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arXiv:submit/2908225 [math.DG] 30 Oct 2019
HIGHER DIMENSIONAL HOLONOMY MAP FOR RULED
SUBMANIFOLDS IN GRADED MANIFOLDS
GIANMARCO GIOVANNARDI
Abstract. The deformability condition for submanifolds of fixed degree im-
mersed in a graded manifold can be expressed as a system of first order PDEs.
In the particular but important case of ruled submanifolds, we introduce a
natural choice of coordinates, which allows to deeply simplify the formal ex-
pression of the system, and to reduce it to a system of ODEs along a character-
istic direction. We introduce a notion of higher dimensional holonomy map in
analogy with the one-dimensional case [22], and we provide a characterization
for singularities as well as a deformability criterion.
Contents
1. Introduction 1
2. Preliminaries 6
2.1. Degree of a submanifold 7
2.2. Admissible variations and admissibility system of PDEs 8
3. Intrinsic coordinates for the admissibility system of PDEs 8
4. Ruled submanifolds in graded manifolds 11
5. The high dimensional holonomy map for ruled submanifolds 13
6. Integrability of admissible vector fields for a ruled regular submanifold 15
References 22
1. Introduction
The goal of this work is to study the deformability of a some particular kind of
submanifolds immersed in an equiregular graded manifold (N , H1,...,Hs), that is
a smooth manifold endowed with a filtration of sub-bundles of the tangent bundle
H1⊂ H2⊂ · · · ⊂ Hs=T N satisfying [Hi,Hj]⊂ Hi+j,i, j >1.
Given p∈N, a vector v∈TpNhas degree iif v∈ Hi
pbut v /∈ Hi−1
p. When
we consider an immersed submanifold Φ : ¯
M→Nand we set M= Φ( ¯
M), the
interaction between the tangent space TpMand the filtration H1
p⊂ H2
p⊂ · · · ⊂ Hs
p
Date: October 30, 2019.
2000 Mathematics Subject Classification. 58H99, 49Q99, 58A17.
Key words and phrases. sub-Riemannian manifolds; graded manifolds; regular and singular
ruled submanifolds; higher-dimensional holonomy map; admissible variations.
The author has been supported by Horizon 2020 Project ref. 777822: GHAIA, MEC-Feder
grant MTM2017-84851-C2-1-P and PRIN 2015 “Variational and perturbative aspects of nonlinear
differential problems” .
2 G. GIOVANNARDI
is embodied by the induced tangent flag
(1.1) TpM∩ H1
p⊂ · · · ⊂ TpM∩ Hs
p,
where p= Φ(¯p), ¯p∈¯
M. The smooth submanifold Mequipped with the induced
filtration pointwise described by (1.1) inherits a graded structure, that is no more
equiregular. M. Gromov in [20] consider the homogeneous dimension of the tangent
flag (1.1) to define the pointwise degree by
degM(p) =
s
X
j=1
j( ˜mj−˜mj−1),
where ˜m0= 0 and ˜mj= dim(TpM∩ Hj
p). In an alternative definition provided in
[27], the authors write the m-tangent vector to M= Φ( ¯
M) as linear combination of
simple m-vectors Xj1∧· · · ∧Xjmwhere (X1,...,Xn) is an adapted basis of T N , see
[4] or (2.3). Then the pointwise degree is the maximum of the degree of the simple
m-vectors whose degree is in turn given by the sum of the single vectors appearing
in the wedge product. The degree of a submanifold deg(M) is the maximum of the
pointwise degree among all points in M= Φ( ¯
M).
In [27] the authors introduced a notion of area for submanifolds immersed in
Carnot groups that later was generalized by [11] for immersed submanifold Min
graded structures. Given a Riemannian metric gin the ambient space N, the area
functional Ad(M) in [11] is obtained by a limit process involving the Riemannian
areas of Massociated to a sequence of dilated metrics grof the original one g. The
density of this area is given by the projection of the m-vector e1∧...∧emtangent
to Monto the space of m-vectors of degree equal to d= deg(M), see equation (2.8).
