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Le -in a ian dis ibu ions di eomo phic o la dis ibu ions
© 2024 he Au ho s
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Nicolussi Golo, Sebas iano; O azzi, Alessand o
Nicolussi Golo, S., & O azzi, A. (2024). Le -in a ian dis ibu ions di eomo phic o la
dis ibu ions. Geome iae Dedica a, 218(2), A icle 56. h ps://doi.o g/10.1007/s10711-024-
00905-3
2024
Geome iae Dedica a (2024) 218:56
h ps://doi.o g/10.1007/s10711-024-00905-3
ORIGINAL PAPER
Le -in a ian dis ibu ions di eomo phic o la
dis ibu ions
Sebas iano Nicolussi Golo1
·Alessand o O azzi2
Recei ed: 14 July 2022 / Accep ed: 14 Feb ua y 2024 / Published online: 13 Ma ch 2024
© The Au ho (s) 2024
Abs ac
Fo a s a i ied g oup G, we cons uc a class o Lie g oups endowed wi h a le -in a ian
dis ibu ion locally di eomo phic o he la dis ibu ion o G.Vice e sa,weshow ha allLie
g oups wi h a le -in a ian dis ibu ion ha is locally di eomo phic o he la dis ibu ion o
Gbelong o he class we cons uc ed, i he Lie algeb a o Ghas ini e Tanaka p olonga ion.
Keywo ds Fla dis ibu ions ·Tanaka p olonga ion ·S a i ied Lie g oups ·Con ac
s uc u es ·Quasi-con o mal maps
Ma hema ics Subjec Classi ica ion (2010) 30L10 ·22E25 ·53C30
Con en s
1 In oduc ion ............................................... 2
2 No a ion and p elimina ies ........................................ 4
2.1 Pola iza ions and Tanaka p olonga ions ............................... 4
2.2 The g oups Pand Qand hei quo ien M............................. 5
2.3 Pola iza ions on G,Pand M.................................... 7
3 Dis ibu ion-p ese ing di eomo phisms o Mwhen Gis igid .................... 8
4 Modi ica ions o s a i ied g oups ....................................11
5 Examples .................................................13
5.1 Modi ica ions o he Heisenbe g g oup ...............................13
Sebas iano Nicolussi Golo has been pa ially suppo ed by he Eu opean Unions Se en h F amewo k
P og amme, Ma ie Cu ie Ac ions-Ini ial T aining Ne wo k, unde G an Ag eemen No. 607643, “Me ic
Analysis Fo Eme gen Technologies (MAnET)”, and by he EPSRC G an "Sub-Ellip ic Ha monic
Analysis" (EP/P002447/1), and by Uni e si y o Pado a STARS P ojec "Sub-Riemannian Geome y and
Geome ic Measu e Theo y Issues: Old and New". Alessand o O azzi has been pa ially suppo ed by he
ARC Disco e y g an DP170103025. Da a sha ing no applicable o his a icle as no da ase s we e
gene a ed o analysed du ing he cu en s udy.
BAlessand o O azzi
a.o [email protected]
Sebas iano Nicolussi Golo
[email p o ec ed]
1Depa men o Ma hema ics and S a is ics, Uni e si y o Jy äskylä, 40014 Jy äskylä, Finland
2School o Ma hema ics and S a is ics, Uni e si y o New Sou h Wales (UNSW), Sydney 2052, Aus alia
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56 Page 2 o 22 Geome iae Dedica a (2024) 218 :56
5.1.1 Rigid mo ions o he plane as a modi ica ion o he Heisenbe g g oup ...........17
5.2 Modi ica ions o he ee nilpo en Lie g oup F24 .........................17
5.3 Modi ica ions o ul a- igid s a i ied g oups ............................20
Re e ences ..................................................21
1 In oduc ion
In his a icle, we conside he ollowing ques ion: gi en a s a i ied g oup G, we wish o
cha ac e ise hose pola ised Lie g oups ha a e equi alen o G.He eapola isa ion on a
Lie g oup is he choice o a le -in a ian and b acke -gene a ing subbundle o he angen
bundle (c . [11]), and wo pola ized Lie g oups a e equi alen i he e is a locally de ined
dis ibu ion-p ese ing di eomo phism be ween hem. S a i ied g oups ca y a canonical
pola isa ion. Gi en a s a i ied g oup, we will cons uc a class o pola ised Lie g oups ha
a e equi alen o G, which we will call modi ica ions o G. The key ool o ou cons uc ion
will be Tanaka p olonga ion heo y.
Be o e di ing in he echnical de ails o ou main esul s, we i s p o ide some amewo k.
Thep oblemunde s udyis ele an indi e en a eas, such as Tanaka p olonga ion heo y,CR
geome y, sub-Riemannian geome y, and con ol heo y. In he se ing o Tanaka’s heo y, i
is known ha he in ini esimal au omo phisms o he pola isa ion associa ed o a s a i ied Lie
algeb aa eencodedbyi s ullTanakap olonga ion(see,e.g.,[13,15,18]).I heLiealgeb ais
no s a i ied, howe e , all we can conclude is ha e e y in ini esimal au omo phism induces
an in ini esimal au omo phism on i s s a i ied symbol. In his pape , we cons uc classes o
pola ised Lie algeb as ha a e no s a i ied, bu ha ha e he same space o in ini esimal
au omo phisms as hei s a i ied symbol.
Ou s udy has po en ial applica ions o geome ic con ol heo y. Gi en a nonholonomic
con ol sys em, he mo ion planning p oblem consis s in inding a cu e angen o he pola i-
sa ion ha connec s wo gi enpoin sin he ambien space.Nilpo en Lie g oupsa e he wides
class o nonholonomic sys ems o which an exac solu ion o he mo ion planning p oblem
is known, see [6]. Dis ibu ion-p ese ing di eomo phisms a e equi alences o mo ion plan-
ning p oblems. Thus, ou me hod de ec s classes o non-nilpo en nonholonomic sys ems
ha a e equi alen o nilpo en ones.
Fu he mo e, ou indings ha e consequences in me ic geome y. On a pola ised Lie
g oup, one may de ine a le -in a ian sub-Riemannian dis ance. In me ic geome y, i is
na u al o s udy he equi alence o me ic spaces up o isome ies, bi-Lipschi z mappings,
con o mal and quasicon o mal mappings. Fo example, i wo s a i ied g oups a e (locally)
quasicon o mal, hen hei Lie algeb as a e isomo phic [14]. I wo nilpo en Lie g oups a e
isome ic, hen hey a e isomo phic [7,9]. I is an open ques ion o de e mine whe he wo
nilpo en Lie g oups ha a e globally bi-Lipschi z o one ano he a e indeed isomo phic. In
[4], he au ho s s udy he Lie g oups ha can be made isome ic o a gi en nilpo en Lie
g oup, endowed wi h a le -in a ian dis ance. (See also [5] o he Riemannian case.) In his
sense, ou wo k ollows [4], because dis ibu ion-p ese ing di eomo phisms a e locally
bi-Lipschi z.
