpoints. As we said, Carnot groups have very rich metric and geometric structures. However, they are easily characterizable as geodesic spaces with self-similarity. Here is an equivalent definition, which is intermediate between the standard one (Section 3.3) and one of the simplest axiomatic ones (Theorem 6.2). A Carnot group is a Lie group G endowed with a left-invariant geodesic distance dadmitting for each λ>0a bijection δλ:G→Gsuch that d(δλ(p),δλ(q)) = λd(p,q),∀p,q∈G.(0.1) In this paper we will pursue an Aristotelian approach: we will begin by discussing the examples, starting from the very basic ones. We first consider Abelian Carnot groups, which are the finite-dimensional normed spaces, and then the basic non-Abelian group: the Heisenberg group. After presenting these examples, in Section 2 we will formally discuss the definitions from the algebraic viewpoint, with some basic properties. In particular, we show the uniqueness of stratifications in Section 2.2. In Section 3 we introduce the homogeneous distances and the general definition of Carnot-Carathéodory spaces. In Section 3.5 we show the continuity of homogeneous distances with respect to the manifold topology. Section 4 is devoted to present Carnot 1Originally, the term homogeneous group has been used by Folland and Stein for those simply connected Lie groups whose Lie algebra is endowed with Lie algebra automorphisms of the form exp (log λ)Aλ>0, where Ais diagonalizable and has positive eigenvalues, see [36, page 4]. The existence of one such a family is equivalent to the presence of a positive grading of the Lie algebra, the layers being the eigenspaces of A. Folland and Stein assume that, up to replacing Awith αA, the smallest of the eigenvalues of Ais 1. In fact, in this case such maps are dilations for some left-invariant distance, see [50]. Brought to you by | Jyväskylän Yliopisto University Authenticated Download Date | 1/30/18 12:38 PM 118 |Enrico Le Donne groups as limits. In particular, we consider tangents of Carnot-Carathéodory spaces. In Section 5 we discuss isometrically homogeneous spaces and explain why Carnot-Carathéodory spaces are the only geodesic examples. In Section 6 we provide a metric characterization of Carnot groups as the only spaces that are locally compact geodesic homogeneous and admit a dilation. Finally, in Section 7 we overview several results that provide the regularity of distance-preserving homeomorphisms in Carnot groups and more generally in homogeneous groups and Carnot-Carathéodory spaces. 1Prototypical examples We start by reviewing basic examples of Carnot groups. First we shall point out that Carnot groups of step one are in fact finite-dimensional normed vector spaces. Afterwards, we shall recall the simplest noncommutative example: the Heisenberg group, equipped with various distances. Example 1.1. Let Vbe a finite-dimensional vector space, so Vis isomorphic to Rn, for some n∈N. In such a space we have •a group operation (sum) p,q7→ p+q, •dilations p7→ λp for each factor λ>0. We are interested in the distances don Vthat are (i) translation invariant: d(p+q,p+q0) = d(q,q0),∀p,q,q0∈V (ii) one-homogeneous with respect to the dilations: d(λp,λq) = λd(p,q),∀p,q∈V,∀λ>0. These are the distances coming from norms on V. Indeed, setting kvk=d(0,v)for v∈V, the axioms of dbeing a distance together with properties (i) and (ii), give that k·kis a norm. A geometric remark: for such distances, straight segments are geodesic. By definition, a geodesic is an isometric embedding of an interval. Moreover, if the norm is strictly convex, then straight segments are the only geodesic. An algebraic remark: each (finite-dimensional) vector space can be seen as a Lie algebra (in fact, a commutative Lie algebra) with Lie product [p,q]=0, for all p,q∈V. Via the obvious identification points/vectors, this Lie algebra is identified with its Lie group (V,+). One of the theorem that we will generalize in Section 7 is the following. Theorem 1.2. Every isometry of a normed space fixing the origin is a linear map. An analytic remark: the definition of (directional) derivatives ∂f ∂y (x) := lim h→0 f(x+hy)−f(x) h of a function fbetween vector spaces makes use of the group operation, the dilations, and