On the Kirchheim-Magnani counterexample to metric differentiability

On the Kirc hheim-Magnani coun terexample to metri c differen tiabilit y Marius Buliga Institute of Mathematics, Romanian Academ y P .O. BO X 1-76 4, R O 014700 Bucure ¸ sti, Romania Marius.Buliga @imar.ro This v ersion: 04.10.2 0 07 In Kirc hh eim-Magnani [7 ] the authors construct a left inv a riant distance ρ on the Heisen b erg group such that the identit y map id is 1-Lipsc h itz but it is n ot metrically differen tiable an ywhere. In this short note w e giv e an in terpretation of the Kirc h heim-Magnani coun terex- ample to metric differen tiabilit y . In fact w e show that they construct something whic h fails shortly from b eing a dilatation stru ctur e. Dilatat ion structures hav e b een introdu ced in [2]. These str uctures are related to conical group [3], wh ic h form a particular class of con tractible groups and are a sligh t generalizat ion of C arnot groups. Carnot groups, in particular the Heisen b erg group, app ear a s infi nitesimal models of sub-r iemann ian manifolds [1], [6]. In [5] w e explain how the formalism of dilatatio n structures applies to sub-riemann ian geometry . F urther on we shall use th e notations, definitions and results concerning dilata- tion structur es, as found in [2], [3] or [5]. W e sh all construct a structure ( H (1) , ρ, ¯ δ ) on H (1) w hic h satisfies all axioms of a dilatation structur e, excepting A3 and A4. W e p ro ve that for ( H (1) , ρ, ¯ δ ) th e axiom A4 implies A3. Finally w e pr o v e that A4 for ( H (1) , ρ, ¯ δ ) is equ iv ale nt with id metrically different iable from ( H (1) , d ) to ( H (1) , ρ ), where d is a left inv ariant CC distance. F or other r elations b et w een dilatation structur es and differen tiabilit y in metric spaces see [4]. 1 Metric differen tiabilit y for conical groups The general definition of metric differen tiabilit y for conical groups is form ulated exactly as the same notion for Carn ot groups. 1 Definition 1.1 L et ( N , d, δ ) b e a c onic al gr oup. A c ont inuous f u nction η : N → [0 , + ∞ ) i s a se minorm if: (a) η ( δ ε x ) = εη ( x ) for any x ∈ N and ε > 0 , (b) η ( xy ) ≤ η ( x ) + η ( y ) for any x, y ∈ N . L et ( N , δ , d ) b e a c onic al gr o up, ( X, ρ ) a metric sp ac e, A ⊂ N an op en set and x ∈ A . A function f : A → X i s metric al ly differ entiable in x if ther e is a seminorm η x : N → [0 , + ∞ ) such that | 1 ε ρ ( f ( xδ ε v ) , f ( x )) − η x ( v ) |→ 0 as ε → 0 , uniformly with r esp e ct to v in c omp act neighb ourho o d of the neutr al element e ∈ N . 2 Kirc h heim-Magnani coun terexample to metric diffe r- en tiabilit y F or the elemen ts of the Heisen b erg group H (1) = R 2 × R w e use the n otation ˜ x = ( x, ¯ x ), with ˜ x ∈ H (1) , x ∈ R 2 , ¯ x ∈ R . In this subsection we s hall u se th e follo wing op eration on H (1): ˜ x ˜ y = ( x, ¯ x )( y , ¯ y ) = ( x + y , ¯ x + ¯ y + 2 ω ( x, y )) , where ω is the c anonical s ymplectic form on R 2 . On H (1) w e consider the left in v arian t distance d uniqu ely determined by th e formula: d ((0 , 0) , ( x, ¯ x )) = max n k x k , p | ¯ x | o . The constr u ction by Kirchheim and Magnani is describ ed fur ther. T ak e an in - v ertible, non d ecreasing function g : [0 , + ∞ ) → [0 , + ∞ ), con tinuous at 0, suc h that g (0) = 0. F or a f unction g wh ic h is w ell chose n , the fu nction ρ : H (1) → [0 , + ∞ ), ρ ( ˜ x ) = max {k x k , g ( | ¯ x | ) } induces a left in v arian t inv arian t distance on H (1) (we use the same symb ol) ρ ( ˜ x, ˜ y ) = ρ ( ˜ x − 1 ˜ y ) . In order to c hec k this it is su ffi cien t to p ro v e that for any ˜ x, ˜ y ∈ H (1) we ha v e ρ ( ˜ x ˜ y ) ≤ ρ ( ˜ x ) + ρ ( ˜ y ) , and that ρ ( ˜ x ) = 0 if and only if ˜ x = (0 , 0). T h e follo wing result is theorem 2.1 [7]. 