In sub-Riemannian geome y, one o he majo open p oblems is o de e mine whe he
he conclusions o Sa d Theo em hold o he endpoin map, which is a canonical map
om an in ini e dimensional pa h space o he unde lying ini e dimensional mani old. The
se o c i ical alues o he endpoin map is also known as abno mal se , being he se
o endpoin s o abno mal ex emals lea ing he base poin . In he con ex o Lie g oups,
pe haps he mos gene al posi i e esul s ha e been p o ed in [11]. He e he au ho s p o e
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Geome iae Dedica a (2024) 218 :56 Page 3 o 22 56
ha he abno mal se has measu e ze o in he case o 2-s ep s a i ied g oups and se e al
o he examples. This p ope y o he abno mal se is p ese ed by dis ibu ion-p ese ing
di eomo phisms be ween sub-Riemannian mani olds. I hen comes ou om ou esul s
ha e e y modi ica ion o a s a i ied g oup sa is ies he Sa d Theo em, i he s a i ied g oup
does.
Now we will p esen ou main esul s in de ail. Recall ha a Tanaka p olonga ion o a
s a i ied Lie algeb a g h ough a Lie subalgeb a g0o he Lie algeb a o de i a ions o g
ha p ese e he s a i ica ion is he maximal nondegene a e g aded Lie algeb a ha con ains
g+g0, whe e nondegene a e means ha he adjoin ac ion o any elemen o posi i e weigh
on gis non i ial.1When g0is chosen as he whole se o s a a p ese ing de i a ions, we
ob ain he ull Tanaka p olonga ion. When he p olonga ion algeb a is ini e dimensional, we
ob ain a g aded Lie algeb a p=g⊕qand we say ha gand any Lie g oup Gwi h Lie algeb a
ga e igid.The e mo ini e ype is also common in he li e a u e o deno e Lie algeb as
wi h ini e Tanaka p olonga ion. Modi ica ions o ga e hen de ined o be subalgeb as so
po he same dimension o g ha a e ans e sal o q. I u ns ou ha he e is a linea map
σ:g→qo which he modi ica ion is he g aph. I g−1is he i s laye o g, hen he
se { +σ( ) : ∈g−1}de ines a pola isa ion on a Lie g oup whose Lie algeb a is he
modi ica ion s.
Ou i s main esul d aws he connec ion be ween modi ica ions o Gand Lie g oups
ha a e equi alen o G.
Theo em A E e y modi ica ion o a s a i ied Lie g oup G is equi alen o G. Vice e sa, i
G is igid, hen e e y pola ised Lie g oup ha is equi alen o G is one o i s modi ica ion.
Theo em Ais es a ed and p o en in Theo ems 4.2 and 4.4. A key ool in he s udy o
local dis ibu ion-p ese ing di eomo phisms is he quo ien mani old M=P/Q,whe eP
and Q<Pa e he Lie g oups wi h Lie algeb as pand q espec i ely. The pola isa ion o G
induces a pola isa ion Mon Mand Gembeds in Mas an open subse , see P oposi ion 2.9.
Mo eo e , i Sis a modi ica ion o G, hen an open neighbo hood o he iden i y in Scan be
also embedded in o M, see Lemma 4.1. The composi ion o such embeddings induce a local
dis ibu ion-p ese ing di eomo phism be ween Gand S. I he g oup Gis igid, i.e., i s ull
Tanaka p olonga ion is ini e dimensional, hen all dis ibu ion-p ese ing di eomo phisms
be ween Gand Sa ise in his way, see Theo em 4.4.
The igid case is pa icula ly a o able because all dis ibu ion-p ese ing di eomo -
phisms o Ga e induced by a ine maps on P,see(3) a page 9. We can exp ess his igidi y
in e ms o local dis ibu ion-p ese ing di eomo phisms o M. Mo e p ecisely, we will
p o e in Theo em 3.3 he ollowing s a emen :
Theo em B Suppose G is igid and le M =P/Q be he mani old desc ibed abo e, whe e
he Lie algeb a o P is he ull anaka p olonga ion o g.Le U ⊂M be open and connec ed
and :U→ (U)⊂M a smoo h map wi h d (M)⊂M. Suppose ha he e exis s
x0∈U such ha d (x0)is non-singula . Then he e exis s a unique dis ibu ion-p ese ing
di eomo phism g :M→M such ha g|U= .
We will also p o e ha , in he hypo hesis o Theo em B, he connec ed componen o he
iden i y in he g oup o dis ibu ion-p ese ing di eomo phisms o Mis isomo phic o Q,
see Theo em 3.6.
1Recall ha a g ading o aLiealgeb a isa ec o spacedecomposi iong=i∈Zgisuch ha [gi,gj]⊂gi+j
o all i,j∈Z. A g ading is a s a i ica ion i g=i≤−1giand [g−1,gj]=gj−1 o all j<0.
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56 Page 4 o 22 Geome iae Dedica a (2024) 218 :56
I emains open whe he he second pa o Theo em Aholds ue wi hou asking ha
he ull Tanaka p olonga ion is ini e. While we canno p o e he heo em in his gene ali y,
examplessugges ha i maybe ue.Mo ep ecisely,inSec .5.1,weshow ha all h eedimen-
sional sub-Riemannian s uc u es a e modi ica ions o he Heisenbe g g oup wi h espec o
a sui able ini e dimensional Tanaka p olonga ion, e en hough he ull p olonga ion o he
Heisenbe g Lie algeb a is in ini e dimensional, see Theo em 5.1. This jus i ies he ollowing
conjec u e.
Conjec u e Suppose ha G is a s a i ied Lie g oup and ha S is a pola ised Lie g oup ha
is equi alen o G. Then he e is a ini e Tanaka p olonga ion o Lie(G)in which Lie(S)is
a modi ica ion o Lie(G).
In Sec . 5.2 we explici ly compu e some modi ica ions o he ee nilpo en Lie g oup
wi h wo gene a o s and s ep ou , F24. I comes ou ha one may cons uc examples o
non-nilpo en Lie g oups ha a e equi alen o F24. We also ind a nilpo en , non-s a i ied,
pola ised Lie g oup ha is equi alen o F24 ia a global dis ibu ion-p ese ing di eomo -
phism, see Theo em 5.5. Finally, in Sec . 5.3, we s udy all he modi ica ions o an ul a- igid
s a i ied g oup, ha is, a s a i ied g oup whose only s a a-p ese ing de i a ion is he
in ini esimal gene a o o dila ions. I u ns ou ha such modi ica ions a e all sol able and
he only nilpo en one is he s a i ied g oup i sel , see Theo em 5.9.
The pape is o ganized as ollows. In Sec .2, we ix he no a ion and es ablish he
amewo k in which we will be wo king. We conside s a i ied algeb as and hei Tanaka
p olonga ions, we de ine he co esponding Lie g oups and ix a pola isa ion on hem. In
Sec .3, we s udy dis ibu ion-p ese ing di eomo phisms o Mas a ine maps o Pand
p o e Theo em B. In Sec .4, we de ine he modi ica ions o a s a i ied algeb a and hose o a
s a i ied g oup, p o ing Theo em A. Finally, we apply ou modi ica ion echnic o a numbe
o examples in Sec .5.