The central issue is that the area functional depends on the degree deg(M) of the
immersed submanifold M. Thus, if we wish to compute the first variation formula
for this area functional we need to deform the original submanifold by variations
Γ(¯p, τ ) that preserve the original degree deg(M). This constraint on the degree
gives rise to a first order system of PDEs that defines the admissibility for vector
fields on M.
The simplest example of immersion is given by a curve γ:I⊂R→N, where
the pointwise degree of the curve γis the degree of its tangent vector γ′(t) at ev-
ery t∈I. In this particular case the admissibility system is a system of ODEs
along the curve γ. This restriction on vector fields produces the phenomenon of
singular curves, that do not admit enough compactly supported variations in the
sub-bundles determined by the original degree of γ. This issue has been addressed
by L. Hsu in [22] and R. Bryant and L. Hsu in [8]. These two works are based on
the Griffiths formalism [18] that studies variational problems using the geometric
theory of exterior differential system [6,7] and the method of moving frames de-
veloped by E. Cartan [9]. In Carnot manifolds (N, H), that are a particular case
of graded manifolds where the flag of sub-bundles is produced by a bracket gener-
ating distribution H, the usual approach to face this problem is by means of the
critical points of the endpoint map [30]. The presence of singular curves is strongly
connected with the existence of abnormal geodesics, firstly established by R. Mont-
gomery in [28,29]. In the literature many papers concerning this topic have been
published, just to name a few we cite [2,1,26,24,31,3,35]. The paper [26] by
HIGHER DIMENSIONAL HOLONOMY MAP FOR RULED SUBMANIFOLDS 3
E. Le Donne, G.P. Leonardi, R. Monti and D. Vittone is specially remarkable be-
cause of the new algebraic characterization of abnormal sub-Riemannian extremals
in stratified nilpotent Lie groups.
More precisely, L. Hsu [22] defines the singular curves as the ones along which the
holonomy map fails to be surjective. This holonomy map studies the controllability
along the curve restricted to [a, b]⊂Iof a system of ODEs embodying the constraint
on sub-bundles determined by the degree. In [10, Section 5] the authors revisited
this construction and defined an admissible vector field as a solution of this system.
A powerful characterization of singular curves in terms of solutions of ODEs is
given by [22, Theorem 6]. On the other hand, when a curve γis regular restricted
to [a, b], [22, Theorem 3] ensures that for any compactly supported admissible
vector field Von [a, b] there exists a variation, preserving the original degree of γ,
whose variational vector field is V. Then, only for regular curves this deformability
theorem allows us to compute the first variation formula for the length functional
deducing the geodesic equations ([10, Section 7]), whereas for singular curves the
situation is more complicated.
The deformability problem of a higher dimensional immersion Φ : ¯
M→N
has been first studied in [11]. The admissibility system of first order linear PDEs
expressing this condition in coordinates is not easy to study. Nonetheless, [11,
Proposition 5.5] shows that only the transversal part V⊥of the vector field V=
V⊤+V⊥affects the admissibility system. Therefore, in the present work we con-
sider an adapted tangent basis E1,...,Emfor the flag (1.1) and then we add
transversal vector fields Vm+1 ,...,Vnof increasing degrees so that a sorting of
{E1,...,Em, Vm+1,...,Vn}is a local adapted basis for N. Then we consider the
metric gthat makes E1,...,Em, Vm+1,...,Vnan orthonormal basis. Hence we
obtain that the admissibility system is equivalent to
(1.2) Ej(fi) = −
n
X
r=m+k+1
bijr fr−
m+k
X
h=m+1
aijh gh,
for i=m+k+ 1, . . . , n and deg(Vi)>deg(Ej). In equation (1.2) the integer k, de-
fined after (3.1), separates the horizontal control of the systems Vh=Pm+k
l=m+1 glVl
from the vertical component Vv=Pn
r=m+k+1 frVr.
The presence of isolated submanifolds and a mild deformability theorem under
the strong regularity assumption are showed in [11]. However, the definition of
singularity for immersed submanifolds, analogous to the one provided by [22] in the
case of curves, is missing. Therefore the natural questions that arise are:
•is it possible to define a generalization of the holonomy map for submani-
folds of dimension grater than one?
•Under what condition does the surjection of these holonomy map still imply
a deformability theorem in the style of [22, Theorem 3]?
