Partial Isometries of a Sub-Riemannian Manifold

0.22 MB·English

by  Mahuya Datta

PARTIAL ISOMETRIES OF A SUB-RIEMANNIAN MANIFOLD
MAHUYADATTA
2
1
0
2

n
Abstract. Inthisarticle,weobtainthefollowinggeneralisationofisometric
a C1-immersion theorem of Nash and Kuiper. Let M be a smooth manifold
J
of dimension m and H a rank k subbundle of the tangent bundle TM with

4 a Riemannian metric gH. Then the pair (H,gH) defines a sub-Riemannian
2 structureonM. WecallaC1-mapf :(M,H,gH)→(N,h)intoaRiemannian
manifold(N,h)apartial isometry if the derivative mapdf restrictedto H is

] isometric, that is if f∗h|H =gH. We prove that if f0 :M →N is a smooth
G map such that df0|H is a bundle monomorphism and f0∗h|H < gH, then f0
D can be homotoped to a C1-map f : M → N which is a partial isometry,
provideddimN >k. Asaconsequenceofthisresult,weobtainthateverysub-
.
h Riemannian manifold (M,H,gH) admits a partial isometry in Rn, provided
t n≥m+k.
a
m
Keywords: Sub-Riemannianmanifold,partialisometry,convexintegration.
[

Mathematics SubjectClassification2000: 53C17,58J99.
3
v
1 1. Introduction
2
2 Let(M,g)be aRiemannianmanifoldandf0 :M Rn be a C∞ mapsuchthat
→
5 f∗h < g (that is, g f∗h is positive definite), where h is the canonical metric on
9. th0e Euclideanspace−Rn0. Nashprovedin[9]thatiff0 isanimmersion(respectively
0 an embedding) then f can be homotoped to an isometric immersion (respectively
0
0 embedding) f : M Rn so that f∗h = g, provided n dimM +2. He further
1 observed that a clo→sed manifold M that immerses (res≥pectively embeds) in Rn
:
v also does so isometrically under the same dimension restriction. Shortly after this,
i Kuiper [8] proved that these results are true even when n dimM + 1. By
X
≥
Whitney’s Immersion Theoremit is knownthat every manifold M of dimension m
ar admits an immersion in R2m and therefore, it admits an isometric C1 immersion
by Nash-Kuiper theorem. Isometric immersions f : (M,g) (N,h) into any
→
Riemannian manifold (N,h) of dimension n can be (locally) seen as solutions to
a system of m(m+1)/2 equations in n variables, which is clearly overdetermined
when n < m(m +1)/2. Therefore, for sufficiently large m, the system remains
overdetermined for n 2m. A remarkable aspect of Nash-Kuiper theorem is in
≥
showing that a overdetermined system may not only be solvable but the solution
space can be ‘very large’.
In this paper, we obtain a generalisation of the Nash-Kuiper isometric C1-
immersion theorem which comes in response to certain observations of Gromov
in [5, 2.4.9(B)]. Let M be a smooth manifold of dimension m and H a rank k
subbundle of the tangent bundle TM with a Riemannian metric g . Then the
H
pair (H,g ) defines a sub-Riemannian structure on M [6]. We call a C1-map
H
f : (M,H,g ) (N,h) into a Riemannian manifold (N,h) a partial isometry if
H
→
1
2 MAHUYADATTA
df is isometric, that is, if f∗h = g . In the special situation, when H is an
H H H
| |
integrable distribution, we obtain a regular foliation on M such that T = H.
F F
The leaves of this foliation, being integral submanifolds of H, inherit Riemannian
structures from the metric g on H. Therefore,a partial isometry in this case can
H
be viewed as a C1 map which restricts to an isometric immersion on each leaf of
the foliation .
F
Partialisometries are also relatedto Carnot-Caratheodorygeometryunderlying
the sub-Riemannian structure (M,H). Let d denote the Carnot Caratheodory
H
metriconM associatedwiththesubbundleH ofTM. Thenforanytwopointsx,y
ofM,d (x,y)= ifthereisnoH-horizontalpathinM connenctingthesepoints.
H
∞
Otherwised (x,y) is the infimum ofthe lengths ofall H-horizontalpaths between
H
x and y. Recall that a piecewise smooth path γ : I M is called H-horizontal if
→
thetangentvectorsγ˙(t)liesinH forallthoset I wherethepathisdifferentiable.
