Bounded geometry and leaves

BOUNDED GEOMETRY AND LEAVES
JES´
US A. ´
ALVAREZ L ´
OPEZ AND RAM´
ON BARRAL LIJ´
O
Abs ac . The main heo em s a es ha any comple e connec ed Riemannian mani old o bounded ge-
ome y can be isome ically ealized as a lea wi h i ial holonomy in a compac Riemannian olia ed
space.
Con en s
1. In oduc ion 1
2. P elimina ies 4
3. (Pa ial) quasi-equi alences 6
4. The C∞ opology on b
M∗(n) 7
5. Folia ed s uc u e o b
M∞
∗,imm(n) 11
6. Uni e sali y 17
7. Realiza ion o mani olds o bounded geome y as lea es 18
8. Open p oblems 20
Re e ences 20
1. In oduc ion
Recall ha a olia ed space X≡(X, F) o dimension nis a opological space Xequipped wi h a pa i ion
Fin o connec ed mani olds (lea es) so ha Xcan be locally desc ibed as a p oduc B×Z, whe e Bis an
open ball in Rnand Zany opological space (local ans e sal), and he slices B× {∗} co espond o open
se s in he lea es. This Fis called a olia ed s uc u e o lamina ion. Folia ed spaces a e usually assumed o
be Polish o ge be e p ope ies. Many basic no ions abou olia ions can be ob iously ex ended o olia ed
spaces, like olia ed cha s, plaques, olia ed a las, holonomy pseudog oup, holonomy g oup and holonomy
co e ing o he lea es, minimali y, ansi i i y, olia ed maps, e c. Some basic esul s can be ex ended as
well; o ins ance, he e is an ob ious e sion o he Reeb local s abili y heo em, and he union o lea es
wi hou holonomy is a meage subse i Xis second coun able. In e es ing classes o olia ed spaces show
up in se e al a eas o ma hema ics, like in dynamics, a i hme ics, essella ions, g aphs and olia ion heo y
(minimal se s).
AC∞ olia ed s uc u e is gi en by a olia ed a las whose changes o coo dina es a e lea wise C∞, wi h
ambien -space-con inuous lea wise de i a i es o a bi a y o de . This gi es ise o he concep o C∞ olia ed
space. To emphasize he di e ence, he olia ed s uc u e unde lying a C∞ olia ed s uc u e may be called
opological. On a C∞ olia ed space X≡(X, F), he concep o C∞ unc ion is de ined by equi ing ha i s
local exp essions, using olia ed coo dina es, a e lea wise C∞, wi h ambien -space-con inuous lea wise pa ial
de i a i es o a bi a y o de . C∞bundles and sec ions also make sense on X, de ined by equi ing ha
hei local desc ip ions a e gi en by C∞ unc ions in he abo e sense. Fo ins ance, he angen bundle TX
(o TF) is he C∞ ec o bundle on X ha consis s o he ec o s angen o he lea es, and a Riemannian
me ic on Xconsis s o Riemannian me ics on he lea es i ing oge he nicely o o m a C∞sec ion on
X. This gi es ise o he concep o Riemannian olia ed space.
1991 Ma hema ics Subjec Classi ica ion. 57R30; 53C12.
Key wo ds and ph ases. C∞con e gence o Riemannian mani olds; bounded geome y; Riemannian olia ed space.
The i s au ho is pa ially suppo ed by MICINN (Spain), g an MTM2011-25656.
1
C∞ olia ed maps be ween C∞ olia ed spaces can be simila ly de ined; in pa icula , C∞ olia ed im-
me sions, subme sions, (local) di eomo phisms and (local) embeddings be ween C∞ olia ed spaces ha e
ob ious meanings. I a homeomo phism be ween C∞ olia ed spaces is C∞and i s es ic ions o he lea es
a e di eomo phisms, hen i is a C∞di eomo phism, as ollows easily om he con inui y o he in e sion o
C∞di eomo phisms be ween C∞mani olds wi h espec o he C∞ opology [18, p. 64, Exe cise 9]. Se e al
esul s abou olia ed spaces ha e ob ious C∞ e sions, like he Reeb local s abili y heo em.
S anda d e e ences abou olia ed spaces a e [23], [4, Chap e 11], [5, Pa 1] and [13]. See also [1,
Sec ion 2.1] o a quick summa y o wha is needed he e.
On he o he hand, ecall ha a Riemannian mani old Mis said o be o bounded geome y when i has a
posi i e injec i i y adius, and he m- h co a ian de i a i e o he cu a u e enso has uni o mly bounded
no m o all o de m; in pa icula , Mis comple e by he posi i i y o he injec i i y adius. The ollowing
a e ypical examples whe e bounded geome y holds: co e ings o closed connec ed Riemannian mani olds,
connec ed Lie g oups wi h le in a ian me ics, and lea es o compac Riemannian olia ed spaces. Mo e
examples can be p oduced by using compac ly suppo ed pe u ba ions o gi en Riemannian mani olds o
bounded geome y. In ac , any smoo h mani old admi s a me ic o bounded geome y [14]. We will ocus
in he case o lea es o compac Riemannian olia ed spaces, showing ha his example indeed cha ac e izes
bounded geome y.
Theo em 1.1. Any connec ed Riemannian mani old o bounded geome y is isome ic o a lea wi h i ial
holonomy o some compac Riemannian olia ed space.
I is commonly accep ed ha such a esul should be ue, and ha i should ollow by using he closu e o
he canonical embedding o he mani old in o he G omo space M∗o poin ed p ope me ic spaces [15], [16,
Chap e 3], o , be e , in o i s smoo h e sion, he space M∞
∗(n) o isome y classes o poin ed comple e
connec ed Riemannian n-mani olds wi h he opology de ined by he C∞con e gence [24, Chap e 10,
Sec ion 3.2], [1, Theo em 1.2]. Howe e , o he au ho s knowledge, no comple e p oo has been gi en so a .
A comple e connec ed Riemannian n-mani old Mis called non-pe iodic ( espec i ely, locally non-pe iodic)
i Iso(M) = {idM}( espec i ely, he canonical p ojec ion M→Iso(M) Mis a co e ing map), whe e Iso(M)
deno es he isome y g oup o M. The non-pe iodic and locally non-pe iodic mani olds de ine subspaces o
M∞
∗(n) espec i ely deno ed by M∞
∗,np(n) and M∞
∗,lnp(n). The e is a canonical map ιM:M→M∞
∗(n), gi en
by ιM(x)=[M, x] ( he isome y class o (M, x)), which induces a con inuous injec ion ¯ιM: Iso(M) M→
M∞
∗(n). The images o all possible maps ιM o m a pa i ion F∗(n) o M∞
∗(n). The es ic ion o F∗(n)
o M∞
∗,lnp(n) is deno ed by F∗,lnp(n). Fo n≥2, M∞
∗,lnp(n) is open and dense in M∞
∗(n), and F∗,lnp(n) is a
Riemannian olia ed space o dimension nso ha each map ιM:M→im ιMis a local isome y and he
holonomy co e ing o he lea im ιM[1, Theo em 1.3]; in pa icula , M∞
∗,np(n) is he union o lea es wi h
i ial holonomy. Mo eo e Cl∞(im ιM) is compac i and only i Mis o bounded geome y [1, Theo em 12.3]
(see also [8], [24, Chap e 10, Sec ions 3 and 4]), whe e Cl∞deno es he closu e ope a o in M∞
∗(n). Then,
analyzing he cases whe e Cl∞(im ιM)⊂M∗,lnp(n), a e sion o Theo em 1.1 ollows assuming es ic ions
on M[1, Theo em 1.5].
To p o e Theo em 1.1 wi h comple e gene ali y, we e ine he abo e a gumen s as ollows. Fix a sepa able
Hilbe space Eand any na u al1n. Conside pai s (M, ) and iples (M, , x), whe e Mis a comple e
connec ed Riemannian n-mani old, ∈C∞(M, E) and x∈M. An equi alence φ: (M, )→(N, h) is an
isome y φ:M→Nsuch ha φ∗h= . I mo eo e dis inguished poin s, x∈Mand y∈N, a e p ese ed,
hen φ: (M, , x)→(N, h, y) is called a poin ed equi alence. The g oup o sel equi alences o (M, ) is
deno ed by Iso(M, ). I he e is a poin ed equi alence (M, , x)→(N, h, y), hen he iples (M, , x) and
(N, h, y) a e decla ed o be equi alen . The equi alence class o each (M, , x) is deno ed by [M, , x]. Le
b
M∗(n) deno e he se 2o such equi alence classes.
De ini ion 1.2. Fo each m∈N, a sequence [Mi, i, xi] in b
M∗(n) is said o be Cmcon e gen o [M, , x]∈
b
M∗(n) i , o each compac domain3Ω⊂Mcon aining x, he e is a poin ed Cm+1 embedding φi: (Ω, x)→
1I is assumed ha 0 is na u al.