2 No a ion and p elimina ies
2.1 Pola iza ions and Tanaka p olonga ions
Gi en a connec ed, smoo h mani old M,apola isa ion o Mis he choice o a subbundle M
o he angen bundle TM ha is b acke gene a ing, i.e., wi h he p ope y ha he sec ions
o Mb acke gene a e all he sec ions o TM. Gi en wo pola ised mani olds (M,M)and
(N,N),adis ibu ion-p ese ing di eomo phism be ween Mand Nis a di eomo phism
:M→Nsuch ha ∗(M)=N.Wedeno eby(TM) he space o ec o ields on
M. A ec o ield V∈(TM)on a pola ised mani old (M,M)is a con ac ec o ield
i i s low is made o dis ibu ion-p ese ing di eomo phisms. Fo a Lie g oup S,weshall
always conside le -in a ian pola isa ions S. The pai (S,S)is called a pola ised g oup.
The iden i y elemen will be deno ed by eS,o simplyei no con usion a ises. We deno e
by Gas a i ied g oup, ha is, a connec ed and simply connec ed Lie g oup whose Lie
algeb a decomposes as g=−1
i=−sgi, wi h [g−1,gj]=gj−1 o e e y −s+1≤j≤−1.
On a s a i ied g oup we will always conside he le -in a ian pola isa ion G o which
(G)eG=g−1. In a s a i ied g oup Gwe conside he s a a p ese ing de i a ions
De (g):= {u∈End(g):u(g−1)⊂g−1,
and u[X,Y]=[u(X), Y]+[X,u(Y)]∀X,Y∈g}.
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Geome iae Dedica a (2024) 218 :56 Page 5 o 22 56
Gi en a subalgeb a g0o De (g),wede ine heTanaka p olonga ion o g h ough g0as
he (possibly in ini e) maximal nondegene a e g aded Lie algeb a P ol(g,g0)=k≥−sgk
which con ains g⊕g0. When g0=De (g), we call P ol(g,g0) he ull Tanaka p olonga ion
o g. I is no di icul o see ha he la e con ains all p olonga ions. We say ha g,o G,is
igid i he ull Tanaka p olonga ion has ini e dimension. When i is clea om he con ex
and he p olonga ion unde conside a ion is ini e dimensional, we shall deno e P ol(g,g0)
by p, he nonnega i e pa k≥0gkby q, and he posi i e pa k>0gkby p+.See[13,15,
18] o u he de ails on Tanaka p olonga ion.
2.2 The g oups Pand Qand hei quo ien M
In he ollowing, we es ablish a numbe o p ope ies o he Lie g oups ha co espond o
he Lie algeb as in oduced abo e. Le ¯
Pbe he connec ed and simply connec ed Lie g oup
whose Lie algeb a is a ini e dimensional Tanaka p olonga ion po a s a i ied Lie algeb a g.
Le ¯
Qbe he connec ed subg oup o ¯
Pwhose Lie algeb a is q.
The se {δλ:λ>0}o mappings on pde ined by δλ(X)=λiX o X∈giis a one pa am-
e e amily o au omo phisms o p. By abuse o no a ion, we w i e δλ o he co esponding
au omo phisms o he g oup ¯
P. Such maps exis because ¯
Pis simply connec ed.
Lemma 2.1 Deno e by expP:p→¯
P he exponen ial map o ¯
P. Then expPis injec i e on
gand on k≥1gk.
P oo Le ,w ∈gsuch ha expP( ) =expP(w).Since ,w ∈g, hen limλ→∞ δλ =
limλ→∞ δλw=0. Le λ≥1 be such ha bo h δλ( ) and δλ(w) belong o a neighbo hood U
o 0 in pon which he exponen ial map expPis injec i e. Then expP(δλ ) =δλ(expP( )) =
δλ(expP(w)) =expP(δλw). By he injec i i y o expPon U,weha eδλ =δλw.Sinceδλ
is a linea isomo phism, we conclude ha =w. A simila a gumen p o es ha expPis
injec i e on k≥1gk.
By Lemma 2.1, he canonical imme sion G→¯
Pinduced by g→pis injec i e. We a e
going o show ha Gis closed in ¯
P. We p o e wo lemmas i s .
Lemma 2.2 The in e sec ion o G wi h ¯
Qis i ial.
P oo Since δλ(g)=gand δλ(q)=q, henδλ(G)=Gand δλ(¯
Q)=¯
Q, o allλ>0. Since
gis nilpo en , G=expP(g).
Le x∈G∩¯
Q; henx=expP( ) o some ∈gand limλ→∞ δλ(x)=
expP(limλ→∞ δλ ) =eP. I ollows ha he cu e γ:(0,1]→ ¯
P,γ( )=δ −1x, ex ends
o a con inuous pa h [0,1]→ ¯
Pconnec ing γ(0)=eP o γ(1)=xand laying in G.Since
δλ(x)∈¯
Q o all λ>0, hen γlies in ¯
Qas well.
Since g⊕q=p, he e a e open neighbo hoods U⊂gand V⊂qo 0 such ha
=expP(U)expP(V)is an open neighbo hood o ePin ¯
Pand he ollowing holds: The
connec ed componen o ∩Gcon aining ePis expP(U), he connec ed componen o
∩¯
Qcon aining ePis expP(V),andexp
P(U)∩expP(V)={eP}.
Since γjoins x o ePcon inuously, hen γ([0,1])∩lies in bo h he connec ed compo-
nen s o ∩Gand ∩¯
Qcon aining eP, i.e., γ([0,1])∩⊂expP(U)∩expP(V)={eP}.
This implies ha x=eP.
Lemma 2.3 (Lemma on Lie g oups) Le G be a Lie subg oup o a Lie g oup P and le
ι:G→P he inclusion. The image ι(G)is no closed in P i and only i he e is a sequence
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{gn}n∈N⊂G such ha limn→∞ gn=∞(i.e., gne en ually escapes e e y compac se o
G) and limn→∞ ι(gn)=eP.
P oo Recall ha Gis closed in Pi and only i ιis an embedding. So, i such a sequence
exis s hen ι(G)is no closed in P. We need o p o e he con e se implica ion.
Le ρbe any le -in a ian Riemannian dis ance on G.Thenρis comple e and in pa icula
closed balls a e compac . Le {gn}n∈N⊂Gbe a sequence such ha limn→∞ ι(gn)=p∈P.
I he e is R>0 such ha ρ(eG,gn)≤R o all n, hen he e is a subsequence gnkcon e ging
o some g∞∈G. Since he imme sion ι:G→Pis con inuous, we ob ain ι(g∞)=p,
hence p∈ι(G).