∈
Observe that a partial isometry preserves the norm of any vector in H, and hence
preservesthe lengths of H-horizontalpaths in M. Thus, if f :(M,g ) (N,h) is
H
→
apartialisometrythenf :(M,d ) (N,d )is necessarilyapath-isometry, where
H h
→
d is the intrinsic metric on N defined by h.
h
The main result of the paper may be stated as follows:
Theorem 1.1. Let M be a manifold with a sub-Riemannian structure (H,g )
H
defined as above and f : M N be a C∞ map into a Riamannian manifold
0
→
(N,h) satisfying the following conditions:
(i) The restriction of df to the bundle H is a monomorphism and
0
(ii) g f∗h is positive definite on H.
H − 0 |H
If dimN > rankH then f can be homotoped to a partial isometry f : (M,g )
0 H
→
(N,h). Furthermore, the homotopy can be made to lie in a given neighbourhood of
f in the fine C0 topology.
0
If we take H = TM in Theorem 1.1 then we obtain the Nash-Kuiper isometric
C1-immersiontheorem. TakingN to be anEuclideanspacewe provethe existence
of partial isometries.
Corollary1.2. Everysub-Riemannianmanifold(M,H,g )admits apartialisom-
H
etry in Rn provided n dimM+rank H.
≥
We also discuss several other consequences of Theorem 1.1 in Corollaries 4.6
and 5.1.
We use the convexintegrationtechnique (see [5], [4])to provethe maintheorem
ofthis paper. It wouldbe appropriateto mentionhere thatGromovdevelopedthe
convex integration theory on the foundation of Kuiper’s technique [8] and applied
this theory to solve many interesting problems which appear in the context of
geometry.
We organise the paper as follows. In Section 2, we outline the proof of The-
orem 1.1, and in Section 3 we briefly discuss the convex integration technique
following the beatiful exposition of Eliashberg and Mishachev [4]. In Section 4 we
prove the main results of the paper stated above and in Section 5 we discuss some
applications of Theorem 1.1.
2. Sketch of the proof
Let (N,h) and (M,H,g ) be as in Section 1 and let g be a fixed Riemannian
H 0
metric on M such that g =g .
0 H H
|
PARTIAL ISOMETRIES OF A SUB-RIEMANNIAN MANIFOLD 3
Definition2.1. AC1 mapfrom M toN is called an H-immersion if itsderivative
restricts to a monomorphism on H (We have borroed this terminology from [2]).
A C1 map f : M N is said to be g -short if g f∗h restricted to H is
0 → H H − 0
positive definite. We use the notation f∗h <g to express g -shortness of f .
0 |H H H 0
It is easy to see that if f is an H-immersionthen f∗h is a Riemannian metric
H
|
on H and conversely. Also note that if f is a partialisometry then it is necessarily
an H-immersion.
We now introduce two real-valued functions on M. The first function will mea-
surethe ‘norm’ofabilinearformonH relativetoaRiemannianmetriconH. The
second function will measure the ‘distance’ between two C1-functions relative to
Riemannian metrics on M and N.
Letg beaRiemannianmetriconH. Foranybilinearformg¯wedefineafunction
n (g¯):M R, as follows:
g
→
g¯ (v,v)
x
n (g¯)(x)= sup | |
g
g (v,v)
v∈Hx\0 x
Given any Riemannian metric g˜ on M and any two C1 maps f,f¯ : M N,
define a function d (f,f¯):M R by →
g˜
→
df (v) df¯(v)
d (f,f¯)(x)= sup k x − x kh
g˜
v
v∈TxM\0 k kg˜
To simplify notations we shall denote n by n and d by d.
gH g0
We now outline the proof of Theorem 1.1. We start with an f as in the hy-
0
pothesisofthetheorem. Firstnotethatwiththenewlyintroducedterminologythe
hypothesisonf readsasfollows: (i) f is anH-immersion,and(ii) f is g -short.
0 0 0 H
(The assumption of g -shortness is not necessary if M is a closed manifold and
H
N is an Euclidean space: For, by multiplying a given H-immersion by a suitable
positive scalar λ we can make it g -short.) The shortness condition means that
H
g f∗h is aRiemannian metric onH. As in the proofofisometric C1-immersion
H− 0
theorem in [9], we need a suitable decomposition of g f∗h on H.
H − 0 |H
Lemma 2.2. Let U λ Λ be an open covering of the manifold M such that (a)
λ
{ | ∈ }
each U is a coordinate neighbourhood in M and (b) for any λ , U intersects at
λ 0 λ0
most c (m) many U ’s including itself. Then g f∗h admits a decomposition
1 λ H − 0 |H
as follows:
∞
g f∗h = φ2(dψ )2 ,
H − 0 |H X i i |H
i=1
where φ and ψ , for i=1,2,..., are C∞ functions on M such that
i i
(i) for each i there exists a λ Λ for which suppφ is contained in U and dψ
i λ i
∈
is a rank 1 quadratic form on U .