2Like in he cases o M∗and M∞
∗(n), wi hou loss o gene ali y, i can be assumed ha he unde lying se o any such Mis
con ained in R, so ha b
M∗(n) becomes a well de ined se .
3He e, a domain in Mis a connec ed C∞submani od, possibly wi h bounda y, o he same dimension as M.
2
(Mi, xi) o each la ge enough isuch ha φ∗
igi→g|Ωand φ∗
i i→ |Ωas i→ ∞ wi h espec o he Cm
opology [18, Chap e 2]. I [Mi, i, xi] is Cmcon e gen o [M, , x] o all m, hen i is said ha [Mi, i, xi]
is C∞con e gen o [M, , x].
I is no comple ely ob ious ha his C∞con e gence sa is ies he condi ions o de ine a opology [21], [17].
Thus he ollowing esul is no i ial.
Theo em 1.3. The C∞con e gence in b
M∗(n)desc ibes a Polish opology.
The opology gi en by Theo em 1.3 will be called he C∞ opology, and he co esponding space is deno ed
by b
M∞
∗(n). The closu e ope a o in his space will be deno ed by c
Cl∞. The ollowing maps a e canonical and
con inuous: a o ge ul map b
M∞
∗(n)→M∞
∗(n), [M, , x]7→ [M, x], and an e alua ion map e : b
M∞
∗(n)→E,
[M, , x]7→ (x). No e ha e : b
M∗(0) →Eis a homeomo phism. Mo eo e , o each comple e connec ed
Riemannian n-mani old Mand any ∈C∞(M, E), he e is a canonical con inuous map ˆιM, :M→b
M∞
∗(n),
gi en by ˆιM, (x) = [M, , x], which induces a con inuous injec ion ¯ιM, : Iso(M, ) M→M∞
∗(n). The
images o he maps ˆιM, o m a na u al pa i ion o b
M∞
∗(n), deno ed by b
F∗(n). Le C∞
imm(M, E) be he se
o C∞imme sions M→E, and le b
M∞
∗,imm(n) be he b
F∗(n)-sa u a ed subspace o b
M∗(n) consis ing o classes
[M, , x] wi h ∈C∞
imm(M, E). The es ic ion o b
F∗(n) o b
M∗,imm(n) is deno ed by b
F∗,imm(n). Obse e
ha he canonical p ojec ion M→Iso(M, ) Mis a co e ing map i ∈C∞
imm(M, E).
On he o he hand, le b
M∞
∗,c(n) ( espec i ely, b
M∞
∗,o(n)) be he b
F∗(n)-sa u a ed subspace o b
M∗(n) con-
sis ing o classes [M, , x] such ha Mis compac ( espec i ely, open). Obse e ha , i [N, h, y] is close
enough o any [M, , x]∈b
M∞
∗,c(n), hen Nis di eomo phic o M. Thus b
M∞
∗,c(n) is open in b
M∗(n), and
he e o e b
M∞
∗,o(n) is closed. Hence hese a e Polish subspaces o b
M∗(n), as well as hei in e sec ions wi h
any Polish subspace. Le b
M∞
∗,imm,c/o(n) = b
M∞
∗,c/o(n)∩b
M∞
∗,imm(n). The es ic ions o b
F∗(n) o b
M∗,c/o(n)
and b
M∗,imm,c/o(n) a e deno ed by b
F∗,c/o(n) and b
F∗,imm,c/o(n), espec i ely.
Theo em 1.4. The ollowing p ope ies hold:
(i)b
M∞
∗,imm(n)is Polish and dense in b
M∞
∗(n).
(ii)b
F∗,imm(n)is a olia ed s uc u e o dimension n.
(iii)b
F∗,imm,o(n)is ansi i e.
(i )The e is a unique C∞ olia ed s uc u e b
F∞
∗,imm(n)on b
M∞
∗,imm(n), whose unde lying opological olia ed
s uc u e is b
F∗,imm(n), such ha e : b
M∞
∗,imm(n)→Eis a C∞imme sion.
( )The e is a unique Riemannian me ic on b
M∞
∗,imm(n)≡(b
M∞
∗,imm(n),b
F∞
∗,imm(n)) such ha ιM, :M→
ˆιM, is a local isome y o all comple e connec ed Riemannian n-mani old Mand ∈C∞
imm(M, E).
( i)Fo all Mand as abo e, he map ˆιM, :M→im ˆιM, is he holonomy co e ing o he lea im ˆιM, .
I is possible o gi e a e sion o Theo em 1.4 close o [1, Theo em 1.3], using he subspace b
M∞
∗,lnp(n)
consis ing o he classes [M, , x] such ha M→Iso(M, ) Mis a co e ing map. Such a esul could be
p o ed wi h he ob ious adap a ion o he p oo o [1, Theo em 1.3], using he exponen ial map o de ine
olia ed cha s. Ins ead, we ha e op ed o s udying b
M∞
∗,imm(n) because, in his case, he imme sions
di ec ly p o ide olia ed cha s.
The ollowing esul s a es ha b
M∞
∗,imm(n) is uni e sal among he class o Polish Riemannian olia ed
spaces ha sa is y a condi ion called co e ing-con inui y (De ini ion 6.1).
Theo em 1.5. A Polish Riemannian olia ed space Xo dimension nwi h comple e lea es is isome ic o
a sa u a ed Riemannian olia ed subspace o b
M∞
∗,imm(n)i and only i Xis co e ing-con inuous.
In Theo em 1.5, when Xconsis s o a single lea M, he isome ic injec ion o Min o b
M∞
∗,imm(n) is ˆιM,
o any C∞embedding :M→E. I mo eo e Mis o bounded geome y, hen can be chosen so ha
c
Cl∞(imˆιM, ) is a compac Riemannian olia ed subspace o b
M∞
∗,imm(n) (P oposi ion 7.1). Then Theo em 1.1
ollows by conside ing he isome ic injec ion ˆιM, :M→c
Cl∞(imˆιM, ).
The e a e examples o Lie g oups wi h le in a ian me ics ha a e no coa sely quasi-isome ic o any
ini ely gene a ed g oup [7], [11]. Applying he abo e a gumen o hose Riemannian mani olds, we ge
3
compac Riemannian olia ed spaces whose lea holonomy co e s a e no coa sely quasi-isome ic o any
ini ely gene a ed g oup.
Theo em 1.1 con as s wi h he examples o connec ed Riemannian mani olds o bounded geome y whose
quasi-isome y ype canno be ealized as lea es o olia ions o codimension one on closed mani olds [2], [37],
[30], [31]. I he me ic is no conside ed, any su ace can be ealized as a lea o a codimension one olia ion on
a closed mani old [6], bu his ails in highe dimension [12], [19], [2], [35], [32]. The s udy o his ealizabili y
p oblem was ini ia ed in [34].
This wo k can be conside ed as a con inua ion o [1], and he e o e many e e ences o [1] a e included.
2. P elimina ies
Le Mbe a Riemannian mani old (possibly wi h bounda y o co ne s). The ollowing s anda d no a ion
will be used. The me ic enso is deno ed by g, he dis ance unc ion on each o he connec ed componen s
o Mby d, he angen bundle by π:TM →M, he Le i-Ci i a connec ion by ∇, and he open and closed
balls o cen e x∈Mand adius > 0 by B(x, ) and B(x, ), espec i ely. I needed, “M” will be added
o all o he abo e no a ion as a subindex o supe index; when a amily o Riemannian mani olds Miis
conside ed, we may add he subindex o supe index “i” ins ead o “Mi”. A co e ing space o Mis assumed
o be equipped wi h he li o g.
Fo m∈Z+, le T(m)M=T· · · TM (m imes); we also se T(0)M=M. I l < m,T(l)Mis iden i ied wi h
a egula submani old o T(m)M ia ze o sec ions, and he e o e, o each x∈M, he no a ion xmay be also
used o he ze o elemen s o TxM,TxTM, e c. Le π:T(m)M→T(l)Mbe he ec o bundle p ojec ion
gi en by composing he angen bundle p ojec ions; in pa icula , we ha e π:T(m)M→M. Gi en any Cm
map be ween Riemannian mani olds, φ:M→N, he induced map T(m)M→T(m)Nwill be deno ed by
φ(m)
∗(o simply φ∗i m= 1).
Hilbe mani olds a e also conside ed in some pa s o he pape , using analogous no a ion.
The Le i-Ci i a connec ion de e mines a decomposi ion T(2)M=H⊕V, as di ec sum o he ho izon al
and e ical subbundles. The Sasaki me ic on T M is he unique Riemannian me ic g(1) so ha H⊥Vand
he canonical iden i ies Hξ≡TξM≡Vξa e isome ies o e e y ξ∈TM [27]. Con inuing by induc ion, o
m≥2, he Sasaki me ic on T(m)Mis g(m)= (g(m−1))(1). The no a ion d(m)is used o he co esponding
dis ance unc ion on he connec ed componen s, and he co esponding open and closed balls o cen e
ξ∈T(m)Mand adius > 0 a e deno ed by B(m)(ξ, ) and B(m)(ξ, ), espec i ely. We may add he
subindex “M” o his no a ion i necessa y, o he subindex “i” ins ead o “Mi” o a amily o Riemannian
mani olds Mi. F om now on, T(m)Mis assumed o be equipped wi h g(m). Fo l < m,T(l)Mbecomes a
o ally geodesic Riemannian submani old o T(m)Mo hogonal o he ibe s o π:T(m)M→T(l)M, which
a e also o ally geodesic [1, Rema k 1-(i)–(iii)] (see also [27, Co olla y o Theo em 13, and Theo ems 14
and 18]).