In the present paper we answer the first question in the cases of ruled m-
dimensional submanifolds whose (m−1) tangent vector fields E2,...,Emhave
degree sand the first vector field E1has degree equal to ι0, where 1 6ι06s−1.
The resulting degree is deg(M) = (m−1)s+ι0. Therefore the ruled submanifold is
foliated by curves of degree ι0out of the characteristic set M0, whose points have
degree strictly less than deg(M). Then, under a logarithmic change of coordinates
4 G. GIOVANNARDI
x= (x1,ˆx), the admissibility system (1.2) becomes
(1.3) ∂F (x)
∂x1
=−B(x)F(x)−A(x)G(x),
where ∂x1is the partial derivative in the direction E1,Gare the horizontal coordi-
nates Vh=Pm+k
l=m+1 glVl,Fare the vertical components given by Vv=Pn
r=m+k+1 frVr
and A, B are matrices defined at the end of Section 4. Therefore, this system of
ODEs is easy to solve in the direction ∂x1perpendicular to the (m−1) foliation
generated by E2,...,Em. We consider a bounded open set Σ0⊂ {x1= 0}in the
foliation, then we build the ε-cylinder Ωε={(x1,ˆx) : ˆx∈Σ0,0< x1< ε}over
Σ0. We consider the horizontal controls Gin the space of continuous functions
compactly supported in Ωε. For each fixed G,Fis the solution of (1.3) vanish-
ing on Σ0. Then we can define a higher dimensional holonomy map Hε
M, whose
image is the solution F, evaluated on the top of the cylinder Ωε. We say that a
ruled submanifold is regular when the holonomy map is surjective, namely we are
able to generate all possible compactly supported continuous vertical functions on
Σε⊂ {x1=ε}by letting vary the control Gin the space of compactly supported
continuous horizontal functions inside the cylinder Ωε. The main difference with
respect to the one dimensional case is that the target space of the holonomy map
is now the Banach space of compactly supported continuous vertical vector fields
on the foliation, instead of the finite vertical space of vectors at the final point γ(b)
of the curve. In Theorem 5.7 we provide a nice characterization of singular ruled
submaifolds in analogy with [22, Theorem 6].
For general submanifolds there are several obstacles to the construction of a
satisfactory generalization of the holonomy map. The main difficulty is that we do
not know how to verify a priori the compatibility conditions [21, Eq. (1.4), Chapter
VI], that are necessary and sufficient conditions for the uniqueness and the existence
of a solution of the admissibility system (1.2) (see [21, Theorem 3.2, Chapter VI]).
In Example 3.3 we show how we can deal with these compatibility conditions in
the particular case of horizontal immersions in the Heisenberg group.
In order to give a positive answer to the second question, we need to consider
two additional assumptions on the ruled submanifold: the first one (i) is that the
vector fields E2, . . . , Emof degree sfill the grading H1⊂... ⊂ Hsfrom the top,
namely dim(Hs)−dim(Hs−1) = m−1, and the second one (ii) is that the ruled
immersion foliated by curves of degree ι0verifies the bound s−36ι06s−1.
Under these hypotheses the space of m-vector fields of degree grater than deg(M)
is reasonably simple, thus in Theorem 6.6 we show that each admissible vector field
on a regular immersed ruled submanifold is integrable in the spirit of [22, The-
orem 3]. This result is sharper than the one obtained for general submanifolds
[11, Theorem 7.3], where the authors only provide variations of the original immer-
sion compactly supported in an open neighborhood of the strongly regular point.
Indeed, since we solve a differential linear system of equations along the charac-
teristics curves of degree ι0, we obtain a global result. On the other hand in [11,
Theorem 7.3] the admissibility system is solved algebraically assuming a pointwise
full rank condition of the matrix A( ¯p). To integrate the vector field V(¯p) on Ωε
we follow the exponential map generating the non-admissible compactly supported
variation Γτ(¯p) = expΦ( ¯p)(τV (¯p)) of the initial immersion Φ, where supp(V)⊂Ωε.