λ
(ii) for all but finitely many i, φ vanishes on an U and
i λ
(iii) there are at most c(m) many φ ’s which are non-vanshing at any point x.
i
Here c(m) and c (m) are positive integers depending on m=dimM.
1
Proof. Choose a subbundle K of TM which is complementary to H and take any
Riemannianmetricg onit. Theng =(g f∗h ) g +f∗hisaRiemannian
K M H− 0 |H ⊕ K 0
metric on TM which clearly restricts to g on H. Moreover, g f∗h > 0 on
H M − 0
4 MAHUYADATTA
M. Then by Nash’s decomposition formula (see [1] and [9]), there exist smooth
functions ψ and φ as described in the lemma such that g f∗h = φ2dψ2.
By restrictinig bothisides to H we get the desired decomposMitio−n.0 Pi i (cid:3)i
Construction of an Approximate solution: Applying Lemma 2.2 we get a
decompositionofg f∗h. WethenusethisdecompositiontoobtainaC∞ mapf˜
H− 0
whichisveryclosetoapartialisometryinthesensethatn(g f˜∗h)issufficiently
H
small. This is achieved following successive deformations f¯, −f¯,...,f¯ ,..., of f
1 2 n 0
such that f¯∗h is approximately equal to f¯∗ h+φ2dψ2 for each i. Each step of
i i−1 i i
deformation involves a convex integration (discussed in Section 3) and the defor-
mation takes place on the open set U containing suppφ in such a way that the
λ i
valueofthederivativedf¯ alongτ =kerdψ isaffectedbyasmallamount. Since
i−1 i i
at most finitely many φ are non-zero on any U , the sequence f¯ is eventually
i λ i
{ }
constant on each U and therefore converges to a C∞ map f˜: M N which is
λ
very close to being a partial isometry. Indeed, for the final map f˜t→he total error
g f˜∗h, estimated by the function n(g f˜∗h), can be made arbitrarily small.
H H
− −
Moreover, d(f ,f˜) can be controlled by the function n(g f∗h). See Lemma 4.1
0 H − 0
and Lemma 4.4 for a detailed proof.
Obtaining a partial isometry: Theprincipalideaistoobtainapartialisometry
asthe limit ofasequence ofC∞ smoothg -shortH-immersionsf :M (N,h)
H j
−→
which is Cauchy in the fine C1-topology and is such that the induced metric f∗h
j
approachesto g on H in the limit. More explicitly, the sequence f will satisfy
H j
{ }
the following relations:
(1) n(g f∗(h)) 1n(g f∗ (h)),
H − j ≈ 2 H − j−1
(2) d(fj,fj−1)<c(m)n(gH −fj∗−1(h))12,
where c(m) is a constant depending on the dimension m of the manifold M. The
j-th map f can be seen as an improvedapproximate solution over f . The con-
j j−1
ditions(1)and(2)togetherguaranteethatthesequence f isaCauchysequence
n
{ }
inthefineC1 topologyandhenceitconvergestosomeC1 mapf :M N. Then
−→
theinducedmetricf∗hmustbeequaltog whenrestrictedtoH bycondition(1).
H
Thus f is the desired partial isometry.
3. Preliminaries of Convex Integration Theory
In this section, we recallfrom[5] and[10] the basic terminologyof the theory of
h-principle and preliminaries of convex integration theory.
LetM andN besmoothmanifoldsandx M. Iff :U N isaCr mapdefined
∈ →
onanopensubset U ofM containing x, then the r-jet of f atx, denotedby jr(x),
f
corresponds to the r-th degree Taylor’s polynomial of f relative to a coordinate
system around x. Let Jr(M,N) denote the space of r-jets of germs of Cr-maps
M N and let pr : Jr(M,N) M be the natural projection map taking jr(x)
→ −→ f
to x, which is a fibration. For any Cr map f : M N, jr is a section of pr.
→ f
Moreover,if r >s then there is a canonical projection pr :Jr(M,N) Js(M,N)
s →
which takes an r jet at x represented by a germ f to the s jet of f at x.
A partial differential relation of order r for Cr maps M N is defined as a
→
subspace of Jr(M,N). If is an open subset then we say that is an open
R R R
relation.