2 Theorem 2.1 (K i r chheim-Magnani) If the function g has the expr ession g − 1 ( t ) = k ( t ) + t 2 for any t > 0 , wher e k : [0 , + ∞ ) → [0 , + ∞ ) is a c onvex function, strictly incr e asing, c ontinuous at 0 , and such that k (0) = 0 , then the function ρ induc es a left invariant distanc e (denote d also by ρ ). Mor e over, the identity function id is 1-Li pschitz fr om ( H (1) , d ) to ( H (1) , ρ ) . 3 In terpretation in terms of dilatatio n structures F urther we shall w ork with a fu n ction g satisfying the hyp othesis of theorem 2.1, and with the asso ciated function ρ d escrib ed in the previous subsection. Definition 3.1 D efine for any ε > 0 , the function ¯ δ ε ( x, ¯ x ) = ( εx, sg n ( ¯ x ) g − 1 ( εg ( | ¯ x | ))) for any ˜ x = ( x, ¯ x ) ∈ H (1) . We define the fol lowing field of dilatations ¯ δ by: for any ε > 0 and ˜ x, ˜ y ∈ H (1) let ¯ δ ˜ x ˜ y = ˜ x ¯ δ  ˜ x − 1 ˜ y  . F or any ε > 0 and ˜ x, ˜ y ∈ H (1) we define ¯ β ε ( ˜ x, ˜ y ) = ¯ δ ε − 1  ¯ δ ε ( ˜ x ) ¯ δ ε ( ˜ y )  . W e wan t to know when ( H (1) , ρ, ¯ δ ) is a dilatation structure. Prop osition 3.2 The structur e ( H (1) , ρ, ¯ δ ) satisfies the axioms A0, A1, A2. Mor e- over, A4 implies A3. Pro of. It is easy to c hec k that for any ε, µ ∈ (0 , + ∞ ) we ha v e ¯ δ ε ¯ δ µ = ¯ δ εµ and that id = δ 1 . Moreo v er, from g non decreasing and con tin u ous at 0 we deduce that lim ε → 0 ¯ δ ε ˜ x = (0 , 0) , uniformly with resp ect to ˜ x in compact sets. Another computation sho ws that ρ ( ¯ δ ε ˜ x ) = ερ ( ˜ x ) 3 for any ˜ x ∈ H (1) and ε > 0. Otherwise stated, the fun ction ρ is h omogeneous with resp ect to ¯ δ . All that is left to p ro ve is that A4 implies A3. Remark that ¯ δ is left inv arian t (in the sense of transp ort b y left translations in H (1)) and the distance ρ is also left in v arian t. Then axio m A4 takes the form: there exists the limit lim ε → 0 ¯ β ε ( ˜ x, ˜ y ) = ¯ β ( ˜ x, ˜ y ) ∈ H (1) (3.0.1) uniform with resp ect to ˜ x, ˜ y ∈ K , K compact set. F rom the homogeneit y of the fun ction ρ with resp ect to ¯ δ w e deduce that for any ˜ x, ˜ y ∈ H (1) we ha v e: 1 ε ρ  ¯ δ ε ( ˜ x ) , ¯ δ ε ( ˜ y )  = ρ ( ¯ β ε ( ˜ x − 1 , ˜ y )) . F rom the left inv ariance of ¯ δ and ρ it follo ws that A4 implies A3.  Theorem 3.3 If the triple ( H (1) , ρ, ¯ δ ) is a dilata tion structur e then id is metric al ly differ entiable fr om ( H (1) , d ) to ( H (1) , ρ ) . Pro of. W e kn o w that the triple ( H (1) , ρ, ¯ δ ) is a dilatation s tr ucture if and only if (3.0.1 ) is true. T aking (3.0.1) as h yp othesis we deduce that the iden tit y fun ction is deriv able from ( H (1) , d, δ ) to ( H (1) , ρ, ¯ δ ). Ind eed, computation sh ows that id deriv able is equiv alen t to the existence of the limit lim ε → 0 ¯ δ ε − 1 δ ε ˜ u = ( u, sg n ( ¯ u ) g − 1  lim ε → 0 1 ε g ( ε 2 | ¯ u | )  ) uniform with resp ect to ˜ u in compact s et. T herefore the function id is deriv able ev erywhere