So, i ι(G)is no closed, hen he e is a sequence {gn}n∈N⊂Gsuch ha limn→∞ ι(gn)=
p∈Pbu gn→∞in G.Le {gnk}kbe a subsequence such ha ρ(gnk,gnk+1)>k o k∈N
and de ine hk=g−1
nkgnk+1.Thenhk→∞in G, because ρ(eG,hk)=ρ(eG,g−1
nkgnk+1)=
ρ(gnk,gnk+1)>k o all k.Howe e ,ι(hk)=ι(g−1
nk)ι(gnk+1)→p−1pin Pas k→∞.
Lemma 2.4 The imme sed g oup G is closed in ¯
P.
P oo We p o e ha , i { n}n∈N⊂gis a sequence so ha limn→∞ expP( n)=eP, hen
limn→∞ n=0. By Lemma 2.3 and expP(g)=G, his claim implies ha Gis closed in P.
Le { n}n∈N⊂gbe a sequence wi h limn→∞ expP( n)=eP.Le U⊂gand W⊂qbe
open neighbo hoods o 0 such ha he map U×W→P,(u,w) → expP(u)expP(w) is
a di eomo phism on o i s image. Then, o nla ge enough, he e a e un∈Uand wn∈W
so ha expP(un)expP(wn)=expP( n). The e o e, expP(un)−1expP( n)=expP(wn)∈
¯
Q∩G. By Lemma 2.2,weha eexp
P(un)=expP( n). By Lemma 2.1,weha eun= n.
Since expP(un)→eP, hen n=un→0.
Co olla y 2.5 The imme sed g oup ¯
Qisclosedin ¯
P.
P oo This is a consequence o Lemma 2.4 and pa (iii) o Lemma 2.15 in [4]
Since ¯
Qis closed, we may conside he homogeneous mani old M:= ¯
P/¯
Qwi h quo ien
p ojec ion π:¯
P→M. The ac ion o ¯
Pmay ha e a non- i ial ke nel
K:= {p∈¯
P:p.x=x∀x∈M}=
p∈¯
P
p¯
Qp
−1.
Lemma 2.6 The ke nel K o he ac ion o ¯
P on M is disc e e and con ained in ¯
Q. Mo eo e ,
i p ∈K, henδλp=p o all λ>0.
P oo Clea ly Kis a no mal and closed subg oup o ¯
Pand i is con ained in ¯
Q.Le ∈
Lie(K), heLiealgeb ao K.Then o someposi i ein ege , wemayw i e = 0+···+ ,
wi h i∈gi o e e y i=0,...,.SinceLie(K)is an ideal in pcon ained in q, i ollows
in pa icula ha o all i=0,...,,
[[...[[ i,y1],y2],...],y+1]∈gi−−1∩q={0},
o e e y y1,...,y+1∈g−1. By de ini ion o Tanaka p olonga ion, his implies ha =0.
The e o e, he Lie algeb a o Kis i ial and so Kis disc e e.
Since K=x∈¯
Px¯
Qx−1, i is clea ha δλ(K)⊂K o all λ>0. Howe e , since
λ→ δλpis a con inuous cu e passing h ough p,wemus ha eδλp=pwhen p∈K.
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F om Lemma 2.6 i ollows ha P:= ¯
P/Kand Q:= ¯
Q/Ka e Lie g oups, ha Qis
closed in Pand M=P/Q. Mo eo e , he maps δλa e au omo phisms o Pas well, o all
λ>0. Since G∩K={e}, he g oup Gis embedded in Pwi h G∩Q={e}.
Rema k 2.7 I we a e gi en Gand Qinside P, o ins ance as ma ix g oups, we may wan
o isualise he ac ion o Pon Mas a local ac ion o Pon G. In o he wo ds, i p∈P,
hen he e may be open subse s Up,Vp⊂Gand a dis ibu ion-p ese ing di eomo phism
p:Up→Vp ha co esponds o he ac ion o pon M, i.e., p(g1)is he only g2∈G,
i i exis s, such ha {g1Q)∩G={g2}. In gene al, such cons uc ion is no possible o all
p∈P,bu i pis nea enough o eP, henUp,Vpand pdo exis . The ac ha such pis a
dis ibu ion-p ese ing di eomo phism will be p o ed in P oposi ion 2.8.
2.3 Pola iza ions on G,Pand M
We deno e by π:P→M he quo ien map, wi h M=P/Q.I p∈Pand m∈M,we
use he no a ion p.mo p(m) o he ac ion o pon m. In such con ex s, we will iden i y
elemen s p∈Pwi h smoo h di eomo phisms p:M→M.
Recall ha on Gwe ha e he pola isa ion Gwi h (G)e=g−1. We de ine on P he
pola isa ion Psuch ha (P)eP=g−1⊕q. No ice ha G=P∩TG.De ineM:=
dπ(P)which is a subse o TM. We shall p o e ha Mis a P-in a ian pola isa ion on
M.
P oposi ion 2.8 The se M⊂T M is a P-in a ian , b acke gene a ing subbundle o M.
In pa icula , (M,M)is a pola ised mani old and he di eomo phisms p :M→M o
p∈P a e dis ibu ion-p ese ing di eomo phisms.
P oo No ice ha Mis a P-in a ian subse o TM.In o de o show ha Mis asubbundle,
we need o p o e ha , i p1,p2∈Pa e such ha π(p1)=π(p2), hen
dπ((P)p1)=dπ((P)p2). (1)
Since p◦π=π◦Lp o all p∈P, hen(1) is equi alen o d(p−1
2◦π◦Lp1)[(P)e]=
dπ[(P)e].Le p=p1and choose q∈Qso ha p2=p1q.Thenp−1
2◦π◦Lp1=π◦Lq−1
and hus (1) is also equi alen o
Adq[(P)e]mod q=(P)emod q.(2)
Since Ad is a homomo phism and e e y q∈Qis he ini e p oduc o exponen ial elemen s,
i ’s enough ha we show (2) o q=exp y,y∈q. Deno e by y0 he p ojec ion o yon g0.
Le w∈g−1⊕qand deno e by w−1i s p ojec ion on g−1.Then
Adqwmod q=ead(y)wmod q
=ead(y0)w−1mod q.
Since ead(y0):g−1→g−1is a bijec ion, we conclude ha Adq[(P)e]mod q=g−1
mod q. This p o es (2) and he e o e (1).
Finally, we need o show ha Mis b acke gene a ing. Recall ha , o an analy ic
subbundle o an analy ic mani old, being b acke gene a ing is equi alen o being connec ed
by cu es angen o he subbundle, and ha quo ien s o Lie g oups and in a ian subbundles
a e all analy ic. Thus, le m0=π(p0)and m1=π(p1)in M.Then he eisaC1-cu e
γ:[0,1]→Psuch ha γ(0)=p0,γ(1)=p1and γ( )∈P o all ∈[0,1]. Hence,
he cu e π◦γ:[0,1]→Mgoes om m0 o m1and is clea ly angen o M.