PARTIAL ISOMETRIES OF A SUB-RIEMANNIAN MANIFOLD 5
A section σ of pr : Jr(M,N) M is said to be a section of if the image of
→ R
σ is contained in . A section of is often referred as a formal solution of . If
R R R
f : M N is such that jr maps M into then f is called a solution of . A
→ f R R
section σ :M is called holonomic if σ =jr for a Cr-map f :M N.
→R f −→
Let Γ( ) denote the space of sections of the r-jet bundle pr : Jr(M,N) M
R −→
whose images lie in . We endow this space with the C0-compact open topology.
R
The space of Cr solutions of is denoted by Sol . We endow it with the Cr
R R
compact-open topology. Then the r-th jet map jr : Sol Γ( ) defined by
R −→ R
jr(f)=jr is continuous relative to the given topologies.
f
Definition 3.1. A relation is said to satisfy the h-principle if given a section σ
R
of there exists a solution f of such that jr is homotopic to σ in Γ( ). If the
R R f R
r-jetmapjr isaweakhomotopy equivalencethen is saidtosatisfytheparametric
R
h-priniple.
Let be a subspace of C0 maps M N. A relation is said to satisfy the C0
U → R
dense h-principle near provided given any f and any neighbourhood N of
U ∈ U
j0(M), everysection σ of which lies over j0 (i.e., pr σ =j0)admits ahomotopy
f R f 0◦ f
σ such that σ lies in (pr)−1( ) and σ is holonomic.
t t 0 U ∩R 1
Given a differential relation , the main problem is to determine whether or
R
not it has a solution. Proving h-principle is a step forward towards this goal. If a
relationsatisfiestheh-principlethenwecannotatoncesaythatthesolutionexists;
however, we can conclude that if has a section (i.e., a formal solution) then it
R
has a solution. Thus, we reduce a differential topological problem to a problem in
algebraictopology. There are severaltechniques due to Gromov which address the
question of h-principle. The convex integration theory is one such. Here we will
review the convex integration theory only for first order differential relations.
Let τ be a codimension 1 integrable hyperplane distribution on M. Let f and g
be germsatx X oftwoC1 smoothmaps fromM to N. We saythatf andg are
∈
-equivalent if
⊥
f(x)=g(x) and Df =Dg .
x τ x τ
| |
The -equivalence is an equivalence relation on the 1-jet space J1(M,N). The
⊥
equivalence class of j1(x) is denoted by j⊥(x) and is called the -jet of f at x.
f f ⊥
Since τ is integrable,we canchooselocalcoordinatesystems (U;x ,...,x ,t)so
1 n−1
that (x ,...,x ,t) : t =const are integral submanifolds of τ. Moreover, we
1 n−1
{ }
canexpress j1(x) as (j⊥(x),∂ f(x)), where j⊥ =( ∂f ,..., ∂f ) and ∂ f denotes
the partial defrivative off f in tthe direction offt. In∂pxa1rticula∂rx,ni−f1M = Rnt, N = Rq
and τ is defined by the codimension one foliation Rn−1 R on Rn, then the 1-jet
spacegetsasplittingJ1(Rn,Rq)=J⊥(Rn,Rq) Rq. Th×esetofall -jets,denoted
× ⊥
by J⊥(M,N), has a manifold structure [10, 6.1.1] and the natural projection map
p1 :J1(M,N) J⊥(M,N), taking a 1-jet to its -equivalence class (relative to
⊥ −→ ⊥
the given τ), defines an affine bundle, in which the fibres are affine spaces of di-
mension n =dimN. The fibres of this affine bundle are called principal subspaces
relative to τ. Note that there is a unique principal subspace through each point of
J1(M,N). In fact, the fibre of J1(M,N) J0(M,N) over any b J0(M,N) is
−→ ∈
foliated by these principal subspaces and the translation map takes principal sub-
spaces onto principal subspaces.
6 MAHUYADATTA
Notation: We shall denote the principal subspace through a J1(M,N) by
∈
R(a,τ). If is a first order relation and a , then the connected component of
R ∈R
a in R(a,τ) will be denoted by (a,τ).
R∩ R
The following theorem, known as the h-Stability Theorem in the literature, ([5,
2.4.2(B)] and [10, Theorems 7.2, 7.17]) is a key result in the theory of convex
integration.