the topology. We shall consider more general spaces on which these operations are defined. Example 1.3. Consider R3with the standard topological and differentiable structure. Consider the operation (¯ x,¯ y,¯ z)·(x,y,z) := ¯ x+x,¯ y+y,¯ z+z+1 2(¯ xy −¯ yx). This operation gives a non-Abelian group structure on R3. This group is called Heisenberg group. Brought to you by | Jyväskylän Yliopisto University Authenticated Download Date | 1/30/18 12:38 PM A Primer on Carnot Groups |119 The maps δλ(x,y,z) = (λx,λy,λ2z)give a one-parameter family of group homomorphisms: δλ(pq) = δλ(p)δλ(q)and δλ◦δµ=δλµ. We shall consider distances that are (i) left (translation) invariant: d(p·q,p·q0) = d(q,q0),∀p,q,q0∈G (ii) one-homogeneous with respect to the dilations: d(δλ(p),δλ(q)) = λd(p,q),∀p,q∈G,∀λ>0. These distances, called ‘homogeneous’, are completely characterized by the distance from a point, in fact, just by the unit sphere at a point. Example 1.3.1. (Box distance) Set dbox(0,p) := kpk:= max ¶|xp|,|yp|,p|zp|©. This function is δλ-homogeneous and satisfies the triangle inequality. To check that it satisfies the triangle inequality we need to show that kp·qk≤kpk+kqk.First, |xp·q|=|xp+xq|≤|xp|+|xq|≤kpk+kqk, and analogously for the ycomponent. Second, p|zp·q|=    zp+zq+1 2(xpyq−xqyp)    ≤p|zp|+|zq|+|xp||yq|+|xq||yp| ≤»kpk2+kqk2+ 2 kpkkqk=kpk+kqk. The group structure induces left-invariant vector fields, a basis of which is X=∂1−y 2∂3,Y=∂2+x 2∂3,Z=∂3. Example 1.3.2. (Carnot-Carathéodory distances) Fix a norm k·kon R2, e.g.,  (a,b) =√a2+b2. Consider d(p,q) := inf ®ˆ1 0 (a(t),b(t)) dt´, where the infimum is over the piecewise smooth curves γ∈C∞ pw([0,1]; R3)with γ(0) = p,γ(1) = q,and ˙γ(t) = a(t)Xγ(t)+b(t)Yγ(t).Such a dis a homogenous distance and for all p,q∈R3there exists a curve γ realizing the infimum. These distances are examples of CC distances also known as subFinsler distances. Example 1.3.3. (Korányi distance) Another important example is given by the Cygan-Korányi distance: dK(0,p) := kpkK:= Ä(x2 p+y2 p)2+ 16 z2 pä1/4. The feature of such a distance is that it admits a conformal inversion, see [26, p.27]. The Korányi distance and the box distance are not CC distances. The space R3with the above group structure is an example of Lie group, i.e., the group multiplication and the group inversion are smooth maps. The space gof left-invariant vector fields (also known as the Lie algebra) has a peculiar structure: it admits a stratification. Namely, setting V1:= span {X,Y}and V2:= span{Z}, we have g=V1⊕V2,[V1,V1] = V2,[V1,V2] = {0}. As an exercise, verify that Z= [X,Y]. Brought to you by | Jyväskylän Yliopisto University Authenticated Download Date | 1/30/18 12:38 PM 120 |Enrico Le Donne 2Stratifications In this section we discuss Lie groups, Lie algebras, and their stratifications. We recall the definitions and we point out a few remarks. In particular, we show that a group can be stratified in a unique way, up to isomorphism. 2.1 Definitions Given a group Gwe denote by gh or g·hthe product of two elements g,h∈Gand by g−1the inverse of g. A Lie groupis a differentiable manifold endowed with a group structure such that the map G×G→G,(g,h)7→ g−1·h is C∞. We shall denote by ethe identity of the group and by Lg(h) := g·hthe left translation. Any vector X in the tangent space at the identity extends uniquely to a left-invariant vector field ˜ X, as ˜ Xg= (dLg)eX, for g∈G. The Lie algebra associated with a Lie group Gis the vector space TeGequipped with the bilinear operation defined by [X,Y] := [˜ X,˜ Y]e, where the last bracket denotes the Lie bracket of vector fields, i.e., [˜ X,˜ Y] := ˜ X˜ Y−˜ Y˜ X. The general notion of Lie algebra is the following: A Lie algebra gover Ris a real vector space together with a bilinear operation [·,·] : g×g→g, called the Lie bracket, such that, for all x,y,z∈g, one has 1. anti-commutativity: [x,y] = −[y,x], 2. Jacobi identity: [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0. All Lie algebras considered here are over Rand finite-dimensional. In what follows, given two subspaces V,Wof a Lie algebra, we set [V,W] := span{[X,Y]; X∈V,Y∈W}. Definition 2.1 (Stratifiable Lie algebras).Astratification of a Lie algebra gis a direct-sum decomposition