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56 Page 8 o 22 Geome iae Dedica a (2024) 218 :56
P oposi ion 2.9 The es ic ion π|G:(G,
G)→(M,M)is a dis ibu ion-p ese ing
di eomo phism on o i s image, which is an open subse o M.
P oo Fi s , we show ha π|Gis injec i e. Le a,b∈Gsuch ha π(a)=π(b).Then
π(e)=π(a−1a)=a−1.π(a)=a−1π(b)=π(a−1b), i.e., a−1b∈Q.Sincea−1b∈Gand
G∩Q={e}, hena=b.
Second, we show ha π|Gis an imme sion. Since ke (dπ|g)=dLg(q)and dLg(q)∩
TgG=DLg(q)∩DLg(g)={0}, hend(π|G)|g=(dπ|g)|TgGis injec i e, o all g∈G.
Thi d,weclaimdπ|G(G)=M∩T(π(G)).SinceG⊂PandsinceM=dπ(P)
by de ini ion, i ollows ha dπ|G(G)⊂M∩T(π(G)). Mo eo e , since Mis P-
in a ian by P oposi ion 2.8, o allx∈M,dim(M)x=dim(M)π(e)=dim((g−1⊕
q)/q)=dim(g−1). The e o e, we ob ain he claim by compa ing he dimensions.
Finally, he ac ha π(G)is open in Mand he ac ha π|Gis an embedding a e
bo h consequences o (π|G)being an imme sion and he ac ha Mand Gha e he same
dimension.
Rema k 2.10 A i s consequence o P oposi ion 2.9 is ha any local dis ibu ion-p ese ing
di eomo phism on Mis in ac a local dis ibu ion-p ese ing di eomo phism on G. Indeed,
by he ac ion o Pon Mand ia he map π|G, any local dis ibu ion-p ese ing di eomo -
phism o Mde ines a local dis ibu ion-p ese ing di eomo phism o G. Simila ly, con ac
ec o ields on Mde ine con ac ec o ields on G.
In case Gis a igid s a i ied g oup and pis he ull Tanaka p olonga ion o g, hese
ela ions a e s onge , see Sec .3.
3 Dis ibu ion-p ese ing diffeomo phisms o Mwhen Gis igid
This sec ion con ains Theo em 3.3 o dis ibu ion-p ese ing di eomo phisms in he igid
case.
Rela i e o a ec o X∈TeP,wedeno eby ˜
X he le -in a ian ec o ield ˜
X(p)=
dLp|e[X], and by X† he igh -in a ian ec o ield X†(p)=dRp|e[X]. Simila ly, we
deno e by ˜
p he Lie algeb a o le -in a ian ec o ields and by p† he Lie algeb a o igh -
in a ian ec o ields on P. Mo eo e , as in he p e ious sec ions, he mani old Mis he
quo ien P/Qand we deno e by o he poin π(e)∈M.
Lemma 3.1 Le :p→pbe a Lie algeb a au omo phism wi h (q)=qand (g−1⊕q)=
g−1⊕q. Then he e is a unique dis ibu ion-p ese ing Lie g oup au omo phism L :P→P
wi h L∗=and a unique dis ibu ion-p ese ing di eomo phism Lπ:M→M wi h
Lπ◦π=π◦L.
P oo I :p→pis a Lie algeb a au omo phism wi h (q)=q, hen he induced Lie g oup
au omo phism ¯
L:¯
P→¯
Phas he p ope y ha ¯
L(K)=K,whe eKis he ke nel o he
ac ion o ¯
Pon M. I ollows ha he e is a Lie g oup au omo phism L:P→Psuch ha
L∗=.
I L:P→Pis a Lie g oup au omo phism wi h L(Q)=Q, hen i is well known ha
he e is a unique di eomo phism Lπ:M→Msuch ha Lπ◦π=π◦L.
Now, suppose ha (g−1⊕q)=g−1⊕q, i.e., L∗(P)e=(P)e.SincePis le -
in a ian , hen Lis a dis ibu ion-p ese ing di eomo phism o (P,P).Finally,wep o e
ha Lπis a dis ibu ion-p ese ing di eomo phism. Le X∈Pand x∈P.Then
dLπ|π(x)[dπ|x[˜
Xx]] = d(Lπ◦π)|x[˜
Xx]=d(π ◦L)|x[˜
Xx]∈M|Lπ(x).
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Geome iae Dedica a (2024) 218 :56 Page 15 o 22 56
Fi s , case (A.1) is no isomo phic o he o he s because in case (A.1) we ha e s(2)=
span{ 3}while in all o he h ee cases we ha e s(2)=span{ 2, 3}.
Second,case(A.4) is no isomo phic o heo he s because incase (A.4) weha e [, s−1] ⊂
s−1while in all o he cases we ha e [, s−1]⊂s−1.
Thi d, o di e en choices o α∈Rin case (A.4) we ge non-isomo phic pola ised Lie
algeb as: To p o e his, we shall show ha he pa ame e αis independen o he choice
o he basis. So, suppose ha g1,g2,g3∈s o m ano he basis wi h s−1=span{g1,g2},
[g1,g2]=g3,[g2,g3]=0and[g1,g3]=αg2+g3. Then one easily shows ha g1=
x
1+y 2,g2=μ 2and g3=λ 3, o somex,y,λ,μ ∈Rwi h xμ
λ=1. Mo eo e ,
[g1,g3]=αxμ
λg2+xg3, which implies x=1andα=α.
Finally, cases (A.2) and (A.3) a e no isomo phic o each o he , because in case (A.2) i
holds ad 1|2
s(2)=Id|s(2), while while in case (A.3) i holds ad 1|2
s(2)=−Id|s(2).
P oo o Theo em 5.1 Le us ix he no a ion o he Heisenbe g Lie algeb a. Fix a basis
e1,e2,e3so ha [e1,e2]=e3, and choose g−1=span{e1,e2}. The space De (g)o he
s a a p ese ing de i a ions o gmay be iden i ied wi h gl(2,R).
Fi s , we conside
g0:= {D∈De (g):D(e1)⊆Re1and D(e2)⊆Re2}.
In his case, P ol(g,g0)=sl(3,R)=g⊕q(see, e.g., [3]), whe e gis iden i ied wi h he Lie
algeb a gene a ed by
e1=⎛
⎝
010
000
000
⎞
⎠,e2=⎛
⎝
000
001
000
⎞
⎠,e3=⎛
⎝
001
000
000
⎞
⎠.(6)
and qis he se o ma ices in sl(3,R)o he o m
⎛
⎝
∗00
∗∗0
∗∗∗
⎞
⎠
The modi ica ions o gin sl(3,R)a e he subalgeb as o sl(3,R)o he o m {X+σ(X):
X∈g}, o some linea map σ:g→q. We show ha all h ee dimensional Lie algeb as
wi h a b acke gene a ing plane a e g aphs o such a σ:
Case (A): I sis sol able, hen de ine σby he assignmen s:
σ(e1)=⎛
⎜
⎝
2β
300
α−β
30
00−β
3
⎞
⎟
⎠,σ(e2)=σ(e3)=0.