By the Implicit Function Theorem there exists a vector field Y(¯p, τ ) on Ωεvanish-
ing on Σ0such that the perturbations ˜
Γτ(¯p) = expΦ( ¯p)(τ V ( ¯p) + Y(τ, ¯p)) of Γ are
HIGHER DIMENSIONAL HOLONOMY MAP FOR RULED SUBMANIFOLDS 5
immersions of the same degree of Φ for each τsmall enough. In general ˜
Γ does
not move points on Σ0but changes the values of Φ on Σε. Finally, the regularity
condition on Φ allows us to produce the admissible variation that fixes the values
on Σεand integrate V. On the other hand, when the bundle of m-vector fields
of degree greater than deg(M) for a general ruled submanifold is larger than the
target space of the higher dimensional holonomy, we lose the surjection in Implicit
Function Theorem that allows us to perturb the exponential map to integrate V.
A direct consequence of this result is that the regular ruled immersions of degree
dthat satisfy the assumption (i) and (ii) are accumulation points for the domain
of degree darea functional Ad(·). Therefore it makes sense to consider the first
variation formula computed in [11, Section 8]. An interesting strand of research is
deducing the mean curvature equations for the critical points of the area functional
taking into account the restriction embodied by the holonomy map. Contrary to
what can be expected, we exhibit in Example 6.7 a plane foliated by abnormal
geodesics of degree one that is regular and is a critical point for the area functional
(since its mean curvature equation vanishes).
Furthermore these ruled surfaces appear in the study of the geometrical struc-
tures of the visual brain, built by the connectivity between neural cells [13]. A
geometric characterization of the response of the primary visual cortex in the pres-
ence of a visual stimulus from the retina was first described by the D. H. Hubel and
T. Wiesel [23], that discovered that the cortical neurons are sensitive to different
features such as orientation, curvature, velocity and scale. The so-called simple
cells in particular are sensitive to orientation, thus G. Citti and A. Sarti in [12]
proposed a model where the original image on the retina is lifted to a 2 dimensional
surface of maximum degree into the three-dimensional sub-Riemannian manifold
SE(2), adding orientation. In [14] they shows how minimal surfaces play an im-
portant role in the completion process of images. Adding curvature to the model,
a four dimensional Engel structure arises, see §1.5.1.4 in [34] and [15]. When in
Example 6.8 we lift the previous 2Dsurfaces in this structure we obtain surfaces
of codimension 2, but their degree is not maximum since we need to take into ac-
count the constraint that curvature is the derivative of orientation. Nevertheless
these surfaces are ruled, regular and verify the assumption (i) and (ii), therefore by
Theorem 6.6 they can be deformed. Hence, there exists a notion of mean curvature
associated to these ruled surfaces and we might ask if the completion process of
images improved for SE(2) based on minimal surfaces can be generalized to this
framework. Moreover, if we lift the original retinal image to higher dimensional
spaces adding variables that encode new possible features, as suggested in [32] fol-
lowing even a non-differential approach based on metric spaces, we may ask if the
lifted surfaces are still ruled and regular.
The paper is organized as follows. In Section 2we recall the definitions of
graded manifolds, degree of a submanifold, admissible variations and admissible
vector fields. In Section 3we deduce the admissibility system (1.2). In Section 4
we provide the definition of ruled submanifolds. Section 5is completely devoted
to the description of the higher-dimensional holonomy map and characterization of
regular and singular ruled submanifolds. Finally, in Section 6we give the proof of
Theorem 6.6.
Acknowledgement. I warmly thank my Ph.D. supervisors Giovanna Citti and Manuel
Ritor´e for their advice and for fruitful discussions that gave rise to the idea of higher
6 G. GIOVANNARDI
dimensional holonomy map. I would also like to thank Noemi Montobbio for an
interesting conversation on proper subspaces of Banach spaces.