g=V1⊕V2⊕· · · ⊕Vs for some integer s≥1, where Vs={0}and [V1,Vj] = Vj+1 for all integers j∈ {1,. . . ,s}and where we set Vs+1 ={0}. We say that a Lie algebra is stratifiable if there exists a stratification of it. We say that a Lie algebra is stratified when it is stratifiable and endowed with a fixed stratification called the associated stratification. A stratification is a particular example of grading. Hence, for completeness, we proceed by recalling the latter. More considerations on this subject can be found in [68]. Definition 2.2 (Positively graduable Lie algebras).Apositive grading of a Lie algebra gis a family (Vt)t∈(0,+∞) of linear subspaces of g, where all but finitely many of the Vt’s are {0}, such that gis their direct sum g=M t∈(0,+∞) Vt and where [Vt,Vu]⊂Vt+u,for all t,u>0. We say that a Lie algebra is positively graduable if there exists a positive grading of it. We say that a Lie algebra is graded (or positively graded, to be more precise) when it is positively graduable and endowed with a fixed positive grading called the associated positive grading. Given a positive grading g=⊕t>0Vt, the subspace Vtis called the layer of degree t(or degree-t layer) of the positive grading and non-zero elements in Vtare said to have degree t. The degree of the grading is the maximum of all positive real numbers t>0such that Vt={0}. Brought to you by | Jyväskylän Yliopisto University Authenticated Download Date | 1/30/18 12:38 PM A Primer on Carnot Groups |121 Given two graded Lie algebras gand hwith associated gradings g=⊕t>0Vtand h=⊕t>0Wt, a morphism of the graded Lie algebras is a Lie algebra homomorphism ϕ:g→hsuch that ϕ(Vt)⊆Wtfor all t>0. Hence, two graded Lie algebras gand hare isomorphic as graded Lie algebras if there exists a bijection ϕ:g→h such that both ϕand ϕ−1are morphisms of the graded Lie algebras. Recall that for a Lie algebra gthe terms of the lower central series are defined inductively by g(1) =g, g(k+1) = [g,g(k)]. A Lie algebra gis called nilpotent if g(s+1) ={0}for some integer s≥1and more precisely we say that gnilpotent of step sif g(s+1) ={0}but g(s)={0}. Remark 2.3. A positively graduable Lie algebra is nilpotent (simple exercise similar to Lemma 2.16). On the other hand, not every nilpotent Lie algebra is positively graduable, see Example 2.8. A stratification of a Lie algebra gis equivalent to a positive grading whose degree-one layer generates gas a Lie algebra. Therefore, given a stratification, the degree-one layer uniquely determines the stratification and satisfies g=V1⊕[g,g].(2.4) However, an arbitrary vector space V1that is in direct sum with [g,g](i.e., satisfying (2.4)) may not generate a stratification, see Example 2.9. Any two stratifications of a Lie algebra are isomorphic, see Section 2.2. A stratifiable Lie algebra with snon-trivial layers is nilpotent of step s. Every 2-step nilpotent Lie algebra is stratifiable. However, not all graduable Lie algebras are stratifiable, see Example 2.7. Stratifiable and graduable Lie algebras admit several different grading: given a grading ⊕Vt, one can define the so-called s-power as the new grading ⊕Wtby setting Wt=Vt/s, where s>0. Moreover, for a stratifiable algebra it is not true that any grading is a power of a stratification, see Example 2.6. Example 2.5. Free-nilpotent Lie algebras are stratifiable. Namely, having fixed r,s∈N, one considers formal elements e1,. . . ,erand all possible formal iterated Lie bracket up to length s, modulo the anti-commutativity and Jacobi relations, e.g., [e1,e2]equals −[e2,e1]and both have length 2. The span of such vectors is the freenilpotent Lie algebra of rank rand step s. The strata of a stratification of such an algebra are formed according to the length of formal bracket considered. Any nilpotent Lie algebra is a quotient of a free-nilpotent Lie algebra, however, the stratification may not pass to this quotient. Example 2.6. The Heisenberg Lie algebra his the 3-dimensional lie algebra spanned by three vectors X,Y,Z and with only non-trivial relation Z= [X,Y], cf Example 1.3. This stratifiable algebra also admits positive gradings that are not power of stratifications. In fact, for α∈(1,+∞), the non-standard grading of exponent αis h=W1⊕Wα⊕Wα+1 where