if and only if the uniform limit, w ith resp ect to ¯ u in compact set: A ( ¯ u ) = lim ε → 0 1 ε g ( ε 2 | ¯ u | ) (3.0.2) exists. W e w ant to sho w th at (3.0.1) implies the existence of this limit. F or th is w e shall use an equiv alen t (isomorphic) description of ( H (1) , ρ, ¯ δ ). Con- sider the f unction F : H (1) → H (1), defin ed by F ( x, ¯ x ) = ( x, sg n ( ¯ x ) g ( | ¯ x | )) . The f u nction F is in ve rtib le b ecause g is inv ertible. F or an y ε > 0 let ˆ δ ε b e the u sual dilatations: ˆ δ ε ( x, ¯ x ) = ( εx, ε ¯ x ) . It is then str aigh tforw ard that ¯ δ ε = F − 1 ˆ δ ε F , for an y ε > 0. 4 The function F can b e mad e into a group isomorphism by r e-defining the group op eration on H (1) ˜ x · ˜ y = F (( x, h ( ¯ x ))( y , h ( ¯ y )) , where h is th e function h ( t ) = s g n ( t )( t 2 + k ( | t | )) . Let µ b e the transp orted left in v ariant distance on H (1), d efi ned by µ ( F ( ˜ x ) , F ( ˜ y )) = ρ ( ˜ x, ˜ y ) . Remark that µ has the simple expr ession µ ((0 , 0) , ( x, ¯ x )) = max {| x | , | ¯ x |} . Exactly as b efore w e can construct the structur e ˆ δ by ˆ δ ˜ x ε ˜ y = ˜ x · ˆ δ ε  ˜ x − 1 · ˜ y  . W e get a dilatation structure ( H (1) , µ, ˆ δ ) isomorp h ic with ( H (1) , ρ, ¯ δ ). The identi ty function id is deriv able from ( H (1) , d, δ ) to ( H (1) , ρ, ¯ δ ) if an d only if the fu nction F is deriv able from ( H (1) , d, δ ) to ( H (1) , µ, ˆ δ ). The axiom A4 for the dilatatio n structure ( H (1) , µ , ˆ δ ) implies that f or any ˜ x, ˜ y ∈ H (1) the limit exists lim ε → 0 1 ε g  | ε 2  1 2 ω ( x, y )+ | ¯ x | ¯ x + | ¯ x | ¯ x  + sg n ( ¯ x ) k ( ε | ¯ x | ) + s g n ( ¯ y ) k ( ε | ¯ y | ) |  , uniform with resp ect to ˜ y in compact set. T ak e in the previous limit ¯ x = ¯ y = 0 and denote ¯ u = 1 2 ω ( x, y ). W e get (3.0.2), therefore we pr o v ed th at id is deriv able f rom ( H (1) , d, δ ) to ( H (1) , ρ, ¯ δ ). Finally , the deriv ability of id implies the metric differen tiabilit y . Indeed, we u se (3.0.2) to compute ν , the metric differen tial of id . W e obtain that ν ˜ x = µ (( x, A ( ¯ x ))) = max {| x | , A ( ¯ u ) } . The pro of is done.  In the coun terexample of Kirc hheim and Magnani the id entit y fun ction id is not metric differentiable, therefore the corresp ondin g triple ( H (1) , ρ, ¯ δ ) is not a dilatation structure. References [1] A. B ella ¨ ıc he, The tangen t space in sub-Riemannian geometry , in: Sub- Riemannian Geometry , A. Bella ¨ ıc he, J.-J. Risler ed s., Pr o gr ess in Mathema tics , 144 , Birkh¨ auser, (1996), 4-78 5 [2] M. Bu liga, Dilatation structures I. F u ndament als, J. Gen. Lie The ory Appl. , V ol 1 (2007), No. 2, 65-95. http:// arxiv.org/abs/math.MG/060 8 536 [3] M. Bu liga, Contract ible groups and linear dilatatio n structures, (2007), h ttp://xxx.arxiv.org/abs/070 5.1440 [4] M. Buliga, Dilatation structures with the Radon-Nik o dym prop erty , (2007 ), h ttp://arxiv.org/abs/070 6.3644 [5] M. Buliga, Dilatation structures in sub-riemannian geometry , (2007 ), h ttp://arxiv.org/abs/070 8.4298 [6] M. Gromo v, Carnot-Carath ´ eo dory spaces seen f rom w ithin, in: Sub- Riemannian Geometry , A. Bella ¨ ıc h e, J.-J. Risler eds ., Progress in Mathematics, 144 , Birkh¨ auser, (1996), 79-323 [7] B. Kirk h heim, V Magnani, A counterexample to metric differentiabilit y , Pr o c. Ed. Math. So c. , 46 (2003), 221-227 6