Le (U;x1, . . . , xn) be a cha o M. As usual, he co esponding me ic coe icien s a e deno ed by gij,
and w i e (gij) = (gij)−1. Iden i y he unc ions xiwi h hei li s o TU. We ge a cha (U(1);x1
(1), . . . , x2n
(1))
o TM wi h U(1) =T U,xi
(1) =xiand xn+i
(1) = i o 1 ≤i≤n, whe e he unc ions igi e he coo dina es
o angen ec o s wi h espec o he local ame (∂1, . . . , ∂n) o T U induced by (U;x1, . . . , xn). By induc ion,
o m≥2, le (U(m);x1
(m), . . . , x2mn
(m)) be he cha o T(m)Minduced by he cha (U(m−1);x1
(m−1), . . . , x2m−1n
(m−1) )
o T(m−1)M.
Le Ω ⊂Mbe a compac domain and m∈N. Fix a ini e collec ion o cha s o M ha co e s Ω,
U={(Ua;x1
a, . . . , xn
a)}, and a amily o compac subse s o Mwi h he same index se as U,K={Ka}, such
ha Ω ⊂SaKa, and Ka⊂Ua o all a. The co esponding Cmno m o a Cm enso Ton Ω is de ined by4
kTkCm,Ω,U,K= max
amax
x∈Ka∩ΩX
|I|≤mX
J,K 
∂|I|TK
a,J
∂xI
a
(x),
whe e TK
a,J a e he coe icien s o Ton Ua∩Ω wi h espec o he ame induced by (Ua;x1
a, . . . , xn
a). Wi h
his no m, he Cm enso s on Ω o a ixed ype o m a Banach space, whose unde lying opology is called
4The s anda d mul i-index no a ion is used he e.
4
he Cm opology. By aking he p ojec i e limi as m→ ∞, we ge he F ´eche space o C∞ enso s o ha
ype, whose unde lying opology is called he C∞ opology (see e.g. [18]). We will always conside he Ck
opology o Ck enso s on Ω o a gi en ype (k∈N∪ {∞}); in pa icula , Ck(Ω) is always assumed o be
equipped wi h he Ck opology. Obse e ha Uand Ka e also quali ied o de ine he no m k kCm,Ω0,U,K
o any compac subdomain Ω0⊂Ω. I is well known ha k kCm,Ω,U,Kis equi alen o he no m k kCm,Ω,g
de ined by
kTkCm,Ω,g = max
0≤l≤mmax
x∈Ω|∇lT(x)|;
i.e., he e is some C≥1, depending on M, Ω, U,K,gand m, such ha
1
Ck kCm,Ω,U,K≤ k kCm,Ω,g ≤Ck kCm,Ω,U,K.(1)
In pa icula , o m= 0 and ∈C∞(M),
k kΩ:= k kC0,Ω,U,K=k kC0,Ω,g = max
x∈Ω| (x)|,(2)
which is independen o he choices U,Kand g.
The no ms k kCm,Ω,U,Kand k kCm,Ω,g ha e s aigh o wa d ex ensions o enso s wi h alues in a sepa able
Hilbe space E, and sa is y he ob ious e sions o (1) and (2), and Ck(M, E) is assumed o be equipped
wi h he Ck opology (k∈N∪ {∞}).
Fo ∈C∞(M, E), ecall ha ∇ =d (i s de Rham di e en ial). Fo each m, he map
(m)
∗≡ (m),1
∗, . . . , (m),2m
∗:T(m)M→T(m)E≡E2m
is also C∞and wi h alues in a sepa able Hilbe space. In he ollowing lemma, we conside he local ep e-
sen a ions o and e e y (m),λ
∗wi h espec o coo dina e sys ems (U, x1, . . . , xn) and (U(m), x1
(m), . . . , x2mn
(m))
o Mand T(m)M. Mo eo e each unc ion on Mo Uis iden i ied wi h i s li o T(m)Mo U(m).
Lemma 2.1. The ollowing p ope ies hold:
(i)The local ep esen a ion o e e y (m),λ
∗is a uni e sal polynomial exp ession o xn+1
(m), . . . , x2mn
(m)and he
pa ial de i a i es up o o de mo he local ep esen a ion o .
(ii)Fo each ρ > 0, he pa ial de i a i es up o o de mo he local ep esen a ion o a e gi en by
uni e sal linea exp essions o he unc ions (σ(m)
ρ,µ )∗ (m),λ
∗ o n+1 ≤µ≤2mn, whe e σ(m)
ρ,µ :U→U(m)
is he sec ion o π:U(m)→Ude e mined by5(σ(m)
ρ,µ )∗xν
(m)=ρδµν o n+ 1 ≤ν≤2mn.
P oo . By using induc ion on m, he esul clea ly boils down o he case m= 1. Bu , in his case, he
s a emen ollows because ∗≡( , d ) : TM →TE≡E2.
By using he sup emum on Ω ins ead o he maximum, he de ini ion o k kCm,Ω,g can be ex ended o
any non-compac n-submani old Ω ⊂M(including Ω = M), wi h possible in ini e alues. The enso s on
Ω wi h ini e no m k kCm,Ω,g a e said o be uni o mly Cm, o Cm
b. Fo a gi en ype, hey o m a Banach
space, and he co esponding p ojec i e limi as m→ ∞ is a F ´eche space, whose elemen s a e said o be
uni o mly C∞, o C∞
b(see e.g. [26, De ini ion 2.7] o [28, De ini ion 3.15]). In pa icula , his gi es ise o
he F ´eche spaces C∞
b(Ω) and C∞
b(Ω,E) when R- alued and E- alued C∞
b unc ions a e conside ed.
Le Nbe ano he Riemannian mani old. Recall ha a C1map φ:M→Nis called a (λ-) quasi-isome y,
o (λ-) quasi-isome ic, i he e is some λ≥1 such ha 1
λ|ξ|≤|φ∗(ξ)| ≤ λ|ξ| o e e y ξ∈T M; in pa icula ,
φis an imme sion. To de ine highe o de quasi-isome ies, le T≤ M={ξ∈TM | |ξ| ≤ } o each > 0. I
Mhas no bounda y, hen T≤ Mis a mani old wi h bounda y; o he wise, i is a mani old wi h co ne s. Also,
de ine T(m),≤ Mby induc ion on m∈Z+, se ing T(1),≤ M=T≤ Mand T(m),≤ M=T≤ T(m−1),≤ M.
I is said ha φ:M→Nis a (λ-) quasi-isome y o o de m∈N, o a (λ-) quasi-isome ic map o o de
m, i i is Cm+1 and φ(m)
∗:T(m),≤1M→T(m)Nis a (λ-) quasi-isome y. I φis a quasi-isome y o o de
m o all m∈N, hen i is called a quasi-isome y o o de ∞. I he e is a quasi-isome ic di eomo phism
M→No o de m∈N∪ {∞}, hen Mand Na e said o be quasi-isome ic wi h o de m. The p ope y
o being a quasi-isome y o o de mis p ese ed by he ope a ions o composi ion o maps and in e sion o
5K onecke ’s del a is used he e.
5

di eomo phisms [1, P oposi ion 3.9], and he e o e i induces an equi alence ela ion be ween Riemannian
mani olds.
Fo m∈N, a pa ial map φ:MNis called a Cmlocal di eomo phism6i dom φand im φa e
open in Mand N, espec i ely, and φ: dom φ→im φis a Cmdi eomo phism. I mo eo e φ(x) = y o
dis inguished poin s, x∈dom φand y∈im φ, hen i is said ha φ: (M, x)(N, y) is a poin ed Cmlocal
di eomo phism. Fo m∈N,R > 0 and λ≥1, a Cm+1 poin ed local di eomo phism φ: (M, x)(N, y) is
called an (m, R, λ)-poin ed local quasi-isome y, o a local quasi-isome y o ype (m, R, λ), i he es ic ion
φ(m)
∗: Ω(m)→T(m)Nis a λ-quasi-isome y o some compac domain Ω(m)⊂dom φ(m)
∗wi h B(m)
M(x, R)⊂
Ω(m)[1, De ini ion 4.2].
3. (Pa ial) quasi-equi alences
Le Mand Nbe Riemannian n-mani olds, le ∈C∞(M, E) and h∈C∞(N, E), and le x∈Mand
y∈N. Recall om Sec ion 1 he concep s o an equi alence (M, )→(N, h), and a poin ed equi alence
(M, , x)→(N, h, y). Obse e ha k (m)
∗kΩ(m)makes sense o any n-submani old Ω(m)⊂T(m)Mbecause
we conside (m)
∗:T(m)M→T(m)E≡E2m, wi h alues in a sepa able Hilbe space. No e also ha
(φ∗h)(m)
∗=h(m)
∗◦φ(m)
∗ o any Cmmap φ:M→N.