2. Preliminaries
Let Nbe an n-dimensional smooth manifold. Given two smooth vector fields
X, Y on N, their commutator or Lie bracket is defined by [X, Y ] := X Y −Y X . An
increasing filtration (Hi)i∈Nof the tangent bundle T N is a flag of sub-bundles
(2.1) H1⊂ H2⊂ · · · ⊂ Hi⊂ · · · ⊆ T N,
such that
(i) ∪i∈NHi=T N
(ii) [Hi,Hj]⊆ Hi+j,for i, j >1,
where [Hi,Hj] := {[X, Y ] : X∈ Hi, Y ∈ Hj}. Moreover, we say that an increasing
filtration is locally finite when
(iii) for each p∈Nthere exists an integer s=s(p), the step at p, satisfying
Hs
p=TpN. Then we have the following flag of subspaces
(2.2) H1
p⊂ H2
p⊂ · · · ⊂ Hs
p=TpN.
Agraded manifold (N, (Hi)) is a smooth manifold Nendowed with a locally
finite increasing filtration, namely a flag of sub-bundles (2.1) satisfying (i),(ii) and
(iii). For the sake of brevity a locally finite increasing filtration will be simply called
a filtration. Setting ni(p) := dim Hi
p, the integer list (n1(p),··· , ns(p)) is called the
growth vector of the filtration (2.1) at p. When the growth vector is constant in a
neighborhood of a point p∈Nwe say that pis a regular point for the filtration.
We say that a filtration (Hi) on a manifold Nis equiregular if the growth vector is
constant in N. From now on we suppose that Nis an equiregular graded manifold.
Given a vector vin TpNwe say that the degree of vis equal to ℓif v∈ Hℓ
pand
v /∈ Hℓ−1
p. In this case we write deg(v) = ℓ. The degree of a vector field is defined
pointwise and can take different values at different points.
Let (N, (H1,...,Hs)) be an equiregular graded manifold. Take p∈Nand con-
sider an open neighborhood Uof pwhere a local frame {X1,··· , Xn1}generating H1
is defined. Clearly the degree of Xj, for j= 1,...,n1, is equal to one since the vector
fields X1,...,Xn1belong to H1. Moreover the vector fields X1,...,Xn1also lie in
H2, we add some vector fields Xn1+1,··· , Xn2∈ H2\ H1so that (X1)p,...,(Xn2)p
generate H2
p. Reducing Uif necessary we have that X1,...,Xn2generate H2in U.
Iterating this procedure we obtain a basis of T M in a neighborhood of p
(2.3) (X1,...,Xn1, Xn1+1,...,Xn2,...,Xns−1+1,...,Xn),
such that the vector fields Xni−1+1,...,Xnihave degree equal to i, where n0:= 0.
The basis obtained in (2.3) is called an adapted basis to the filtration (H1,...,Hs).
Given an adapted basis (Xi)16i6n, the degree of the simple m-vector field Xj1∧
...∧Xjmis defined by
deg(Xj1∧...∧Xjm) :=
m
X
i=1
deg(Xji).
Any m-vector Xcan be expressed as a sum
Xp=X
J
λJ(p)(XJ)p,
HIGHER DIMENSIONAL HOLONOMY MAP FOR RULED SUBMANIFOLDS 7
where J= (j1, . . . , jm), 1 6j1<··· < jm6n, is an ordered multi-index, and
XJ:= Xj1∧...∧Xjm. The degree of Xat pwith respect to the adapted basis
(Xi)16i6nis defined by
max{deg((XJ)p) : λJ(p)6= 0}.
It can be easily checked that the degree of Xis independent of the choice of the
adapted basis and it is denoted by deg(X).
If X=PJλJXJis an m-vector expressed as a linear combination of simple
m-vectors XJ, its projection onto the subset of m-vectors of degree dis given by
(2.4) (X)d=X
deg(XJ)=d
λJXJ,
and its projection over the subset of m-vectors of degree larger than dby
Πd(X) = X
deg(XJ)>d+1
λJXJ.
In an equiregular graded manifold with a local adapted basis (X1,...,Xn), de-
fined as in (2.3), the maximal degree that can be achieved by an m-vector, m6n,
is the integer dm
max defined by
(2.5) dm
max := deg(Xn−m+1) + ···+ deg(Xn).
2.1. Degree of a submanifold. Let Mbe a submanifold of class C1immersed in
an equiregular graded manifold (N , (H1,...,Hs)) such that dim(M) = m < n =
dim(N). Following [25,27], we define the degree of Mat a point p∈Mby
degM(p) := deg(v1∧...∧vm),
where v1,...,vmis a basis of TpM. The degree deg(M) of a submanifold Mis the
integer
deg(M) := max
p∈MdegM(p).