Theorem 3.2. Let be an open relation and let f :M N be a C1 map such
0
R −→
that
(1) j⊥ lifts to a section σ of and
f0 0 R
(2) j1 (x) lies in the convex hull of (σ (x),τ ) for every x M.
f0 R 0 x ∈
Let be any neighbourhood of j⊥(M). Then there exists a homotopy σ :M ,
N f t →R
t [0,1], such that
∈
(i) σ =j1 for some C1 map f :M N, so that f is a solution of and
1 f1 1 → 1 R
(ii) (p1 σ )(M) for all t [0,1]. In particular f is close to f in the
⊥ ◦ t ⊂ N ∈ 1 0
fine C0 topology.
Further, if the initial map f is a solution on OpK for some closed set K then
0
the homotopy remains constant on OpK.
Remark 3.3. Since C∞(M,N) is dense in C1(M,N) relative to the fine C1-
topology and is open, we can perturb any C1-solution of to obtain a C∞
R R
solution.
Definition 3.4. A connected subset S in a vector space (or in an affine space) V
is said to be ample if the convex hull of S is all of V. The subset defined by the
polynomial x2+y2 z2 =0 in R3 is an example of an ample subset. However, the
complement of a 2-−dimensional vector subspace in R3 is not ample.
A relation is said to be ample if for every hyperplane distribution τ on M,
R
(a,τ) is ample in R(a,τ) for all a .
R ∈R
Theorem 3.5. ([5, 2.4.3, Theorem (A)]) Every open ample relation satisfies the
C0-dense parametric h-principle.
We end this section with an application of Theorem 3.5 to the H-immersion
relation; (see [4, 8.3.4] for an alternative proof).
Proposition 3.6. Let M be a smooth manifold and H a subbundle of TM. Then
H-immersions f : M N satisfy the C0-dense parametric h-principle provided
→
dimN > rankH. In other words, every bundle map (F ,f ):TM TN suchthat
0 0
→
F is a monomorphism is homotopic through such bundle maps to an (F,f) :
0 H
|
TM TN such that F =df provided dimN > rankH.
→
Proof. The H-immersionsf :M N aresolutions to the firstorderpartialdiffer-
→
ential relation
= (x,y,α) J1(M,N) α :H T N is injective linear .
R { ∈ | |Hx x → y }
First of all, we prove that is an open relation. Recall that if (U,φ) and (V,ψ)
R
are coordinate charts in M and N respectively then the bijection τ : J1(U,V)
J1(φ(U),ψ(V)=φ(U) ψ(V) L(Rm,Rn) defined by →
× ×
τ(j1(x))=(φ(x),ψ(f(x)),d(ψfφ−1) )
f φ(x)
PARTIAL ISOMETRIES OF A SUB-RIEMANNIAN MANIFOLD 7
is a coordinate chart for the total space J1(M,N) of the 1-jet bundle [7], where
m=dimM andn=dimN. SinceH isasubbundleofTM wecanfurtherchoosea
trivialisationΦ:TM U RmofthebundleTM (possiblyaftershrinkingU),
U U
suchthat Φ maps H |ont→o U ×Rk. Then τ¯:J1(U,V|) φ(U) ψ(V) L(Rm,Rn)
by τ¯(j1(x))=(φ(x),ψ(f(x))×,d(ψf) Φ¯−1 ) is a diffe→omorph×ism, whe×re Φ¯ =(φ
f x◦ φ(x) ×
Id) Φ:TM φ(U) Rm.
U
N◦ow, consid|er→the rest×rictionmorphismr :L(Rm,Rn) L(Rk,Rn) whichtakes
a linear transformation L L(Rm,Rn) onto its restrictio→n LRk. Let Lk(Rm,Rn)
denote the inverse image u∈nder r of the set of all monomorph|isms Rk Rn. This
→
is clearly an open set and it is easy to see that τ¯ maps J1(U,V) diffeomorphi-
cally onto φ(U) ψ(V) L (Rm,Rn). Consequently Ri∩s open in the 1-jet space
k
× × R
J1(M,N).
Next, we shall show that is an ample relation. To see this, consider a codi-
R
mension 1 subspace τ of TM for some x M and take a 1-jet j1(x) . We
x x ∈ f ∈ R
need to show that the principal subspace
R(j1(x),τ )= (x,f(x),β) J1(M,N)β =df on τ
f x { ∈ | x x}
intersectstherelation inapathconnectedsetandmoreovertheconvexhullofthe
R
intersection, denoted by (j1(x),τ ), is all of R(j1(x),τ ). There are two possible
R f x f x
cases:
Case 1. H τ . In this case,the principalsubspace is completely containedin
x x
⊂
. Thus (j1(x),τ ) is equal to the principal subspace itself.