De ini ion 3.1. Le λ≥1 and ε≥0, and le φ:M→Nbe a C1map. I is said ha φ: (M, )→(N, h)
is a ((λ, ε)-) quasi-equi alence o o de m∈Ni i is Cm+1,φ(m)
∗:T(m),≤1M→T(m)Nis a (λ-) quasi-
isome y, and k (m)
∗−(φ∗h)(m)
∗kT(m)M≤ε. I mo eo e dis inguished poin s xand ya e p ese ed, hen
φ: (M, , x)→(N, h, y) is called a poin ed quasi-equi alence o o de m. I he e is a quasi-equi alence
(M, )→(N, h) ( espec i ely, (M, , x)→(N, h, y)), hen (M, ) and (N, h) ( espec i ely, (M, , x) and
(N, h, y)) a e called quasi-equi alen .
Rema k 1.(i) Any (λ, ε)-quasi-equi alence o o de m≥1 is a (λ, ε)-quasi-equi alence o o de m−1.
(ii) Fo in ege s 0 ≤m0≤m, i φis a (λ, ε)-quasi-equi alence o o de m, hen φ(m0)
∗is a (λ, ε)-quasi-
equi alence o o de m−m0.
Fo a submani old Ω ⊂Mand ∈C∞(M, E), he no a ion (Ω, ) is used o (Ω, |Ω).
P oposi ion 3.2. The ollowing p ope ies hold o any m∈N,λ, µ ≥1and ε, δ ≥0:
(i)The e is some ν≥1, depending on m,λand µ, such ha , i φ: (M, )→(N, h)is a (λ, ε)-
quasi-equi alence and ψ: (N, h)→(L, u)a(µ, δ)-quasi-equi alence, bo h o hem o o de m, hen
ψ◦φ: (M, )→(L, u)is a (ν, ε +δ)-quasi-equi alence o o de m.
(ii)The e a e some ν0≥1, depending on mand λ, such ha , i φ: (M, )→(N, h)is a (λ, ε)-quasi-
equi alence o o de mand a di eomo phism, hen φ−1: (N, h)→(M, )is a (ν0, ε)-quasi-equi alence
o o de m.
P oo . By [1, P oposi ion 3.9], we only ha e o check he condi ions on he E- alued unc ions. Thus (i)
ollows because, o each ξ∈T(m)M, we ha e
 (m)
∗(ξ)−((ψ◦φ)∗u)(m)
∗(ξ)
≤ (m)
∗(ξ)−(φ∗h)(m)
∗(ξ)+h(m)
∗φ(m)
∗(ξ)−(ψ∗u)(m)
∗φ(m)
∗(ξ)≤ε+δ .
Simila ly, (ii) ollows because, o each ζ∈T(m)N,
h(m)
∗(ζ)−((φ−1)∗ )(m)
∗(ζ)=(φ∗h)(m)
∗(φ−1)(m)
∗(ζ)− (m)
∗(φ−1)(m)
∗(ζ)≤ε . 
Co olla y 3.3. “Being quasi-equi alen wi h o de m” is an equi alence ela ion on he se s o pai s (M, )
and iples (M, , x).
Now, suppose ha Mand Na e connec ed, comple e and wi hou bounda y.
6The e m “Cmlocal di eomo ism” (m≥1) is also used in he s anda d sense, e e ing o any Cmmap M→Nwhose
angen map is an isomo phism a e e y poin o M. The con ex will always cla i y his ambigui y.
6
De ini ion 3.4. Fix m∈N,R > 0, λ≥1 and ε≥0. Le φ: (M, x)(N, y) be a Cm+1 poin ed local
di eomo phism, and le ∈C∞(M, E) and h∈C∞(N, E). I is said ha φ: (M, , x)(N, h, y) is an
(m, R, λ, ε)-poin ed local quasi-equi alence, o a local quasi-equi alence o ype (m, R, λ, ε), i he e is some
compac domain Ω(m)⊂dom φ(m)
∗such ha B(m)
M(x, R)⊂Ω(m)and φ(m)
∗: (Ω(m), (m)
∗)→(T(m)N, h(m)
∗) is
a (λ, ε)-quasi-equi alence.
Rema k 2.(i) Any poin ed local quasi-equi alence (M, , x)(N, h, y) o ype (m, R, λ, ε) is also o ype
(m0, R0, λ0, ε0) o 0 ≤m0≤m, 0 < R0< R,λ0> λ and ε0> ε.
(ii) Conside in ege s 0 ≤m0≤m, any poin ed Cm+1 local di eomo phism φ: (M, x)(N, y), and any
∈C∞(M, E) and h∈C∞(N, E). Then φ: (M, , x)(N, h, y) is a poin ed local quasi-equi alence
o ype (m, R, λ, ε) i and only i φ(m0)
∗: (T(m0)M, (m0)
∗, x)(T(m0)N, h(m0)
∗, y) is a poin ed local
quasi-equi alence o ype (m−m0, R, λ, ε).
(iii) I he e is an (m, R, λ, ε)-poin ed local quasi-equi alence (M, , x)(N, h, y), hen, o all R0< R,
λ0> λ and ε0> ε, he e is a C∞(m, R0, λ0, ε0)-poin ed local quasi-equi alence (M, , x)(N, h, y)
by [18, Theo em 2.7].
Lemma 3.5. The ollowing p ope ies hold:
(i)I φ: (M, , x)(N, h, y)and ψ: (N, h, y)(L, u, z)a e poin ed local quasi-equi alences o ypes
(m, R, λ, ε)and (m, λR, λ0, ε0), espec i ely, hen ψ◦φ: (M, , x)(L, u, z)is an (m, R, λλ0, ε +ε0)-
poin ed local quasi-equi alence.
(ii)I φ: (M, , x)(N, h, y)is an (m, λR, λ, ε)-poin ed local quasi-isome y, hen φ−1: (N, h, y)
(M, , x)is an (m, R, λ, ε)-poin ed local quasi-isome y.
P oo . To p o e (i), ake compac domains, Ω(m)⊂T(m)Mand Ω0(m)⊂T(m)N, such ha B(m)
M(x, R)⊂
Ω(m),B(m)
N(x, λR)⊂Ω0(m),φ(m)
∗: (Ω(m), (m)
∗)→(T(m)N, h(m)
∗)isa(λ, ε)-quasi-equi alence, and ψ(m)
∗:
(Ω0(m), h(m)
∗)→(T(m)L, u(m)
∗) is a (λ0, ε0)-quasi-equi alence. Acco ding o he p oo o [1, Lemma 4.3-(i)],
he e is a compac domain Ω(m)
0⊂T(m)Msuch ha B(m)
M(x, R)⊂Ω(m)
0and φ(m)
∗(Ω(m)
0)⊂Ω0(m). Then
(ψ◦φ)(m)
∗: Ω(m)
0→T(m)Lis a λλ0-quasi-isome y by [1, Rema k 2-( )]. Mo eo e , o each ξ∈Ω(m)
0,
 (m)
∗(ξ)−((ψ◦φ)∗u)(m)
∗(ξ)
≤ (m)
∗(ξ)−(φ∗h)(m)
∗(ξ)+h(m)
∗φ(m)
∗(ξ)−(ψ∗u)(m)
∗φ(m)
∗(ξ)≤ε+ε0.
So ψ◦φ: (M, , x)(L, u, z) is an (m, R, λλ0, ε +ε0)-poin ed local quasi-equi alence.
To p o e (ii), le Ω(m)⊂T(m)Mbe a compac domain such ha B(m)
M(x, R)⊂Ω(m), and φ(m)
∗:
(Ω(m), (m)
∗)→(T(m)N, h(m)
∗) is a (λ, ε)-quasi-equi alence. Acco ding o he p oo o [1, Lemma 4.3-(ii)], he
compac domain Ω0(m):= φ(m)
∗(Ω(m))⊂T(m)Ncon ains B(m)
N(y, R). Then (φ−1)(m)
∗= (φ(m)
∗)−1: Ω0(m)→
T(m)Mis a λ-quasi-isome y by [1, Rema k 2-( i)]. Mo eo e , o each ξ∈Ω0(m),
h(m)
∗(ξ)−((φ−1)∗ )(m)
∗(ξ)≤(φ∗h)(m)
∗(φ−1)(m)
∗(ξ)− (m)
∗(φ−1)(m)
∗(ξ)≤ε .
So φ−1: (N, h, y)(M, , x) is an (m, R, λ, ε)-poin ed local quasi-equi alence. 
4. The C∞ opology on b
M∗(n)
De ini ion 4.1. Fo m∈Nand R, > 0, le b
Um
R, be he se o pai s ([M, , x],[N, h, y]) ∈b
M∗(n)×b
M∗(n)
such ha he e is some (m, R, λ, ε)-poin ed local quasi-equi alence (M, , x)(N, h, y) o some λ∈[1, e )
and ε∈(0, ).