I is easy o check ha ec o s i:= ei+σ(ei),i=1,2,3, sa is y he b acke ela ions o
case (A) in P oposi ion 5.2.
Case (B):
Fo his case, we choose
σ(e1)=⎛
⎝
000
−100
000
⎞
⎠,σ(e2)=⎛
⎝
000
000
0−10
⎞
⎠,σ(e3)=⎛
⎝
000
000
−100
⎞
⎠.
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56 Page 16 o 22 Geome iae Dedica a (2024) 218 :56
Case (C): we ob ain he b acke s in (C) by choosing
σ(e1)=0,σ(e2)=⎛
⎝
000
1/200
000
⎞
⎠,σ(e3)=⎛
⎝
1/200
0−1/20
000
⎞
⎠.
Case (D): In his case we use he ini e p olonga ion su(2,1)o he Heisenbe g algeb a,
as in [8, p313]. Le
J=⎛
⎝
10 0
0−10
00−1⎞
⎠.
The Lie algeb a su(2,1)isgi enby3×3 complex ma ices Awi h ze o ace and such
ha A∗J+JA =0, whe e A∗is he he mi ian anspose o A. De ine he Lie algeb a
au omo phism θ:su(2,1)→su(2,1),θA:= JAJ.De ine
X=⎛
⎝
0i0
−i0−i
0−i0⎞
⎠Y=⎛
⎝
010
101
0−10
⎞
⎠Z=⎛
⎝
2i02i
000
−2i0−2i⎞
⎠
H=⎛
⎝
001
000
100
⎞
⎠U=⎛
⎝
i00
0−2i0
00i⎞
⎠
θX=⎛
⎝
0−i0
i0−i
0−i0⎞
⎠θY=⎛
⎝
0−10
−101
0−10
⎞
⎠θZ=⎛
⎝
2i0−2i
00 0
2i0−2i⎞
⎠
The g ading o su(2,1)is
g−2(g)=span{Z}
g−1(g)=span{X,Y}
g0(g)=span{H,U}
g1(g)=span{θX,θY}
g2(g)=span{θZ},
whe e g−2(g)⊕g−2(g)=gis he Heisenbe g Lie algeb a: no ice ha [X,Y]=Zwhile
[X,Z]=[Y,Z]=0. So, q=span{H,U,θX,θY,θZ}.De ineσ:g→qby se ing
σX:= − 1
16θX+i9
16θY=⎛
⎝
0−i1
20
−i5
80i5
8
0−i1
20⎞
⎠,
σY:= −i9
16θX−1
16θY=⎛
⎝
0−1
20
5
80−5
8
0−1
20⎞
⎠,
σZ:= −i9
4H+1
4U−5
16θZ=⎛
⎝
−i3
80−i13
8
0−i1
20
−i23
80i7
8
⎞
⎠.
One can easily check ha 1=X+σX, 2=Y+σYand 3=Z+σZ o m a basis o a
Lie subalgeb a o su(2,1)sa is ying he ela ions o Case (D).
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Geome iae Dedica a (2024) 218 :56 Page 17 o 22 56
Rema k 5.4 The map σabo e can easily be ound using he so wa e Maple and i is no
unique.
5.1.1 Rigid mo ions o he plane as a modi ica ion o he Heisenbe g g oup
We conclude his sec ion discussing mo e in de ail he case o he g oup o igid mo ions
o he plane as a modi ica ion o he Heisenbe g g oup. A a g oup le el, we may ep esen
poin s in he Heisenbe g g oup Has ma ices in SL(3,R)by
H(x1,x2,x3):= ⎛
⎝
1x1x3
01x2
00 1
⎞
⎠,
o x1,x2,x3∈R.
The Lie algeb a o he he g oup o igid mo ions o he plane E(2)co esponds o he
case (A) wi h α=−1andβ=0. The co esponding ep esen a ion in sl(3,R)gi enin he
p e ious heo em is he span o he ec o s
1=⎛
⎝
010
−100
000
⎞
⎠, 2=⎛
⎝
000
001
000
⎞
⎠, 3=⎛
⎝
001
000
000
⎞
⎠.
A he g oup le el, he poin s o E(2)inside SL(3,R)a e pa ame ized by
R(y1,y2,y3):= ⎛
⎝
cos y1sin y1y3
−sin y1cos y1y2
001
⎞
⎠
whe e y1∈R/(2πZ)and y2,y3∈R.
Wi h he p ocedu e desc ibed in Rema k 4.3, we ind he mapping E(2)→H:
R(y1,y2,y3)→ H( an y1,y2,y3),
which is de ined on he domain (−π/2,π/2)×R2.
5.2 Modi ica ions o he ee nilpo en Lie g oup F24
We conside he ee nilpo en Lie algeb a 24 =span{ei:i=1,...,8}o ank 2 and s ep
4 and he co esponding simply connec ed Lie g oup F2,4. We will p o e he ollowing esul
Theo em 5.5 The e exis s a nilpo en Lie g oup S, no isomo phic o F2,4, ha is a modi i-
ca ion o F2,4and is globally equi alen o F2,4.
P oo The Lie b acke s in 24 a e
[e2,e1]=e3,[e3,e1]=e4,[e3,e2]=e5,
[e4,e1]=e6,[e5,e1]=e7,[e4,e2]=e7,[e5,e2]=e8.
I isknown ha he ullTanakap olonga iono 24 isp= 24⊕De (g),wi hDe (g)≃gl(2,R)
(see [17]). The e o e, he modi ica ions o 24 a e subalgeb as o p ha a e g aphs o some
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56 Page 18 o 22 Geome iae Dedica a (2024) 218 :56
linea map σ: 24 →gl(2,R). He e we only conside σ ha on he basis o 24 is ze o excep
o σ(e1). Imposing ha he g aph is a Lie algeb a, a di ec compu a ion shows ha
σ(e1)=a0
cb
,
whe e a,b,c∈R. We ob ain a h ee pa ame e amily s(a,b,c)o Lie algeb as wi h basis
1,..., 8,whe e 1=e1+σ(e1)and i=ei o i=2,...,8, and b acke s
[ 2, 1]= 3−b 2,[ 3, 1]= 4−(a+b) 3,[ 3, 2]= 5,
[ 4, 1]= 6−c 5−(2a+b) 4,[ 4, 2]= 7,[ 5, 1]= 7−(a+2b) 5,
[ 1, 6]=2c 7+(3a+b) 6,[ 1, 7]=c 8+2(a+b) 7,
[ 1, 8]=(a+3b) 8,[ 5, 2]= 8.
In pa icula , se ing a=b=0 gi es a one pa ame e amily o nilpo en Lie algeb as
s(c). We now ind he dis ibu ion-p ese ing di eomo phism  om S(c) o F24 when
c=1, as in Rema k 4.3. E e y poin in S(1)is o he o m expP(xi i). Following
[12], expP(xi i)=(EF24 (x1σ(e1);xiei), expGL(x1σ(e1)) ∈F24 GL(2,R),whe e
EF24 (x1e1;xiei)=γ(1)and γ:[0,1]→F24 is he solu ion o
γ( )=dLγ( )expGL( x1σ(e1))(xiei)
γ(0)=eF24 .