R R f x
Case 2. H τ is a codimension 1 subspace of H . Choose a vector v H
x x x x
∩ ∈
which is transverse to H τ . First observe that R(j1(x),τ ) is affine isomorphic
x∩ x f x
to T N since any 1-jet (x,y,β) is completely determined by β(v). Therefore,
f(x)
(j1(x),τ ) is affine equivalent to the subset
R f x
S(j1(x))= w T N w df (τ H ) .
f { ∈ f(x) | 6∈ x x∩ x }
Sinceτ H hasdimensionk 1anddf isinjective onH ,the subspacedf (τ
x x x x x x
∩ − ∩
H ) is of codimension at least 2 in T N provided dimN > k. Hence S(j1(x))
x f(x) f
is path-connected and its convex hull is all of T N. In other words, the convex
f(x)
hull of (j1(x),τ ) is all of R(j1(x),τ ).
R f x f x
Thisprovesthat isanopen,amplerelation. Hence,wecanapplyTheorem3.5
to conclude that Rsatisfies the C0-dense parametric h principle. (cid:3)
R
Corollary 3.7. Suppose that f : M N is a smooth map. If dimN dimM+
0
→ ≥
rankH,thenf canbehomotopedwithinitsC0-neighbourhoodtoaC∞ H-immersion
0
f :M N.
→
Proof. Inviewoftheabovepropositionitisenoughtoshowthatf canbecovered
0
by a monomorphism F : H TN if dimN dimM+ rank H. It is well-known
→ ≥
that the obstruction to the existence of such an F lies in certain homotopy groups
of the Stiefel manifold V (Rn), namely in π (V (Rn)) for 0 i m 1, where
k i k
m=dimM, n=dimN and k = rankH. Since V (Rn) is n ≤k ≤1 con−nected the
k
obstructions vanish for n m+k. This proves the corollary−. − (cid:3)
≥
Remark 3.8. The set of smooth H-immersions M N is an open, dense subset
→
of C∞(M,N) relative to the fine C∞ topology when dimN dimM+ rank H [2,
≥
Proposition 2.2].
8 MAHUYADATTA
4. Proof of Theorem 1.1 and Corollary 1.2
Let (N,h) be a smooth Riemannian manifold of dimension n and (M,H,g )
H
be as in Section 1. Let g be a Riemannian metric on M such that g = g .
0 0 H H
|
Suppose that dimN > rank H.
Lemma4.1. (MainLemma)Letg beaRiemannianmetriconH suchthatg <g .
H
Suppose that f : M N is a smooth H-immersion and g f∗h = φ2dψ2 on H,
→ −
where φ and ψ are smooth real valued functions on M such that suppφ is contained
in a coordinate neighbourhood U and dψ has rank 1 on U. Given any two positive
functions ε and δ on M, f can be homotoped to a C∞ map f :M N in a given
→
C0 neighbourhood of f such that f coincides with f outsidee U and satisfies the
following properties: e
(i) f is an H-immersion,
(ii) 0e n (g f∗h)<δ,
(iii) dg≤0(f,gfH)<−ngeH(g−f∗h)12 +ε.
e
Remark 4.2. Observe that inequality (iii) in the lemma above is independent of
any particular choice of g .
0
Proof. We will prove the lemma by an application of Theorem 3.2. First observe
that the partial isometries (M,H,g) (N,h) are solutions to a first order differ-
→
ential relation given by:
I
= (x,y,α) J1(M,N) α:T M T N is linear and α∗h=g on H .
x y x
I { ∈ | → }
Let f be as in the hypothesis: g f∗h = φ2dψ2 on H, where dψ is of rank 1 on
−
U. Then the kernel of dψ defines a codimension 1 integrable distribution on U
which we will denote by τ. We construct -jet bundle J⊥(M,N) M relative
⊥ →
to this hyperplane distribution τ as described in Section 3. Recall that two 1-jets
(x,y,α) and (x,y,β) in J1(M,N) are equivalent if α =β , and the -jets are
|τx |τx ⊥
equivalence classes of 1-jets under the above equivalence relation. The set of all
1-jets equivalent to (x,y,α) is an affine subspace of J1 (M,N). This is referred
(x,y)
as the principal subspace containing α and is denoted by R(α,τ). We claim that
for all x U
∈
(a) R(j1(x),τ ) is a non-empty path-connected set.
f x ∩I
(b) j1(x) belongs to the convex hull of R(j1(x),τ ) .
f f x ∩I
(c) There is a section σ of such that σ (x) R((j1(x),τ ).