We define the singular set of a submanifold Mby
(2.6) M0={p∈M: degM(p)<deg(M)}.
Singular points can have different degrees between mand deg(M)−1. Following
[20, 0.6.B] an alternative way to define the pointwise degree is by means of the
formula
degM(p) =
s
X
j=1
j( ˜mj−˜mj−1),
setting ˜m0= 0 and ˜mj= dim(TpM∩ Hj
p). Namely, the degree is the homogenous
dimension of the flag
(2.7) ˜
H1
p⊂˜
H2
p⊂ · · · ⊂ ˜
Hs
p=TpM,
where ˜
Hj
p:= TpM∩ Hj
p. As we pointed out in [11, Section 3] the area functional
associated to an immersed sumbanifold depends on the degree.
Definition 2.1. Let Mbe a C1immersed submanifold of degree d= deg(M) in an
equiregular graded manifold (N , H1,...,Hs) endowed with a Riemannian metric
8 G. GIOVANNARDI
g. Let µbe a Riemannian metric in Mand e1,...,embe an µ-orthonormal basis.
Then the degree darea Adis defined by
(2.8) Ad(M′) = ZM′
|(e1∧...∧em)d|gdµ(p).
for any bounded measurable set M′⊂M. In the previous formula (·)ddenotes the
projection onto the subset of m-vectors of degree ddefined in (2.4).
2.2. Admissible variations and admissibility system of PDEs. Given a
graded manifold (N , H1,...,Hs), we consider a generic Riemannian metric g=h·,·i
on T N . Let Φ : ¯
M→Nbe a C1immersion in N, we set M= Φ( ¯
M) and
d= deg(M). Let (Xi)ibe a local adapted basis around p∈M. Following [11,
Section 5] we recall the following definitions
Definition 2.2. A smooth map Γ : ¯
M×(−ε, ε)→Nis said to be an admissible
variation of Φ if Γt:¯
M→N, defined by Γt(¯p) := Γ( ¯p, t), satisfies the following
properties
(i) Γ0= Φ,
(ii) Γt(¯
M) is an immersion of the same degree as Φ( ¯
M) for small enough t, and
(iii) Γt(¯p) = Φ(¯p) for ¯poutside a given compact subset of ¯
M.
Definition 2.3. Given an admissible variation Γ, the associated variational vector
field is defined by
(2.9) V(¯p) := ∂Γ
∂t ( ¯p, 0).
Since it turns out that variational vector fields associated to an admissible vari-
ations satisfy the system (2.10) (see [11, Section 5]) we are led to the following
definition
Definition 2.4. Given an immersion Φ : ¯
M→N, a vector field V∈X0(¯
M , N) is
said to be admissible if it satisfies the system of first order PDEs
(2.10) 0 = he1∧...∧em,∇V(p)XJi+
m
X
j=1
he1∧...∧ ∇ejV∧...∧em, XJi
where XJ=Xj1∧...∧Xjm, deg(XJ)> d and e1,...,emis basis of TpM.
Definition 2.5. We say that an admissible vector field V∈X0(¯
M , N) is integrable
if there exists an admissible variation such that the associated variational vector
field is V.
3. Intrinsic coordinates for the admissibility system of PDEs
Let Φ : ¯
M→Nbe a C1,1immersion in a graded manifold, M= Φ( ¯
M) and
d= deg(M). By [11, Proposition 6.4] we realize that the admissibility of a vector
field Vis independent of the metric. Therefore we can use any metric in order
to study the system. Let pbe a point in M\M0, that is an open set thanks to
[11, Corollary 2.4]. We consider e1,...,ema basis of TpMadapted to the flag
(2.7). Then we complete this basis to a basis of the ambient space TpNadding
vm+1,...,vnof increasing degree such that a sorting of {e1,...,em, vm+1,...,vn}
is an adapted basis of TpN. Thus we extend e1,...,em, vm+1,...,vnto vector fields
E1,...,Em, Vm+1,...,Vnso that their sorting is still adapted in a neighborhood of
p. Since the immersion is C1,1, the vector fields E1,...,Emare Lipschitz, thus the
HIGHER DIMENSIONAL HOLONOMY MAP FOR RULED SUBMANIFOLDS 9
vector fields Vm+1,...,Vnare also Lipschitz. Then we consider the metric g=h·,·i
that makes E1, . . . , Em, Vm+1,...,Vnan orthonormal basis in a neighborhood Uof
p.