W1:= span{X},Wα:= span{Y},Wα+1 := span{Z}. Up to isomorphisms of graded Lie algebras and up to powers, these non-standard gradings give all the possible positive gradings of hthat are not a stratification. Example 2.7. Consider the 7-dimensional Lie algebra ggenerated by X1,. . . ,X7with only non-trivial brackets [X1,X2] = X3,[X1,X3] = 2X4,[X1,X4] = 3X5 [X2,X3] = X5,[X1,X5] = 4X6,[X2,X4] = 2X6 [X1,X6] = 5X7,[X2,X5] = 3X7,[X3,X4] = X7. This Lie algebra gadmits a grading but it is not stratifiable. Example 2.8. There exist nilpotent Lie algebras that admit no positive grading. For example, consider the Lie algebra of dimension 7 with basis X1,. . . ,X7with only non-trivial relations given by [X1,Xj] = Xj+1 if 2≤j≤6,[X2,X3] = X6,[X2,X4] = −[X2,X5] = [X3,X4] = X7. Brought to you by | Jyväskylän Yliopisto University Authenticated Download Date | 1/30/18 12:38 PM 122 |Enrico Le Donne Example 2.9. We give here an example of a stratifiable Lie algebra gfor which one can find a subspace Vin direct sum with [g,g]but that does not generate a stratification. We consider gthe stratifiable Lie algebra of step 3 generated by e1,e2and e3and with the relation [e2,e3] = 0. Then dim g= 10 and a stratification of g is generated by V1:= span{e1,e2,e3}. Taking V:= span{e1,e2+[e1,e2],e3}, one has (2.4), but since [V,V] and [V,[V,V]] both contain [e2,[e1,e3]],Vdoes not generate a stratification of g. Definition 2.10 (Positively graduable, graded, stratifiable, stratified groups).We say that a Lie group Gis a positively graduable (respectively graded,stratifiable,stratified)group if Gis a connected and simply connected Lie group whose Lie algebra is positively graduable (respectively graded, stratifiable, stratified). For the sake of completeness, in Theorem 2.14 below we present an equivalent definition of positively graduable groups in terms of existence of a contractive group automorphism. The result is due to Siebert [95]. Definition 2.11 (Dilations on graded Lie algebras).Let gbe a graded Lie algebra with associated positive grading g=⊕t>0Vt. For λ>0, we define the dilation on g(relative to the associated positive grading) of factor λas the unique linear map δλ:g→gsuch that δλ(X) = λtX∀X∈Vt. Dilations δλ:g→gare Lie algebra isomorphisms, i.e., δλ([X,Y]) = [δλX,δλY]for all X,Y∈g. The family of all dilations (δλ)λ>0is a one-parameter group of Lie algebra isomorphisms, i.e., δλ◦δη=δλη for all λ,η>0. Exercise 2.12. Let gand hbe graded Lie algebras with associated dilations δg λand δh λ. Let ϕ:g→hbe a Lie algebra homomorphism. Then ϕis a morphism of graded Lie algebras if and only if ϕ◦δg λ=δh λ◦ϕ,for all λ>0. [Solution: Let g=⊕t>0Vtbe the grading of g. If x∈Vtthen ϕ(δλx) = ϕ(λtx) = λtϕ(x), which gives the equivalence.] Given a Lie group homomorphism ϕ:G→H, we denote by ϕ*:g→hthe associated Lie algebra homomorphism. If Gis simply connected, given a Lie algebra homomorphism ψ:g→h, there exists a unique Lie group homomorphism ϕ:G→Hsuch that ϕ*=ψ(see [100, Theorem 3.27]). This allows us to define dilations on Gas stated in the following definition. Definition 2.13 (Dilations on graded groups).Let Gbe a graded group with Lie algebra g. Let δλ:g→gbe the dilation on g(relative to the associated positive grading of g) of factor λ>0. The dilation on G(relative to the associated positive grading) of factor λis the unique Lie group automorphism, also denoted by δλ:G→G, such that (δλ)*=δλ. For technical simplicity, one keeps the same notation for both dilations on the Lie algebra gand the group G. There will be no ambiguity here. Indeed, graded groups being nilpotent and simply connected, the exponential map exp : g→Gis a diffeomorphism from gto G(see [27, Theorem 1.2.1] or [36, Proposition 1.2]) and one has δλ◦exp = exp ◦δλ(see [100, Theorem 3.27]), hence dilations on gand dilations on Gcoincide in exponential coordinates. For the sake of completeness, we give now an equivalent characterization of graded groups due to Siebert [95]. If Gis a topological group and τ:G→Gis a continuous group isomorphism, we say that τis contractive if, for all g∈G, one has limk→∞τk(g) = e. We say that Gis contractible if Gadmits a contractive isomorphism. For graded groups, dilations of factor λ<1are contractive isomorphisms, hence