P oposi ion 4.2. The ollowing p ope ies7hold o all m, m0∈Nand R, S, , s > 0:
7The ollowing s anda d no a ion is used o a se Xand ela ions U, V ⊂X×X:
U−1={(y, x)∈X×X|(x, y)∈U},
V◦U={(x, z)∈X×X| ∃y∈Xso ha (x, y)∈Uand (y, z)∈V}.
Mo eo e he diagonal o X×Xis deno ed by ∆.
7
(i) (b
Um
e R, )−1⊂b
Um
R, .
(ii)b
Um0
R0, 0⊂b
Um
R, ∩b
Um0
S,s, whe e m0= max{m, m0},R0= max{R, S}and 0= min{ , s}.
(iii) ∆ ⊂b
Um
R, .
(i )b
Um
R, ◦b
Um
e R,s ⊂b
Um
R, +s.
P oo . P ope ies (ii) and (iii) a e elemen a y, and (i) and (i ) a e consequences o Lemma 3.5. 
P oposi ion 4.3. TR, >0b
Um
R, = ∆ o all m∈N.
P oo . We only ha e o p o e “⊂” by P oposi ion 4.2-(iii). Fo ([M, , x],[N, h, y]) ∈TR, >0b
Um
R, , he e
is a sequence o poin ed local quasi-equi alences φi: (M, , x)(N, h, y), wi h co esponding ypes
(m, Ri, λi, εi), such ha Ri↑ ∞,λi↓1 and εi↓0 as i→ ∞. Acco ding o he p oo o [1, P oposi-
ion 5.3], o each i, he e is some subsequence φk(i,l)whose es ic ion o BM(x, Ri) con e ges o some
poin ed isome ic imme sion ψi: (BM(x, Ri), x)→(N, y) in he weak Cm opology, ψi+1|BM(x,Ri)=ψi o
all i, and he combina ion o he maps ψiis a poin ed isome y ψ: (M, x)→(N, y). Fo e e y x0∈M
and ε > 0, he e a e some iand δ > 0 so ha x0∈BM(x, Ri), εi≤ε/2, and kh(y0)−h(y00)k< ε/2 i
dN(y0, y00)< δ o all y0, y00 ∈BM(x, Ri). Mo eo e he e is some lsuch ha dN(φk(i,l)(x0), ψi(x0)) < δ.
Hence
k (x0)−h◦ψ(x0)k≤k (x0)−h◦φk(i,l)(x0)k+kh◦φk(i,l)(x0)−h◦ψ(x0)k< εi+ε/2≤ε.
Since x0and εa e a bi a y, i ollows ha ψ: (M, , x)→(N, h, y) is an equi alence, and he e o e
[M, , x]=[N, h, y]. 
By P oposi ions 4.2 and 4.3, he se s b
Um
R, o m a base o en ou ages o a sepa a ing uni o mi y on b
M∗(n),
which is called he C∞uni o mi y.
De ini ion 4.4. Fo R, > 0 and m∈N, le b
Dm
R, be he se o pai s ([M, , x],[N, h, y]) ∈b
M∗(n)×
b
M∗(n) such ha he e is some Cm+1 poin ed local di eomo phism φ: (M, x)(N, y) so ha kgM−
φ∗gNkCm,Ω,gM< and k −φ∗hkCm,Ω,gM< o some compac domain Ω ⊂dom φwi h BM(x, R)⊂Ω.
Rema k 3.By (1), and i s e sion o E- alued unc ions, a sequence [Mi, i, xi]∈b
M∗(n) is C∞con e gen
o [M, , x]∈b
M∗(n) i and only i i is e en ually in8b
Dm
R, (M, , x) o a bi a y m∈Nand R, > 0.
P oposi ion 4.5. The ollowing p ope ies hold:
(i)Fo all R, > 0, i 0< 0≤min{1−e−2 , e2 −1, }, hen b
D0
R, 0⊂b
U0
R, .
(ii)Fo all m∈Z+,R, > 0and [M, , x]∈b
M∗(n), he e is some 0>0such ha b
Dm
R, 0(M, , x)⊂
b
Um
R, (M, , x).
P oo . Le us show (i). I ([M, , x],[N, h, y]) ∈b
D0
R, 0, hen he e is a C1poin ed local di eomo phism φ:
(M, x)(N, y) such ha 0
0:= kgM−φ∗gNkC0,Ω,gM< 0and ε0:= k −φ∗hkC0,Ω,gM< 0 o some compac
domain Ω ⊂dom φwi h BM(x, R)⊂Ω. Take some λ∈[1, e ) such ha 0
0≤min{1−λ−2, λ2−1}. Acco ding
o he p oo o [1, P oposi ion 6.4-(i)], φ: Ω →Nis a λ-quasi-isome y. Since mo eo e k −φ∗hkΩ≤ε0, i
ollows ha φis a (0, R, λ, 0
0, ε0)-poin ed local quasi-equi alence, ob aining ha ([M, , x],[N, h, y]) ∈b
U0
R, .
Le us p o e (ii). Take m∈Z+,R, > 0 and [M, , x]∈b
M∗(n). Le Ube a ini e collec ion o cha s
o Mwi h domains Ua, and le K={Ka}be a amily o compac subse s o M, wi h he same index se
as U, such ha Ka⊂Ua o all a, and BM(x, R)⊂In (K) o K=SaKa. Le 0>0, o be ixed la e .
Fo any [N, h, y]∈b
Dm
R, 0(M, x), he e is a Cm+1 poin ed local di eomo phism φ: (M, x)(N, y) so ha
kgM−φ∗gNkCm,Ω,gM< 0and k −φ∗hkCm,Ω,gM< 0 o some compac domain Ω ⊂dom φ∩In (K) wi h
BM(x, R)⊂Ω. By con inui y, he e is ano he compac domain Ω0⊂dom φ∩In (K) such ha Ω ⊂In (Ω0),
kgM−φ∗gNkCm,Ω0,gM< 0and k −φ∗hkCm,Ω0,gM< 0. Acco ding o he p oo o [1, P oposi ion 6.4-(i)], i
0is small enough (depending on m,R, and [M, x]), hen he e is some compac domain Ω(m)⊂T(m)M
8Gi en a se X, o U⊂X×Xand x∈X, le U(x) = {y∈Y|(x, y)∈U}. In he case o U⊂b
M∗(n)×b
M∗(n) and
[M, , x]∈b
M∗(n), we simply w i e U(M, , x).
8
such ha B(m)
M(x, R)⊂Ω(m)⊂π−1(Ω0), whe e π:T(m)M→M, and φ(m)
∗: Ω(m)→T(m)Nis a λ-quasi-
isome y o some λ∈[1, e ). Gi en ε∈(0, ), choose some C≥1 sa is ying (1) o E- alued unc ions wi h
U,K, Ω0and g, and, acco ding o Lemma 2.1-(i), choose some ε0>0 such ha
k −φ∗hkCm,Ω0,U,K< ε0=⇒ k (m)
∗−(φ∗h)(m)
∗kΩ(m)<ε.
Suppose ha 0≤ε0/C. Then
k −φ∗hkCm,Ω0,gM< 0=⇒ k −φ∗hkCm,Ω0,U,K< C 0≤ε0=⇒ k (m)
∗−(φ∗h)(m)
∗kΩ(m)<ε.
Hence φis an (m, R, λ, ε)-poin ed local quasi-equi alence (M, , x)(N, h, y), and he e o e [N, h, y]∈
b
U(m)
R, (M, , x). 
P oposi ion 4.6. The ollowing p ope ies hold:
(i)Fo all R, > 0, i e2 0−e−2 0≤ , hen b
U0
R, 0⊂b
D0
R, .
(ii)Fo all m∈Z+,R, > 0and [M, , x]∈b
M∗(n), he e is some 0>0such ha b
Um
R, 0(M, , x)⊂
b
Dm
R, (M, , x).
P oo . Le us show (i). I ([M, , x],[N, h, y]) ∈b
U0
R, 0, hen he e is a (0, R, λ, ε)-poin ed local quasi-
equi alence φ: (M, , x)(N, h, y) o some λ∈[1, e 0) and ε∈(0, 0). Thus he e is some compac
domain Ω ⊂dom φsuch ha BM(x, R)⊂Ω and φ: (Ω, )→(N, h)isa(λ, ε)-quasi-equi alence. Acco ding
o he p oo o [1, P oposi ion 6.5-(i)], kgM−φ∗gNkC0,Ω,g < . So ([M, , x],[N, h, y]) ∈b
D0
R, .