The image o his poin ia is going o be ha elemen p∈F24 such ha gQ =
expP(xi i)Q, i.e.,
expPxi i=EF24 (x1e1;xiei).
To compu e his, we i s obse e ha
:= expGL( x1σ(e1)) xiei
=x1,x2
1 +x2,x3,x4,x5+ x1x4,x6,x7+2 x1x6,x8+ x1x7+ 2x2
1x6.
Second, we need o compu e dLγ using he Bake –Campbell–Hausdo o mula:
dLγ =d
dhh=0
exp−1(exp(γ ) exp(h )) = +1
2[γ, ]+ 1
12[γ,[γ, ]].
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Geome iae Dedica a (2024) 218 :56 Page 19 o 22 56
The sys em o di e en ial equa ions ˙γ=dLγ ha we ob ain is
˙γ1=x1
˙γ2= x2
1+x2
˙γ3=−1
2 x2
1γ1−1
2x2γ1+1
2x1γ2+x3
˙γ4=1
12 x2
1γ2
1+1
12x2γ2
1−1
12(γ1γ2−6γ3)x1−1
2x3γ1+x4
˙γ5=1
12x2γ1γ2−1
12x1γ2
2+1
12x2
1γ1γ2+6x2
1γ3+12x1x4 −1
2x3γ2
+1
2x2γ3+x5
˙γ6=1
12x3γ2
1−1
12(γ1γ3−6γ4)x1−1
2x4γ1+x6
˙γ7=1
6x3γ1γ2−1
12x2γ1γ3−1
12 x2
1γ1γ3+6x1x4γ1−6x2
1γ4−24x1x6
−1
12 (γ2γ3−6γ5)x1−1
2x5γ1−1
2x4γ2+1
2x2γ4+x7
˙γ8= 2x2
1x6+1
12x3γ2
2−1
12x2γ2γ3−1
12 x2
1γ2γ3+6x1x4γ2−6x2
1γ5−12x1x7
−1
2x5γ2+1
2x2γ5+x8.
Thi d, we need o in eg a e his sys em o ODEs wi h ini ial condi ions γi(0)=0 o
e e y i=1,...,8. The solu ion is
γ1( )= x1
γ2( )=1
2 2x2
1+ x2
γ3( )=−1
12 3x3
1+ x3
γ4( )= x4
γ5( )=− 1
240 5x5
1+1
12 3x2
1x3+1
2 2x1x4+ x5
γ6( )=1
720 5x5
1+ x6
γ7( )=1
720 6x6
1+1
360 5x4
1x2+ 2x1x6+ x7
γ8( )=1
5040 7x7
1+1
720 6x5
1x2+1
720 x3
1x2
2+3x4
1x3 5
−1
12 x1x2x4−x2
1x5−4x2
1x6 3+1
2 2x1x7+ x8.
The e o e, he mapping om S(1) o Gis :expP(xi i)→ γ(1), which is a global,
su jec i e smoo h dis ibu ion-p ese ing di eomo phism.
Finally, S(1)is no isomo phic o F2,4because S(1)has nilpo ency s ep 5 ins ead o 4, as
one can easily see om he exp ession o he Lie b acke s in s(1).
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56 Page 20 o 22 Geome iae Dedica a (2024) 218 :56
Rema k 5.6 The mapping om S(1) o Gdesc ibed abo e is in pa icula bi-Lipschi z on
e e y compac se , when he g oups a e endowed wi h le -in a ian sub-Riemannian dis-
ances. No ice, howe e , ha his is no a global quasicon o mal mapping.
5.3 Modi ica ions o ul a- igid s a i ied g oups
A s a i ied Lie algeb a gis called ul a- igid i he only au omo phisms o gp ese ing he
s a i ica ions a e dila ions, see [10]. In pa icula , he ull Tanaka p olonga ion o such gis
p=gR, as semi-di ec p oduc o Liealgeb as. In his sec ion wedesc ibe all modi ica ions
in gRand hei equi alence ela ion. Many esul s do no need he assump ion o gbeing
ul a- igid, so we assume his hypo hesis only when needed.
Le g=−1
j=−sgjbe a s a i ied Lie algeb a. Le D:g→gbe he linea map wi h
D =j o ∈g−j. No ice ha Dis a de i a ion o g ha p ese es he laye s and ha
δe =e D :g→ga e he dila ions.
The semi-di ec p oduc p:= gRis he Lie algeb a whose Lie b acke s a e
[(0,a), (Y,0)]=(aDY,0)hence [(X,a), (Y,b)]=([X,Y]+aDY −bDX,0).
P oposi ion 5.7 Le σ:g→Rbe a linea map and se s:= {(X,σX):X∈g}. The ec o
space sis a Lie subalgeb a o gRi and only i −s
j=−2gj⊂ke σ.
P oo Fi s , we no e ha sis a Lie algeb a i and only i , o all X,Y∈g,
σ([X,Y])+(σ X)(σ DY)−(σY)(σ DX)=0.(7)
⇒Suppose sis a Lie algeb a, i.e., (7) holds o all X,Y∈g. We p o e −s
j=−2gj⊂
ke σby induc ion on j.I X,Y∈g−1, henDX =Xand DY =Y, hus (7) implies
σ([X,Y])=0. Since g−2=[g−1,g−1], i ollows ha g−2⊂ke σ. Now, suppose ha
g−k⊂ke σ o k≥2. I X∈g−1and Y∈g−k, hen(7) implies ha σ([X,Y])=0. Since
g−k−1=[g−1,g−k], i ollows ha g−k−1⊂ke σ. We conclude ha −s
j=−2gj⊂ke σ.
⇐Suppose −s
j=−2gj⊂ke σ. By he bilinea i y o he exp ession, we need o show
ha (7) holds only when X∈giand Y∈gj o some iand j.Sinceσis non-ze o only on he
i s laye , he only non- i ial ins ance o (7)is o X,Y∈g−1. In his case, σ([X,Y])=0,
and (σ X)(σ DY)−(σY)(σ DX)=(σ X)(σY)−(σ Y)(σ X)=0. The e o e, (7) is sa is ied
and sis a Lie algeb a.
Lemma 5.8 The Lie algeb a au omo phisms φ:p→psuch ha φ({0}×R)={0}×R
and φ(g−1×R)=g−1×Ra e exac ly hose o he o m φ(X,a)=(φ1X,a) o some Lie
algeb a au omo phism φ1:g→g ha p ese es he laye s.
P oo On he one hand, i φ1:g→gis a Lie algeb a au omo phism ha p ese es he
laye s, hen φ(X,a)=(φ1X,a)is clea ly a Lie algeb a au omo phism φ:p→pwi h
φ({0}×R)={0}×Rand φ(g−1×R)=g−1×R, because φ1D=Dφ1.