positively graduable groups are contractible. Conversely, Siebert proved (see Theorem 2.14 below) that if Gis a connected locally compact group and τ:G→Gis a contractive isomorphism then Gis a connected and simply connected Lie group and τinduces a positive grading on the Lie algebra gof G(note however that τitself may not be a dilation relative to the induced grading). Theorem 2.14. [95, Corollary 2.4] A topological group Gis a positively graduable Lie group if and only if Gis a connected locally compact contractible group. Brought to you by | Jyväskylän Yliopisto University Authenticated Download Date | 1/30/18 12:38 PM A Primer on Carnot Groups |123 Sketch of the proof. Regarding the non-trivial direction, by the general theory of locally compact groups one has that Gis a Lie group. Hence, the contractive group isomorphism induces a contractive Lie algebra isomorphism ϕ. Passing to the complexified Lie algebra, one considers the Jordan form of ϕwith generalized eigenspaces Vα,α∈C. The t-layer Vtof the grading is then defined as the real part of the span of those Vα with −log |α|=t. Remark 2.15. A distinguished class of groups in Riemannian geometry are the so-called Heintze groups. They are those groups that admit a structure of negative curvature. It is possible to show that such groups are precisely the direct product of a graded group Ntimes Rwhere Racts on Nvia the grading, see [Hei74]. 2.2 Uniqueness of stratifications We start by observing the following simple fact. Lemma 2.16. If g=V1⊕· · · ⊕Vsis a stratified Lie algebra, then g(k)=Vk⊕· · · ⊕Vs. In particular, gis nilpotent of step s. Now we show that the stratification of a stratifiable Lie algebra is unique up to isomorphism. Hence, also the structure of a stratified group is essentially unique. Proposition 2.17. Let gbe a stratifiable Lie algebra with two stratifications, V1⊕· · · ⊕Vs=g=W1⊕· · · ⊕Wt. Then s=tand there is a Lie algebra automorphism A:g→gsuch that A(Vi) = Wifor all i. Proof. We have g(k)=Vk⊕· · ·⊕Vs=Wk⊕· · ·⊕Wt. Then s=t. Moreover, the quotient mappings πk:g(k)→ g(k)/g(k+1) induce linear isomorphisms πk|Vk:Vk→g(k)/g(k+1) and πk|Wk:Wk→g(k)/g(k+1). For v∈Vk define A(v) := (πk|Wk)−1◦πk|Vk(v). Explicitly, for v∈Vkand w∈Wkwe have A(v) = w⇐⇒ v−w∈g(k+1). Extend Ato a linear map A:g→g. This is clearly a linear isomorphism and A(Vi) = Wifor all i. We need to show that Ais a Lie algebra morphism, i.e., [Aa,Ab] = A([a,b]) for all a,b∈g. Let a=Ps i=1 aiand b=Ps i=1 biwith ai,bi∈Vi. Then A([a,b]) = s X i=1 s X j=1 A([ai,bj]) [Aa,Ab] = s X i=1 s X j=1 [Aai,Abj], therefore we can just prove A([ai,bj]) = [Aai,Abj]for ai∈Viand bj∈Vj. Notice that [ai,bj]belongs to Vi+j and [Aai,Abj]belongs to Wi+j. Therefore we have A([ai,bj]) = [Aai,Abj]if and only if [ai,bj]−[Aai,Abj]∈ g(i+j+1). On the one hand, we have ai−Aai∈g(i+1) and bj∈Vj, so [ai−Aai,bj]∈g(i+j+1). On the other hand, we have Aai∈Wiand Abj−bj∈g(j+1), so [Aai,Abj−bj]∈g(i+j+1). Hence, we have [ai,bj]−[Aai,Abj] = [ai−Aai,bj]−[Aai,Abj−bj]∈g(i+j+1). Brought to you by | Jyväskylän Yliopisto University Authenticated Download Date | 1/30/18 12:38 PM 130 |Enrico Le Donne Examples of intrinsic distances are given by length structures, subRiemannian structures, Carnot-Carathéodory spaces, and geodesic spaces. A metric space whose distance is intrinsic is called geodesic if the infimum in Definition 5.4 is always attained. The significance of the next result is that CC-spaces are natural objects in the theory of homogeneous metric spaces. Theorem 5.5 (Berestovskii, [13]).If a homogeneous Lie space G/His equipped with an admissible G-invariant intrinsic distance d, then dis subFinsler, i.e., there exist a G-invariant bracket-generating subbundle ∆and a G-invariant norm k·ksuch that dis the CC distance associated to ∆and k·k. How to prove Berestovskii’s result. In three steps: Step 1. Show that locally dis ≥to some Riemannian distance dR. Step 2. Deduce that curves that have finite length with respect to dare rectifiable (in any coordinate system) and define the horizontal bundle ∆using velocities of such curves. Similarly, define k·kusing tangents of finite-length curves. Step 3. Conclude that dis the CC distance for ∆and k·k. A posteriori we know that ∆is