Le us p o e (ii). Le m∈Z+,R, > 0 and [M, , x]∈b
M∗(n). Take U,Kand Klike in he p oo o
P oposi ion 4.5-(ii). Le 0>0, o be ixed la e . Fo any [N, h, y]∈b
Um
R, 0(M, x), he e is an (m, R, λ, ε)-
poin ed local quasi-equi alence φ: (M, , x)(N, h, y) o some λ∈[1, e 0) and ε∈(0, 0). Thus he e
is a compac domain Ω(m)⊂dom φ(m)
∗∩In (K(m)) so ha B(m)
M(x, R)⊂Ω(m)and φ(m)
∗: (Ω(m), (m)
∗)→
(T(m)N, h(m)
∗) is a (λ, ε)-quasi-equi alence. Acco ding o he p oo o [1, P oposi ion 6.5-(ii)], he e a e
compac domains, Ω0(m)⊂dom φ(m)
∗and Ω ⊂M, such ha Ω(m)⊂In (Ω0(m)), Ω(m)∩M⊂Ω⊂In (Ω0(m)),
and kgM−φ∗gNkCm,Ω,g < i 0is small enough; in pa icula , BM(x, R)⊂Ω because Mis a o ally
geodesic Riemannian submani old o T(m)M. Take some C≥1 sa is ying (1) o E- alued unc ions wi h U,
K, Ω and gM. Wi h he no a ion o Sec ion 2, o ρ > 0 and n+ 1 ≤µ≤2mn, le σ(m)
a,ρ,µ :Ua→U(m)
abe he
sec ion o each π:U(m)
a→Uao he ype used in Lemma 2.1-(ii). Since Ω ⊂In (Ω0(m)), he e is some ρ > 0
so ha σ(m)
ρ,µ (Ka∩Ω) ⊂Ω0(m) o all aand µ. Thus, by Lemma 2.1-(ii), he e is some ε0>0, depending on
and ρ, such ha
k (m)
∗−(φ∗h)(m)
∗kΩ0(m)< ε0=⇒ k ∗−φ∗hkCm,Ω,U,K< /C .
Suppose ha mo eo e 0< ε0, and he e o e ε<ε0. Then
k (m)
∗−(φ∗h)(m)
∗kΩ0(m)≤ε<ε0=⇒ k −φ∗hkCm,Ω,U,K< /C =⇒ k −φ∗hkCm,Ω,g < ,
showing ha [N, h, y]∈b
D(m)
R, (M, , x). 
Co olla y 4.7. The C∞con e gence in b
M∗(n)desc ibes he opology induced by he C∞uni o mi y.
P oo . This is a di ec consequence o Rema k 3 and P oposi ions 4.5 and 4.6. 
Acco ding o Co olla y 4.7, he C∞uni o mi y induces wha was called he C∞ opology in Sec ion 1.
Recall ha he co esponding space is deno ed by b
M∞
∗(n), and he no a ion c
Cl∞is used o he closu e
ope a o in b
M∞
∗(n).
P oposi ion 4.8. b
M∞
∗(n)is sepa able.
P oo . Acco ding o he p oo o [1, P oposi ion 7.1], he e is a coun able amily Co C∞compac mani olds
con aining exac ly one ep esen a i e o e e y di eomo phism class, and, o e e y M∈C, he e is a coun able
9
Mand ∈C∞
imm(M), he map ˆιM, :M→imˆιM, is a C∞local di eomo phism wi h espec o he C∞
s uc u e induced by Gon he lea im ˆιM, because e is a C∞imme sion and e ◦ˆιM, = , which is a
C∞local embedding. Thus he es ic ion o χ:N2→B o each plaque is a C∞di eomo phism. Using
again [18, p. 64, Exe cise 9], i ollows ha Φ : N2→B×Zis also C∞ olia ed di eomo phism wi h espec
o he es ic ion o Gand he C∞p oduc olia ed s uc u e o B×Z. This shows ha G=b
F∞
∗,imm(n),
comple ing he p oo o (i). 
Conside a lea im ˆιM, o b
F∞
∗,imm(n). E e y x∈Mhas an open neighbo hood Uin Mso ha :U→E
is an embedding, ob aining ha φ(U)∩U=∅ o all φ∈Iso(M, ) {idM}. The e o e he subg oup
Iso(M, )⊂Iso(M) is disc e e, he quo ien p ojec ion M→Iso(M, ) Mis a co e ing map, and he e is a
unique Riemannian s uc u e on he mani old Iso(M, ) Mso ha M→Iso(M, ) Mis a local isome y.
Mo eo e ˆιM, :M→imˆιM, induces a di eomo phism ¯ιM, : Iso(M, ) M→imˆιM, . Thus ˆιM, :M→
imˆιM, is a co e ing map, and im ˆιM, has a unique Riemannian me ic so ha ˆιM, :M→im ˆιM, is a
local isome y, and he e o e ¯ιM, : Iso(M, ) M→imˆιM, becomes an isome y.
P oposi ion 5.14. The abo e Riemannian me ics on he lea es o b
F∞
∗,imm(n) o m a C∞Riemannian
me ic on (b
M∞
∗,imm(n),b
F∞
∗,imm(n)).
P oo . Le Φ = (χ, Θ) : N2→B×Zbe de ined by any choice o (V, e, ρ, κ, σ) as abo e, and le [Mi, i, xi]→
[M, , x] be a con e gen sequence in Z. Le ¯gMand ¯gibe he me ics on B ha co espond o gMand gi
by he di eomo phisms
χM, :P:= BM(x, 2ρ)∩χ−1
M, (B)→B , χMi, i:Pi:= Bi(xi,2ρ)∩χ−1
Mi, i(B)→B ,
espec i ely (see Lemma 5.7). Acco ding o he p oo o P oposi ion 5.13-(ii), we ha e o p o e ha ¯gi→¯gM
as i→ ∞ in he weak C∞ opology.
Gi en m∈N,R, > 0, o each ila ge enough, he e is an (m, R, λi, εi)-poin ed local quasi-equi alence
φi: (M, , x)(Mi, i, xi) o some λi∈(1, e ) and εi∈(0, ). Assuming R > 2e ρ, we ge BM(x, 2ρ)⊂
BM(x, R) and Bi(xi,2ρ)⊂φi(BM(x, R)), like in he p oo o P oposi ion 5.12. Take a compac domain
Ω(m)
i⊂dom φ(m)
i∗such ha B(m)
i(xi, R)⊂Ω(m)
iand φ(m)
i∗: Ω(m)
i→T(m)Mis a (λi, εi)-quasi-isome y. Le
Ξ⊂Bbe a compac domain, and le Ξ(m)be a compac domain con ained in T(m)Bsuch ha
Ξ⊂In (Ξ(m)),(χ−1
M, )(m)
∗(Ξ(m))∩T(m)P⊂Ω(m)
i,(χ−1
Mi, i)(m)
∗(Ξ(m))∩T(m)Pi⊂φ(m)
i∗(Ω(m)
i).
Like in (10), he e is some ν≥1, independen o i, such ha
d(m)
i(φ−1
i◦χ−1
Mi, i)(m)
∗(ξ),(χ−1
M, )(m)
∗(ξ)< ν ,
o all ξ∈Ξ(m). Since he choice o Ξ(m)is alid o all small enough, i ollows ha φ−1
i◦χ−1
Mi, i→χ−1
M,
in Cm(Ξ, M) by he ob ious e sion o Lemma 2.1 o maps be ween mani olds. Since he choice o Ξ is
alid o all m, i ollows ha his con e gence also holds in C∞(Ξ, M). Take a compac domain Ω ⊂M
such ha BM(x, R)⊂Ω and φ∗
igi→gMon Ω wi h espec o he C∞ opology. We ge
(φ−1
i◦χ−1
Mi, i)∗(φ∗
igi−gM)→(χ−1
M, )∗0 = 0
on Ξ wi h espec o he C∞ opology. So
¯gi−¯gM= (χ−1
Mi, i)∗gi−(χ−1
M, )∗gM
= (φ−1
i◦χ−1
Mi, i)∗(φ∗
igi−gM)+(φ−1
i◦χ−1
Mi, i)∗gM−(χ−1
M, )∗gM→0
on Ξ wi h espec o he C∞ opology. Since e e y poin in Bbelongs o some domain Ξ as abo e i is
chosen small enough, i ollows ha ¯gi−¯gM→0 on Bwi h espec o he weak C∞ opology. 
P oposi ion 5.15. The holonomy co e ing o any lea im ˆιM, o b
F∗,imm(n)is ˆιM, :M→imˆιM, .
This p oposi ion ollows di ec ly om he ob ious e sion o [1, Lemma 11.9] o b
M∞
∗,imm(n).
16

6. Uni e sali y
De ini ion 6.1. Le Xbe a sequen ial Riemannian olia ed space wi h comple e lea es, and le Lxdeno e
he lea h ough e e y x∈X, whose holonomy co e ing is deno ed by e
Lhol
x. I is said ha Xis co e ing-
con inuous when he e is a connec ed poin ed co e ing (e
Lx,˜x) o (Lx, x) o all x∈Xsuch ha [e
Lxi,˜xi]
is C∞con e gen o [e
Lx,˜x] i xi→xis a con e gen sequence in X. When his condi ion is sa is ied wi h
e
Lx=e
Lhol
x o all x∈X, i is said ha Xis holonomy-con inuous.