Definition 3.1. Letting ι0be the integer defined by
(3.1) ι0(U) = max
p∈Umin
16α6s{α: ˜mα(p)6= 0},
we set
(3.2) k:= nι0−˜mι0
Given a generic vector field Wtransversal to T M , the only simple m-vectors of
degree strictly greater than dwhose scalar product with
(3.3) E1∧ · · · ∧
(j)
W∧ · · · ∧ Em
are candidate to be different from zero are
E1∧ · · · ∧
(j)
Vi∧ · · · ∧ Em
for i=m+ 1, . . . , n and deg(Vi)>deg(Ej). Since (Vi)iand (Ej)jhave increasing
degree, we obtain deg(Vi)>deg(E1) = ι0if and only if i=m+k+ 1,...,n, where
kis defined in (3.2). Therefore we deduce that the candidates simple m-vector of
degree strictly greater than dwhose scalar product against (3.3) is different from
zero are
E1∧ · · · ∧
(j)
Vi∧ · · · ∧ Em
for i=m+k+ 1,...,n and deg(Vi)>deg(Ej). Furthermore by [11, Proposition
5.5] we know that Vis admissible if and only if
(3.4) V⊥=
m+k
X
h=m+1
ghVh+
n
X
r=m+k+1
frVr
is admissible. Therefore putting V⊥in (2.10) we obtain
(3.5)
m
X
j=1n
X
r=m+k+1
˜cijrα Ej(fr) +
m+k
X
h=m+1
˜cijhα Ej(gh)
+
n
X
r=m+k+1
˜
bijrα fr+
m+k
X
h=m+1
˜aijhα gh= 0,
where
˜cijtα =hE1∧ · · · ∧
(j)
Vt∧ · · · ∧ Em, E1∧ · · · ∧
(α)
Vi∧ · · · ∧ Emi
˜aijhα =hE1∧...∧
(j)
[Ej, Vh]∧...∧Em, E1∧ · · · ∧
(α)
Vi∧ · · · ∧ Emi
˜
bijrα =hE1∧...∧
(j)
[Ej, Vr]∧...∧Em, E1∧ · · · ∧
(α)
Vi∧ · · · ∧ Emi,
for t=m+ 1, . . . , n,r=m+k+ 1,...,n,h=m+ 1,...,m+k,α= 1,...,m,
i=m+k+ 1, . . . , n and deg(Vi)>deg(Eα). Then we have that ˜cij tα is equal
to 1 for i=t > m +k,α=jand deg(Vi)>deg(Ej) or equal to zero otherwise.
10 G. GIOVANNARDI
Moreover, we notice that ˜aij hα and ˜
bijrα are different from zero only when α=j
and in particular we have
(3.6) aijh := ˜aijhj =hVi,[Ej, Vh]i,
for h=m+ 1, . . . , m +k,i=m+k+ 1,...,n, deg(Vi)>deg(Ej) and
(3.7) bijr := ˜
bijrj =hVi,[Ej, Vr]i,
for i, r =m+k+ 1,...,n and deg(Vi)>deg(Ej). Therefore Vis admissible if and
only if
(3.8) Ej(fi) = −
n
X
r=m+k+1
bijr fr−
m+k
X
h=m+1
aijh gh,
for i=m+k+ 1,...,n and deg(Vi)>deg(Ej).
Remark 3.2. Notice that the coefficients aijh and bij r defined in (3.6) and (3.7) are
defined almost everywhere. Indeed the vector fields E1,...,Em, Vm+1 ,...,Vnare
Lipschitz, then thanks to [16] the Lie brackets [Ej, Vh] and [Ej, Vr] for j= 1,...,m,
h=m+ 1,...,m+kand r=m+k+ 1,...,n are defined almost everywhere.