bracket-generating, since dis finite-valued. The core of the argument is in Step 1. Hence we describe its proof in an exemplary case. Simplified proof of Step 1. We only consider the following simplification: G=G/H=R2, i.e., dis a translation invariant distance on the plane. Let dEbe the Euclidean distance. We want to show that dE/dis locally bounded. If not, there exists pn→0such that dE(pn,0) d(pn,0) >n. Since the topologies are the same, there exists r>0such that the closure of Bd(0,r)is contained in BE(0,1). Since pn→0, there is hn∈Nsuch that eventually hnpn∈BE(0,1) \Bd(0,r). Hence, 0<r<d(hnpn,0) ≤hnd(pn,0) ≤hn ndE(pn,0) =1 ndE(hnpn,0) ≤1 n→0, which gives a contradiction. 6A metric characterization of Carnot groups 6.1 Characterizations of Lie groups Providing characterization of Lie groups among topological groups was one of the Hilbert problems: the 5th one. Nowadays, it is considered solved in various forms by the work of John von Neumann, Lev Pontryagin, Andrew Gleason, Deane Montgomery, Leo Zippin, and Hidehiko Yamabe. See [78] and references therein. For the purpose of characterizing Carnot groups among homogeneous metric spaces, we shall make use of the following. Theorem 6.1 (Gleason - Montgomery - Zippin. 1950’s).Let Xbe a metric space that is connected, locally connected, locally compact, of finite topological dimension, and isometrically homogeneous. Then its isometry group Iso(X)is a Lie group. As a consequence, the isometrically homogeneous metric spaces considered in Theorem 6.1 are all of the form discussed in Section 5 after Definition 5.1. How to prove Theorem 6.1. In many steps: Step 1. Iso(X)is locally compact, by Ascoli-Arzelá Theorem. Brought to you by | Jyväskylän Yliopisto University Authenticated Download Date | 1/30/18 12:38 PM A Primer on Carnot Groups |131 Step 2. There exists an open (and closed) subgroup Gof Iso(X)that can be approximated by Lie groups: G= lim ←−Gi,(inverse limit of continuous epimorphisms with compact kernel). This step is called Main Approximation Theorem in the structure of locally compact groups. It is due mainly to Gleason and Yamabe and it is based on Peter-Weyl Theorem, see [103]. Step 3. GyXtransitively, by Baire Category Theorem; so X=G/H= lim ←−Gi/Hi. Step 4. Since the topological dimension of Xis finite and Xis locally connected, for ilarge Gi/Hi→G/His a homeomorphism. Step 5. G=Gifor ilarge, so Gis a Lie group so Gis NSS, i.e., Ghas no small subgroups. Step 6. the fact that Gis NSS implies that Iso(X)is NSS. Step 7. Locally compact groups with NSS are Lie groups, by Gleason Theorem [39]. 6.2 A metric characterization of Carnot groups In what follows, we say that a metric space (X,d)is self-similar if there exists λ>1such that the metric space (X,d)is isometric to the metric space (X,λd). In other words, there exists a homeomorphism f:X→Xsuch that d(f(p),f(q)) = λd(p,q),for all p,q∈X. When this happens for all λ>0(and all maps f=fλfix a common point) Xis said to be a cone. Homogeneous groups are examples of cones. The following result is a corollary of the work of Gleason-Montgomery-Zippin, Berestovskii, and Mitchell, see [62]. It gives a metric characterization of Carnot groups. Theorem 6.2. SubFinsler Carnot groups are the only metric spaces that are 1. locally compact, 2. geodesic, 3. isometrically homogeneous, and 4. self-similar. Sketch of the proof. Each such metric space Xis connected and locally connected. Using the conditions of local compactness, self-similarity, and homogeneity, one can show that Xis a doubling metric space. In particular, Xis finite dimensional. By the result of Gleason-Montgomery-Zippin (Theorem 6.1) the space Xhas the structure of a homogeneous Lie space G/Hand, by Berestovskii’s result (Theorem 5.5), as a metric space Xis an equiregular subFinsler manifold. By Mitchell’s result (Theorem 4.6), the tangents of Xare subFinsler Carnot groups. Since Xis self-similar, Xis isometric to its tangents. 