0 I 0 ∈ f x
Let V denote the open subset of U consisting of all points x such that g
−
f∗h = 0. We shall first prove the statements (a), (b) and (c) for points in V.
|Hx 6
If x V then H is not contained in τ because g f∗h = φ2dψ2 on H , and
x x x
∈ −
therefore H τ is of codimension 1 in H . Choose a smooth unit vector field v
x x x
∩
on the open set V such that v H and is g-orthogonal to H τ for all x V.
x x x x
∈ ∩ ∈
Observe that every β R(j1(x),τ ) is then completely determined by its value on
∈ f x
the vector v . In fact, we candefine anaffine isomorphismR(j1(x),τ ) T N by
x f x → y
β β(v ) which maps R(j1(x),τ ) onto the set
7→ x I∩ f x
S = w T N w df (H τ ), w =1 .
x f(x) h x x x h
{ ∈ | ⊥ ∩ k k }
PARTIAL ISOMETRIES OF A SUB-RIEMANNIAN MANIFOLD 9
Denote the rank of H by k. Since df is injective linear, S represents the
x|Hx x
unit sphere in a codimension (k 1) subspace of T N. Hence, for n > k, S is
y x
−
path-connected. This proves (a).
Also,df (v )liesintheconvexhullofS . Indeed,theconditiong f∗h=φ2dψ2
x x x
−
onH impliesthath(df (v ),df (w))=0forallw H τ andtherefore,df (v )
x x x x x x x x
∈ ∩
is h-orthogonal to df (H τ ). Moreover, as g f∗h > 0 and f is an H-
x x ∩ x − |Hx
immersionit alsofollowsthat0< df (v ) <1. Hence, df (v )lies inthe convex
x x x x
k k
hull of S proving (b).
x
To prove (c) note that df (v ) is orthogonal to df (H τ ). Therefore, if we
x x x x x
∩
define w (x)=df (v )/ df (v ) then w (x) S . Let σ (x) denote the 1-jet in
0 x x x x h 0 x 0
k k ∈
R(j1(x),τ ) which correspondsto w (x). Thus, σ is a continuous section of
f x ∩I 0 0 I
over V as mentioned in (c).
If x U V, then either φ(x) = 0 or dψ = 0 i.e., H is contained in
∈ \ x|Hx x
τ = kerdψ . If φ(x) = 0 then proceeding as in the above case we can prove
x x
that (a) and (b) are true. If H τ then the principal subspace R(j1(x),τ ) is
x ⊂ x f x
completely contained in . Therefore (a) and (b) are clearly true in this case also.
I
Further, x U V implies that j1(x) and we can choose σ (x) = j1(x) on
∈ \ f ∈ I 0 f
U V sothat (c)is provedonallof U. This completes the proofofthe claimmade
\
above.
Infact,wehaveprovedthatthemapf :M N satisfiesboth(1)and(2)ofthe
→
hypothesis of Theorem 3.2 relative to the relation . Indeed, by our construction
I
σ lifts j⊥. Further, j1(x) lies inthe convexhullof (σ (x),τ ) for all x U. This
0 f f I 0 x ∈
follows from (a) and (b) since R(j1(x),τ ) = R(σ (x),τ ) = (σ (x),τ )
f x ∩I 0 x ∩I I 0 x
(see section 3). However, we cannot apply Theorem 3.2 to (f, ), since is not
I I
open. To surpass this difficulty, we consider a small open neighbourhood Op of
I
in the H-immersion relation and apply Theorem 3.2 to the pair (f,Op ) to
I R I
obtainasmoothH-immersionf˜:M N whichisasolutionofOp . Bychoosing
Op sufficiently small we canensure→that f˜∗h is arbitrarilycloseIto g. Thus, we
H
I |
prove (i) and (ii) as stated in the theorem.
Inorderthatf˜satisfiescondition(iii)aswell,weneedtomodifytherelationOp
I
further. Consider the subset S′ = w S h(w,df (v )) h(df (v ),df (v ) of
x { ∈ x| x x ≥ x x x x }
S (see [4, 21.5]). This is path-connected, symmetric about w (x) and contains
x 0
§
df (v ) in its convex hull. Moreover, for any vector w in S′, w df (v )
x x x k − x x k ≤
1 df (v ) 2. Let ′ denote the subset of defined by S′, x M. Now,
p −k x x k I I x ∈
applying Theorem 3.2 to (Op ′,f) we obtain a C∞ map f˜ : M N which is
I →
homotopic to f and is a solutionof Op ′. As we have alreadyobserved, f˜satisfies
I
(i) and (ii) as stated in the theorem. Further, we have,
df˜(v ) df (v ) 1 df (v ) 2 +ε
k x x − x x kh ≤ p −k x x kh
= (g f∗h)(v ,v )+ε
p − x x
where the ‘error term’ ε appears because of enlarging . Since g = g and
0 H H
I |
v H, dividing out both sides by v we obtain from the above that
x ∈ k xkg0
df˜(v ) df (v )
k x xv− x x kh ≤qngH(g−f∗h)+ε.