Rema k 5.Obse e he ollowing:
(i) Co e ing-con inui y and holonomy-con inui y a e weake han co e ing-de e mina ion and holonomy-
de e mina ion [1, De ini ion 12.1], which we e de ined by using “i and only i ” ins ead o “i ”.
(ii) The condi ion o being co e ing-con inuous is he edi a y (by sa u a ed subspaces).
(iii) Co e ing/holonomy-con inui y/de e mina ion ha e ob ious gene aliza ions o a bi a y Riemannian
olia ed spaces by using ne s ins ead o sequences.
Example 6.2. The ollowing simple examples cla i y De ini ion 6.1:
(i) The Reeb olia ion on S3wi h he s anda d me ic is co e ing-con inuous, bu i is no holonomy-
con inuous wi h any Riemannian me ic. I he me ic is modi ied a ound he compac lea T2=S1×S1
so ha he di eomo phism (x, y)7→ (y, x) o T2is no an isome y, hen his olia ion becomes non-
co e ing-con inuous.
(ii) The Riemannian olia ed space o [22, Example 2.5] is co e ing-de e mined bu no holonomy-con inuous.
This example can be easily ealized as a sa u a ed subspace o a Riemannian olia ed space whe e he
holonomy co e ings o he lea es a e isome ic o R. So holonomy-con inui y is no he edi a y.
(iii) b
M∞
∗,imm(n) is holonomy-con inuous. Howe e i is no holonomy-de e mined o n≥1 by [1, Rema k 10-
(iii)], since he e a e di e en poin s wi h isome ic poin ed holonomy co e s o he co esponding
poin ed lea es. To see his, ake any connec ed comple e Riemannian n-mani old M, and some x∈M
and , 0∈C∞
imm(M, E) such ha (x)6= 0(x). Then ˆιM, (x)6= ˆιM, 0(x), bu (M, x) is isome ic o
he holonomy co e s o he poin ed lea es (imˆιM, ,ˆιM, (x)) and (im ˆιM, 0,ˆιM, 0(x)).
P oposi ion 6.3 (C . [4, Theo em 11.4.4]).Fo any Polish C∞ olia ed space Xwi h comple e lea es, he e
is a C∞embedding X→E.
P oo . This is an adap a ion o he usual a gumen o show he exis ence o C∞embeddings o C∞mani olds
in Euclidean spaces [18, Theo em 1.3.4]. Le n= dim X(as olia ed space), and le B =BRn(0, ) and
B =BRn(0, ) o each > 0.
Claim 3.Le Zbe a Polish space, and conside he C∞ olia ed s uc u e on U:= B2×Zwi h lea es
B2× {∗}. Le Vand Wbe open subse s o Usuch ha V⊂Wand W⊂B1×Z. Then he e is some
h∈C∞(U) such ha h= 1 on Vand supp h⊂W.
Since B1is compac , i easily ollows ha each z∈Zhas an open neighbo hood Pzin Zsuch ha ,
o some open subse s Gz, Hz⊂B2wi h Gz⊂Hzand Hz⊂B1, we ha e V∩(B1×Pz)⊂Gz×Pzand
Hz×Pz⊂W. Le {λi}be a pa i ion o uni y o Zsubo dina ed o he open co e {Pz|z∈Z}; in
pa icula , o e e y i, he e is some zi∈Zso ha supp λi⊂Pzi. Le hi∈C∞(B2) such ha hi= 1 on
Gziand supp hi⊂Hzi. Then hiλi∈C∞(U), hiλi=λion Gzi×Pziand supp(hiλi)⊂Hzi×Pzi. I ollows
ha h=Pihiλisa is ies he p ope ies s a ed in Claim 3.
Now, le Ube a coun able collec ion o C∞ olia ed cha s φi:U2,i →B2×Zio Xsuch ha he open
se s U1,i := φ−1
i(B1×Zi) co e X. Using he pa acompac ness and egula i y o X, a s anda d a gumen
gi es locally ini e open co e s, V={Vi}and W={Wi}, wi h he same index se as U, such ha Vi⊂Wi
and Wi⊂U1,i. Fo each i, le Eibe a copy o E. Take embeddings ψi:Zi→Ei[9, Co olla y IX.9.2]. Thus
each composi e
U2,i
φi
−−−−→ B2×Zi
id ×ψi
−−−−→ B2×Ei,→Rn×Ei=: e
Ei
17
is a C∞embedding wi h espec o he es ic ion o F, which will be deno ed by ˜
φi. By Claim 3, he e
a e unc ions hi∈C∞(U2,i) such ha hi= 1 on Viand supp hi⊂Wi. Then a C∞embedding9 :X→
c
Lie
Ei∼
=Eis de ined by (x) = Paha(x)˜
φika.

P oo o Theo em 1.5. The Polish Riemannian olia ed space b
M∞
∗,imm(n) has comple e lea es and is holonomy-
con inuous (Example 6.2-(iii)). Thus any Polish Riemannian olia ed subspace o b
M∞
∗,imm(n) is also co e ing-
con inuous (Rema k 5-(ii)).
Le Xbe any co e ing-con inuous Polish Riemannian olia ed space wi h comple e lea es. By P opo-
si ion 6.3, he e is a C∞embedding :X→E. Wi h he no a ion o De ini ion 6.1, suppose ha he
co e ing-con inui y o Xis sa is ied wi h he connec ed poin ed co e ings (e
Lx,˜x)→(Lx, x) (x∈X). Le
ˆιX, :X→b
M∞
∗,imm(n) be de ined by ˆιX, (x) = [e
Lx,˜
x,˜x], whe e ˜
xis he li o |Lx o e
Lx. This map is
well de ined because he lea es o Xa e comple e. Mo eo e i is ob iously olia ed and con inuous by he
de ini ions o co e ing-con inui y and he opology o b
M∞
∗,imm(n).
To show ha ˆιX, is C∞, ake a olia ed cha Φ = (χ, Θ) : N2→B×Zo b
F∞
∗,imm(n) de ined by any
choice o (V, e, ρ, κ, σ) as abo e. Le Ube he domain o a olia ed cha o Xsuch ha ˆιX, (U)⊂N2. Then
he composi e
UˆιX,
−−−−→ N2
χ
−−−−→ B
is equal o ΠV◦( −e), and he e o e i is C∞.
Finally, ˆιX, is a C∞embedding because he composi e
XˆιX,
−−−−→ b
M∞
∗,imm(n)e
−−−−→ E
equals he C∞embedding .
7. Realiza ion o mani olds o bounded geome y as lea es
P oposi ion 7.1. Le Mbe any connec ed, comple e Riemannian n-mani old o bounded geome y. Then
he e is a C∞embedding :M→Esuch ha c
Cl∞(imˆιM, )is a compac subspace o b
M∞
∗,imm(n).
P oo . Le B =BRn(0, ) o each > 0. By he bounded geome y o M, he e is some > 0, smalle
han he injec i i y adius o M, such ha he ollowing p ope ies hold:
(i) Fo he no mal pa ame iza ions κx:B →BM(x, ) (x∈M), he co esponding me ic coe icien s,
gij and gij, as a amily o C∞ unc ions on B pa ame ized by x,iand j, lie in a bounded subse o
he F ´eche space C∞
b(B ) [28, Theo em A.1], [29, Theo em 2.5] (see also [26, P oposi ion 2.4], [10]).
(ii) The e is some coun able subse {xi|i∈N} ⊂ Mand some c∈Nsuch ha he amily o balls
BM(xi, /2) co e s M, and BM(x, ) mee s a mos cse s BM(xi, ) o all x∈M[33, A1.2 and A1.3],
[29, P oposi ion 3.2].
Le κi=κxi o each i.
Claim 4.The e is a pa i ion o Nin o ini ely many se s, I1, . . . , Ic+1, such ha BM(xi, )∩BM(xj, ) = ∅
o i∈Ikand j∈Ilwi h k6=l.
This claim ollows by conside ing he g aph Gwhose se o e ices is N, and such ha he e is a unique
edge connec ing wo di e en e ices, iand j, i and only i BM(xi, )∩BM(xj, )6=∅. Since he e a e a
mos cedges mee ing a each e ex acco ding o (ii), Gis c+ 1-colo able10; i.e., he e is a pa i ion o N
in o subse s, I1, . . . , Ic+1, such ha he e is no edge joining any pai o di e en e ices in any Ik.
Le Sbe an isome ic copy in Rn+1 o he s anda d n-dimensional sphe e con aining he o igin 0. Choose
some sphe ically symme ic C∞ unc ion ρ∈C∞(Rn) such ha ρ(x)=1i |x| ≤ /2 and ρ(x)=0i |x| ≥ .
Take also some C∞map τ:Rn→Rn+1 ha es ic s o a di eomo phism B →S {0}and maps Rn B
9The no a ion c
LiFiis used o he Hilbe space di ec sum o a amily o Hilbe spaces Fi; i.e., he Hilbe space comple ion
o LiFiwi h he scala p oduc h( i),(wi)i=Pih i, wii.