7Isometries of metric groups 7.1 Regularity of isometries for homogeneous spaces Let Xbe a metric space that is connected, locally connected, locally compact, with finite topological dimension, and isometrically homogeneous. By Montgomery-Zippin, G:= Iso(X)has the structure of (analytic) Lie group, which is unique. Fixing x0∈X, the stabilizing subgroup H:= StabG(x0)is compact. Therefore, G/H has an induced structure of analytic manifolds. We have that Xis homeomorphic to G/Hand Gacts on G/H analytically. One may wonder if Xcould have had a different differentiable structure. The next result says that this cannot happen. Brought to you by | Jyväskylän Yliopisto University Authenticated Download Date | 1/30/18 12:38 PM 132 |Enrico Le Donne Theorem 7.1 (LD, Ottazzi, [65]).Let M=G/Hbe a homogeneous manifold equipped with an admissible G- invariant distance d. Then the isometry group Iso(M)is a Lie group, the action Iso(M)×M→M(7.2) (F,p)7→ F(p) is analytic, and for every p∈Mthe space Isop(M)is a compact Lie group. The above result is just a consequence of Theorem 6.1 and the uniqueness of analytic structures for homogeneous Lie spaces, see [65, Proposition 4.5]. 7.2 Isometries of Carnot groups We now explain in further details what are the isometries of Carnot groups and some of their generalizations. We summarize our knowledge with the following results. •[78] implies that the global isometries are smooth, since Carnot groups are homogeneous spaces. •[25] implies that isometries between open subsets of Carnot groups are smooth, since every 1-quasi- conformal map is. •[32] implies that isometries between equiregular subRiemannian manifolds are smooth. •[65] implies that isometries between open subsets of subFinsler Carnot groups are affine, i.e., composition of translations and group homomorphisms. •[55] implies that isometries of nilpotent connected metric Lie groups are affine. In the rest of this exposition we give more explanation on the last three points. 7.3 Local isometries of Carnot groups Theorem 7.3 (LD, Ottazzi, [65]).Let G1,G2be subFinsler Carnot groups and for i= 1,2consider Ωi⊂Gi open sets. If F:Ω1→Ω2is an isometry, then there exists a left translation τon G2and a group isomorphism ϕbetween G1and G2, such that Fis the restriction to Ω1of τ◦ϕ, which is a global isometry. Sketch of the proof. In four steps: Step 1. We may assume that the distance is subRiemannian, regularizing the subFinsler norm. Step 2. Fis smooth; this is a PDE argument using the regularity of the subLaplacian, see the next section. Alternatively, one can use [25]. Step 3. Fis completely determined by its horizontal differential (dF)HeG, see the following Corollary 7.6. Step 4. The Pansu differential (PF)eexists and is an isometry with the same horizontal differential as Fat e, see [85]. We remark that in Theorem 7.3 the assumption that Ωiare open is necessary, unlike in the Euclidean case. However, these open sets are not required to be connected. 7.4 Isometries of subRiemannian manifolds The regularity of subRiemannian isometries should be thought of as a two-step argument, where as an intermediate result one obtains the preservation of a good measure: the Popp measure. A good introduction to the notion of Popp measure can be found in [19]. Both of the next results are due to the author in collaboration with Capogna, see [32]. Theorem 7.4 (Capogna, LD).Let F:M→Nbe an isometry between two subRiemannian manifolds. If there exist two C∞volume forms volMand volNsuch that F*volM= volN, then Fis a C∞diffeomorphism. Brought to you by | Jyväskylän Yliopisto University Authenticated Download Date | 1/30/18 12:38 PM A Primer on Carnot Groups |133 Theorem 7.5 (Capogna, LD).Let F:M→Nbe an isometry between equiregular subRiemannian manifolds. If volMand volNare the Popp measures on Mand N, respectively, then F*volM= volN. How to prove Theorem 7.5. In two steps: Step 1. Carnot group case: Popp is a Haar measure and hence a fixed multiple of the Hausdorff measure. The latter is a metric invariant. Step 2. There is a representation formula ([2, pages 358-359], [38, Section 3.2]]) for the Popp volume volMin terms of the Hausdorff measure SQ Mand the tangent measures Np(volM)at pin the tangent metric spaces Np(M), which are Carnot groups: d volM= 2−QNp(volM)(BNp(M)(e,1))dSQ M, where Qis the Hausdorff dimension. One then makes use of Step 1. How to prove Theorem 7.4. In many steps: Step 1. Fis an isomorphism of metric measure spaces. Thus, the map Fpreserves minimal upper gradients, Dirichlet energy, harmonic