On he o he hand, i φ:p→pis a Lie algeb a au omo phism, hen φ(g×{0})=g×{0}
because g×{0}=[p,p]. Suppose also ha φ({0}×R)={0}×Rand φ(g−1×R)=g−1×R.
Then φ(X,a)=φ(X,0)+φ(0,a)=(φ1(X), 0)+(0,φ
2(a)) and φ1(g−1)=g−1.This
implies ha φ1(gj)=gj o all j, as one can p o e by induc ion on j. No ice ha , o all
X∈gand all a∈R,
φ2(a)Dφ1X=[(0,φ
2(a)), (φ1X,0)]=φ([(0,a), (X,0)])=φ(aDX,0)=aφ1DX.
Fo e e y X∈g−1,DX =Xand Dφ1X=φ1X, hence φ2(a)φ1X=aφ1X, i.e., φ2(a)=a.
123

Geome iae Dedica a (2024) 218 :56 Page 21 o 22 56
Theo em 5.9 Suppose ha gis ul a- igid, i.e., p=gRis i s ull Tanaka p olonga ion.
The se o all non-isomo phic modi ica ions o gis pa ame ized by g∗
−1/R>0. Mo eo e , all
modi ica ions o gin pa e sol able and he only nilpo en one is gi sel .
P oo The se o all modi ica ions o gin pcan be iden i ied wi h g∗
−1by P oposi ion 5.7,
whe e σ∈g∗
−1is iden i ied wi h σ(j j)=σ( −1) o j j∈gand he modi ica ion
sσ:= {(X,σX):X∈g}⊂p.Sincegis igid, by Theo em 4.6 wo modi ica ions σ, τ ∈
g∗
−1a e isomo phic i and only i he e is a Lie algeb a au omo phism φ:p→pwi h
φ({0}×R)={0}×Rand φ(g−1×R)=g−1×Rsuch ha φ(sσ)=sτ. The e o e, by
Lemma 5.8, wo modi ica ions σ, τ ∈g∗
−1a e isomo phic i and only i he e is a Lie algeb a
au omo phism φ1:g→gsuch ha , o all X∈g,
(φ1X,σX)=(φ1X,τφ
1X),
i.e., σX=τφ1X o all X∈g−1.Now,sincegis ul a igid, φ1=δλ o some λ>0.
The e o e, wo modi ica ions σ, τ ∈g∗
−1a e isomo phic i and only i he e is λ>0such
ha σ=λτ.
Finally, no ice ha all modi ica ions o gin pa e sol able, because pi sel is sol able.
Mo eo e , he only nilpo en modi ica ion is gi sel . Indeed, i s= g, hen he e is X∈g−1
wi h σX= 0, so ha , i Y∈g−sis nonze o, hen [(X,σX), (Y,0)]=sσX(Y,0),whe e
sis he s ep o g. The e o e, we ob ain ha (Y,0)∈[s,[...,[s,s]...]] o any o de o
b acke s, ha is, sis no nilpo en .
Au ho Con ibu ions Sebas ianoNicolussi GoloandAlessand oO azziwe eequallyin ol edin he esea ch
ha led o his manusc ip and in w i ing i .
Funding Open Access unding enabled and o ganized by CAUL and i s Membe Ins i u ions.
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Re e ences
1. Ag ache , A., Ba ila i, D.: Sub-Riemannian s uc u es on 3D Lie g oups. J. Dyn. Con ol Sys . 18(1),
21–44 (2012)
2. Baudoin, F., Cecil, M.: The subellip ic hea ke nel on he h ee-dimensional sol able Lie g oups. Fo um
Ma h. 27(4), 2051–2086 (2015)
3. Cowling, M., e al.: Con ac and con o mal maps on Iwasawa Ng oups. In: A i Accad. Naz. Lincei
Cl. Sci. Fis. Ma . Na u . Rend. Lincei (9) Ma . Appl. 13.3-4 (2002). Ha monic analysis on complex
homogeneous domains and Lie g oups (Rome, 2001), pp. 219–232
4. Cowling, M.G., e al.: F om homogeneous me ic spaces o Lie g oups. a Xi :1705.09648 e-p in s (2017)
5. Go don,C.S.,Wilson,E.N.:Isome yg oupso Riemanniansol mani olds.T ans.Am.Ma h.Soc.307(1),
245–269 (1988)
123
56 Page 22 o 22 Geome iae Dedica a (2024) 218 :56
6. Jean, F.: Con ol o Nonholonomic Sys ems: F om Sub-Riemannian Geome y o Mo ion Planning.
Sp inge B ie s in Ma hema ics. Sp inge , Cham, (2014), pp. x+104
7. Ki ioja, V., Le Donne, E.: Isome ies o nilpo en me ic g oups. J. É c. Poly ech. Ma h. 4, 473–482 (2017)
8. Ko ányi, A., Reimann, H.M.: Quasicon o mal mappings on he Heisenbe g g oup. In en . Ma h. 80(2),
309–338 (1985)
9. Le Donne, E., O azzi, A.: Isome ies o Ca no g oups and sub-Finsle homogeneous mani olds. J. Geom.
Anal. 26(1), 330–345 (2016)
10. Le Donne, E., O azzi, A., Wa hu s , B.: Ul a igid angen s o sub-Riemannian nilpo en g oups. Ann.
Ins . Fou ie (G enoble) 64(6), 2265–2282 (2014)
11. Le Donne, E., e al.: Sa d p ope y o he endpoin map on some Ca no g oups. Ann. Ins . H. Poinca é
Anal. Non Lin é ai e 33(6), 1639–1666 (2016)
12. Moskowi z, M., Sacks ede , R.: The exponen ial map and di e en ial equa ions on eal Lie g oups. J. Lie
Theo y 13(1), 291–306 (2003)
13. O azzi, A., Wa hu s , B.: Con ac and 1-quasicon o mal maps on Ca no g oups. J. Lie Theo y 21(4),
787–811 (2011)
14. Pansu, P.: Mé iques de Ca no -Ca a héodo y e quasiisomé ies des espaces symé iques de ang un. Ann.
Ma h. (2) 129(1), 1–60 (1989)
15. Tanaka, N.: On di e en ial sys ems, g aded Lie algeb as and pseudog oups. J. Ma h. Kyo o Uni . 10,
1–82 (1970)
16. Šnobl, L., Win e ni z, P.: Classi ica ion and iden i ica ion o Lie algeb as. Vol. 33. CRM Monog aph
Se ies. Ame ican Ma hema ical Socie y, P o idence, RI (2014), pp. xii+306
17. Wa hu s , B.: Tanaka p olonga ion o ee Lie algeb as. Geom. Dedica a. 130, 59–69 (2007)
18. Yamaguchi, K.: Di e en ial sys ems associa ed wi h simple g aded Lie algeb as’. In: P og ess in Di e -
en ial Geome y. Vol. 22. Ad . S ud. Pu e Ma h. Ma h. Soc. Japan, Tokyo, pp. 413–494 (1993)
Publishe ’s No e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps and
ins i u ional a ilia ions.
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