10This easily ollows by induc ion, assigning o each ia colo di e en om he colo s o he p e ious e ices ha a e
neighbo s o i, which is possible because he e a e a mos co hem (see [3]).
18
o 0. Le ˜ρibe he ex ension by ze o o ρ◦κ−1
i o he whole o M, and le ˜ρk=Pi∈Ik˜ρi. Fo each k, de ine
k:M→Rn+2 by
k(x) = (0 i x /∈Si∈IkBM(xi, )
˜ρk(x)/i, ˜ρk(x)·τ◦κ−1
i(x)i x∈BM(xi, ) o some i∈Ik.
So k◦κi= (ρ/i, ρ ·τ), ob aining ha , o e e y mul i-index α, he unc ion |∂α( k◦κi)|is uni o mly
bounded o e B by a cons an depending only on |α|. Le = ( 1, . . . , c+1) : M→R(c+1)(n+2). We ha e
supM|∇m |<∞ o each m∈Nby (i). Mo eo e k◦κi= (1/i, τ) on B /2, ob aining ha is a C∞
embedding, and in M|Vnd |>0 by (i). By aking any isome ic linea embedding o R(c+1)(n+2) in o E,
we can conside R(c+1)(n+2)- alued unc ions as E- alued unc ions; in pa icula , his applies o .
Claim 5.c
Cl∞(imˆιM, )⊂b
M∞
∗,imm(n).
This claim is ue because, o all [N, h, y]∈c
Cl∞(imˆιM, ), i is easy o see ha in N|Vndh| ≥
in M|Vnd |>0, ob aining ha his an imme sion.
Claim 6.c
Cl∞(imˆιM, ) is compac .
This asse ion ollows by showing ha any sequence in im ˆιM, has a subsequence ha is con e gen in
b
M∞
∗(n). Assume i s ha he sequence is o he o m [M, , xip] o some sequence o indices ip. Since
Cl∞(im ιM) is compac in M∞
∗(n) by [1, Theo em 12.3], we can suppose ha [M, xip] con e ges o some
poin [N, y] in M∞
∗(n). Take a sequence o compac domains Ωqin Nsuch ha BN(y, q + 1) ⊂Ωq. Fo each
q, he e a e poin ed local embeddings φq,p : (N, y)(M, xip), o pla ge enough, such ha Ωq⊂dom φq,p
and φ∗
q,pgM→gNon Ωqwi h espec o he C∞ opology. Le hq,p =φ∗
q,p on Ωq. I is easy o see ha , o
all na u als qand m, he sequence khq,pkCm,Ωq,gNis uni o mly bounded. Hence he unc ions hq,p o m a
compac subse o C∞(Ωq,R(c+1)(n+2)) wi h he C∞ opology by [1, P oposi ion 3.11]. So some subsequence
hq,p(q,`)is con e gen o some hq∈C∞(Ωq,R(c+1)(n+2)) wi h he C∞ opology. In ac , a guing induc i ely
on q, i is easy o see ha we can assume ha each hq+1,p(q+1,`)is a subsequence o hq,p(q,`), and he e o e
hq+1 ex ends hq. Thus he unc ions hqcan be combined o de ine a unc ion h∈C∞(M, R(c+1)(n+2)). Take
sequences o in ege s, `q↑ ∞ and mq↑ ∞, so ha
kh−φ∗
q,p(q,`q) kCmq,Ωq,gN=khq−hq,p(q,`q)kCmq,Ωq,gN→0.
Then, conside ing has an E- alued unc ion, we ge ha [M, , xip(q,`q)]→[N, h, y] in b
M∞
∗(n) as q→ ∞.
Now ake an a bi a y sequence [M, , x0
p] in im ˆιM, . By (ii), he e is a sequence o na u als, ip, such
ha dM(x0
p, xip)< /2. By he abo e case in he p oo , a e aking a subsequence i necessa y, we can
assume ha [M, , xip] is con e gen o some poin [N, h, y] in b
M∞
∗(n). Thus, gi en sequences, mj↑ ∞
in N, and Sj↑ ∞ and sj↓0 in R+, he e is some sequence pj↑ ∞ in Nsuch ha he e exis s some
(mj, Sj+esj /2, λj, εj)-poin ed local quasi-equi alence φj: (N, h, y)(M, , xipj) o some λj∈[1, esj)
and εj∈(0, sj). Since y0
j:= φ−1
j(x0
pj)∈BN(y, esj /2), i ollows ha φj: (N, h, y0
j)(M, , x0
pj) is
an (mj, Sj, λj, εj)-poin ed local quasi-equi alence, showing ha [M, , x0
pj]∈b
Umj
Sj,sj(N, h, y0
j). On he o he
hand, since he sequence y0
jis bounded in N, we can suppose ha i is con e gen o some y0∈Nby aking
a subsequence i necessa y. Hence [N, h, y0
j]→[N, h, y0] in b
M∞
∗(n) by he con inui y o ˆιN,h. Hence he e a e
sequences, nj↑ ∞ in N, and Tj↑ ∞ and j↓ ∞ in R+, such ha [N, h, y0
j]∈b
Unj
esjTj, j(N, h, y0) o jla ge
enough. So
[M, , x0
pj]∈b
Umj
Sj,sj◦b
Unj
esjTj, j(N, h, y0)⊂b
Umin{mj,nj}
min{Sj,Tj},sj+ j(N, h, y0)
o pla ge enough by P oposi ionn 4.2-(i ). This shows ha [M, , x0
pj]→[N, h, y0] in b
M∞
∗(n), comple ing
he p oo o Claim 6. 
P oo o Theo em 1.1. Gi en a connec ed, comple e Riemannian n-mani old Mo bounded geome y, by
P oposi ion 7.1, and Theo ems 1.3 and 1.4, c
Cl∞(imˆιM, ) is a compac Riemannian olia ed subspace o
b
M∞
∗,imm(n). Mo eo e ˆιM, :M→im ˆιM, is an isome y because is an embedding. 
19
8. Open p oblems
Ques ion 8.1. In Theo em 1.1, is i possible o ge he Riemannian olia ed space so ha i s lea es ha e
i ial holonomy?
Ques ion 8.1 can be educed o he ollowing ques ion, in he same way as Theo em 1.1 ollows om
P oposi ion 7.1.
Ques ion 8.2. In P oposi ion 7.1, is i possible o ge such ha mo eo e 11 Iso(N, h) = {idM}i im ˆιN,h ⊂
c
Cl∞(imˆιM, )?
In u n, Ques ion 8.2 can be educed o he ollowing g aph e sion. Conside only connec ed g aphs wi h
a coun able se o e ices, all o hem wi h ini e deg ee. These g aphs a e p ope pa h me ic spaces in a
canonical way so ha each edge is o leng h one. Thus hey de ine a subspace G∗o he G omo space M∗
o poin ed p ope me ic spaces. Deco a e such g aphs wi h maps o hei e ex se o N. This gi es ise
o a space b
G∗o isomo phism classes o poin ed deco a ed g aphs, like in he case o b
M∞
∗(n). Le c
Cl deno e
he closu e ope a o in b
G∗. Fo each deco a ed g aph (G, α), le Iso(G, α) deno e i s g oup o isomo phisms.
The e is a canonical map ˆιG,α :G→b
G∗, like he abo e map ˆιM, . I is said ha Gis o bounded geome y
i he e is a uni o m uppe bound o he deg ee o i s e ices.
Ques ion 8.3. Fo any g aph Go bounded geome y, does he e exis a ini e alued deco a ion αso ha
Iso(H, β) = {id} o all deco a ed g aph (H, β) wi h imˆιH,β ⊂c
Cl(imˆιG,α)?
The e a e ape iodic ilings o R(like he Fibonacci iling), o elemen s o {0,1}Z, gi ing ise o examples
o deco a ions o he Cayley g aph o Zsa is ying he condi ion o Ques ion 8.3 (see e.g. [25]). I Ques ion 8.3
had an a i ma i e answe , hen, in he p oo o P oposi ion 7.1, we could ake a ini e alued deco a ion α
o Gsa is ying he condi ion o Ques ion 8.3, and modi y he de ini ion o so ha
k(x) = ˜ρk(x)·(α(i)+1/i),˜ρk(x)·τ◦κ−1
i(x)
i x∈BM(xi, ) o some i∈Ik. This would gi e a i ma i e answe s o Ques ions 8.2 and 8.1.
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Depa amen o de Xeome ´
ıa e Topolox´
ıa, Facul ade de Ma em´
a icas, Uni e sidade de San iago de Compos ela,
Campus Vida, 15782 San iago de Compos ela, Spain
E-mail add ess:[email p o ec ed]
Depa amen o de Xeome ´
ıa e Topolox´
ıa, Facul ade de Ma em´
a icas, Uni e sidade de San iago de Compos ela,
Campus Vida, 15782 San iago de Compos ela, Spain
E-mail add ess:[email p o ec ed]
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