maps, and subLaplacian. We recall the horizontal gradient: If X1,. . . ,Xm is an orthonormal frame for the horizontal bundle, define ∇Hu:= (X1u)X1+. . . + (Xmu)Xm. From the horizontal gradient one defines the subLaplacian: ∆Hu=gmeans ˆM gv d volM=ˆMh∇Hu,∇Hvid volM,∀v∈Lipc(M). We remark that ∆H(·)depends on the choice of the measure volM. Step 2. Hajlasz and Koskela’s result, [49, page 51 and Section 11.2]: k∇Hukcoincides almost everywhere with the minimal upper gradient of u. Consequently, since F*volM= volN, ∆Hu=g=⇒∆H(u◦F) = g◦F, where the first subLaplacian is with respect to volNand the second one with respect to volM. Step 3. Rothschild and Stein’s version of Hörmander’s Hypoelliptic Theorem, [90, Theorem 18]: Let X0,X1, ..., Xr bracket generating vector fields in Rn. Let ube a distributional solution to the equation (X0+ Pr i=1 X2 i)u=gand let k∈N∪ {0}and 1<p<∞. Then, considering the horizontal local Sobolev spaces Wk,p H,loc, g∈Wk,p H(Rn,Ln) =⇒u∈Wk+2,p H,loc (Rn,Ln). Corollary: If volMis a C∞volume form, ∆Hu∈Wk,p H,loc(M,volM) =⇒u∈Wk+1,p H,loc (M,volM). Step 4. Bootstrap argument: Fisometry implies that F∈W1,p H. Taking xjsome C∞coordinate system, we get that ∆Hxi=: gi∈C∞⊂Wk,p H,loc, for all kand p. Then, by Step 3 ∆H(xi◦F) = gi◦F∈W1,p H=⇒xi◦F∈W2,p H. Then we iterate Rothschild and Stein’s regularity of Step 3: gi◦F∈W2,p H,loc =⇒xi◦F∈W3,p H,loc and by induction we complete. Brought to you by | Jyväskylän Yliopisto University Authenticated Download Date | 1/30/18 12:38 PM 134 |Enrico Le Donne 7.5 SubRiemannian isometries are determined by the horizontal differential Corollary 7.6. Let Mand Nbe two connected equiregular subRiemannian manifolds. Let p∈Mand let ∆be the horizontal bundle of M. Let f,g:M→Nbe two isometries. If f(p) = g(p)and df|∆p=dg|∆p, then f=g. The proof can be read in [65, Proposition 2.8]. Once we know that the isometries fixing a point are a compact Lie group of smooth transformations, the argument is an easy exercise in differential geometry. 7.6 Isometries of nilpotent groups The fact that Carnot isometries are affine (Theorem 7.3) is a general feature of the fact that we are dealing with a nilpotent group. In fact, isometries are affine whenever they are globally defined on a nilpotent connected group. Here we obviously require that the distances are left-invariant and induce the manifold topology. For example, this is the case for arbitrary homogeneous groups. Theorem 7.7 (LD, Kivioja).Let N1and N2be two nilpotent connected metric Lie groups. Any isometry F:N1→ N2is affine. This result is proved in [55] with algebraic techniques by studying the nilradical, i.e., the biggest nilpotent ideal, of the group of self-isometries of a nilpotent connected metric Lie group. The proof leads back to a Riemannian result of Wolf, see [101]. 7.7 Two isometric non-isomorphic groups In general isometries of a subFinsler Lie group Gmay not be affine, not even in the Riemannian setting. As counterexample, we take the universal covering group ˜ Gof the group G=E(2) of Euclidean motions of the plane. This group is also called roto-translation group. One can see that there exists a Riemannian distance on ˜ Gthat makes it isometric to the Euclidean space R3. In particular, they have the same isometry group. However, a straightforward calculation of the automorphisms shows that not all isometries fixing the identity are group isomorphisms of ˜ G. As a side note, we remark that the group ˜ Gadmits a left-invariant subRiemannian structure and a map into the subRiemannian Heisenberg group that is locally biLipschitz. However, these two spaces are not quasi-conformal, see [35]. Examples of isometric Lie groups that are not isomorphic can be found also in the strict subRiemannian context. There are three-dimensional examples, see [1]. Also the analogue of the roto-translation construction can be developed. Acknowledgement: The author would like to thank E. Breuillard, S. Nicolussi Golo, A. Ottazzi, A. Prantl, S. Rigot for help in preparing this article. This primer was written after the preparation of a mini course held at the Ninth School on ‘Analysis and Geometry in Metric Spaces’ in Levico Terme in July 2015. 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