k xkg0
10 MAHUYADATTA
Moreover,by Theorem 3.2 we can choose f˜so that the directional derivatives of f˜
along τ are arbitrarily close to the corresponding derivatives of f. Thus we obtain
that d (f,f˜) n (g f∗h)+ε. (cid:3)
g0 ≤p gH −
Remark 4.3. In the above lemma we started with a C∞ map f satisfying f∗h <
g < g and given any δ > 0 obtained an f˜satisfying the condition n(g f˜∗h) <
H
−
δ. Therefore, if we choose δ sufficiently small then f˜ can be made to satisfy the
inequality f∗h<f˜∗h<g .
H
We nowfix acountableopencovering = U λ Λ ofthe manifoldM which
λ
U { | ∈ }
has the following properties:
(a) each U is a coordinate neighbourhood in M and
λ
(b) for any λ , U intersects atmost c (m) many U ’s including itself,
0 λ0 1 λ
where c (m) is an integer depending on m = dimM. This open convering will
1
remain fixed througout. All decompositions of Riemannian metrics on H will be
considered with respect to this covering.
Lemma 4.4. Let f : M N be a smooth H-immersion such that f∗h < g on
0 → 0 H
H. Then f can be homotoped to a C∞ map f in any given C0 neighbourhood of
0 1
f such that
0
(i) f is an H-immersion and f∗h<g on H,
1 1 H
(ii) 0<n (g f∗h)< 2n (g f∗h),
gH H − 1 3 gH H − 0
(iii) d (f ,f )<c(m) n (g f∗h),
g0 0 1 p gH H − 0
where c(m) is a constant which depends on the dimension m of M.
Proof. Since g f∗h >0 we get a decomposition as follows:
H − 0 |H
g f∗h=2 ∞ φ2dψ2 on H,
H − 0 Pk=1 k k
whereφ andψ areasdescribedinLemma 2.2. Itfurtherfollowsfromthe lemma
k k
that all but finitely many φ vanish on any U and and at most c(m) number of
i p
φ are non-vanishing at any point x. Define a sequence of Riemannian metrics
i
on H as follows: g¯ = f∗h and g¯ = g¯ +φ2dψ2 . Then each g¯ < g and
0 0 |H k k−1 k k|H k H
lim g¯ =f∗h+1(g f∗h). Insuccessivestepsweaimtoincrementtheinduced
k→∞ k 0 2 − 0
metriconH byφ2dψ2 ,k =1,2,.... However,intheprocessofachievingthiswe
k k|H
admitanerrorineachstep;theerrorinstepk isdenotedbyδ . Thusattheendof
k
the k-th step we will have a map f¯ such that f¯∗h =g¯ + k δ , k =1,2,....
Explicitly, we will construct a sequkence of smookth|Hmapsk f¯P,i=k1=i1,2,..., such
k
that f¯∗h =g¯ +δ¯ , k =1,2,... which satisfy the follow{ing}conditions:
k |H k k
(1) f¯ is a g -short H-immersion, and f¯ =f¯ outside U ,
k H k k−1 k
(2) 0 n (δ¯ δ¯ )<δ′,
≤ gH k− k−1 k
(3) d (f¯ ,f¯)<n (g¯ g¯ )1/2+ε ,
(4) g¯g0 +k−δ¯1 <k g . gH k− k−1 k
k+1 k H
where f¯ =f and δ =0 and ε < ,
0 0 0 Pk≥1 k ∞
Taking g = g¯ and f = f in Lemma 4.1 we can prove the first step of the
1 0
inductionfork=1. Letf¯∗h=g¯ +δ ,whereδ issuchthatg¯ +δ <g . Suppose
1 1 1 1 2 1 H
we have obtained f¯ satisfying (1)–(4)at the end of the k-th step, where we can
k
writef¯∗h =g¯ +δ¯ . Inthenextstepwewanttoincrementtheinducedmetricon
H by akqu|Hantitykφ2 kdψ2 , that is, we want to induce g¯ +δ¯ on H. Taking
k+1 k+1|H k+1 k
g =g¯ +δ¯ andf =f¯ inLemma4.1weobtainasmoothmapf¯ whichclearly
k+1 k k k+1