Example 3.3 (Horizontal submanifolds).Given n > 1 we consider the Heisenberg
group Hn, defined as R2n+1 endowed with the distribution Hgenerated by
Xi=∂
∂xi
+yi
2
∂
∂t
, Yi=∂
∂yi
−xi
2
∂
∂t
i= 1,...,n.
The Reeb vector fields is provided by T=∂t= [Xi, Yi] for i= 1,...,n
and has degree equal to 2. Let g=h·,·i be the Riemannian metric that make
(X1,...,Xn, Y1, . . . , Yn, T ) an orthonormal basis. We rename the horizontal vector
fields X1,...,Xn, Y1,...,Ynwith Z1,...,Zn, Zn+1 ,...,Z2n. Let Ω be an open set
of Rm, with m6n. Here we consider a C1,1immersion Φ : Ω →Hnsuch M= Φ(Ω)
is a Lagrangian submanifold. Let E1,...,Embe an orthonormal local frame of T M ,
then there exist Zi1,...,Zinsuch that [Ziα, Ziβ] = 0 for each α, β = 1,...,n and
(3.9) Ej=
n
X
α=1
aα
j(¯p)Ziαfor j= 1,...,m,
where the matrix A= (aα
j(¯p))α=1,...,n
j=1,...,m has full rank equal to m, for each ¯p∈Ω.
Therefore a vector field V=Pn
i=1 giXi+gi+nYi+f T is admissible if and only if
it satisfies the system (3.8), that in this case is given by
Ej(f) = −h[Ej, T ], T if−
n
X
i=1
(h[Ej, Xi], T igi+h[Ej, Yi], T igi+n),
for j= 1,...,m. A straightforward computation shows that this system is equiva-
lent to
(3.10) Ej(f) = −
n
X
α=1
aα
jgiαfor j= 1,...,m.
A necessary and sufficient conditions for the uniqueness and the existence of a
solution of the admissibility system (3.10) (see [21, Theorem 3.2, Chapter VI]) are
HIGHER DIMENSIONAL HOLONOMY MAP FOR RULED SUBMANIFOLDS 11
given by
(3.11) EjEν(f)−EνEj(f) = −Ej
n
X
β=1
aβ
νgiβ
+Eν n
X
α=1
aα
jgiα!,
for each j, ν = 1, . . . , m. These are the so called integrability condition [21, Eq.
(1.4), Chapter VI]. A straightforward computation shows that the right hand side
of is equal to
(3.12)
n
X
α,β=1
aβ
νaα
j(Ziβ(giα)−Ziα(giβ)) −aα
jZiα(aβ
ν)giβ+aβ
νZiβ(aα
j)giα.
Moreover, the left hand side is equal to
[Ej, Eν](f) =
m
X
k=1
ck
jν Ek(f) = −
m
X
k=1
ck
jν
n
X
γ=1
aγ
kgiγ
=−
n
X
γ=1
h[Ej, Eν], Ziγigiγ
=
n
X
β,α=1
aβ
νZiβ(aα
j)giα−aα
jZiα(aβ
ν)giβ.
Therefore the compatibility (or integrability) conditions are given by
(3.13)
n
X
α,β=1
aβ
νaα
j(Ziβ(giα)−Ziα(giβ)) = 0,
for each ν, j = 1, . . . , m. Moreover, when the horizontal submanifold is a Lagrangian
manifold of dimension m=n, the compatibility conditions (3.13) are equivalent to
Ziβ(giα) = Ziα(giβ)
for each α, β = 1, . . . , n.
Remark 3.4. Notice that if we want to find a solution fof (3.10), the controls
gi1,...,ginhave to verify the compatibility conditions (3.13). Therefore to obtain
a suitable generalization of the holonomy map (defined for curves in [10, Section 5])
we need to consider the subspace of the space of horizontal vector fields on Mthat
verify (3.13). We recognize that studying the holonomy map for these horizontal
immersions is engaging problem that have been investigated by [19,33], but in the
present work we will consider different kind of immersions that allow us to forget
these compatibility conditions in the construction of the high dimensional holonomy
map.
4. Ruled submanifolds in graded manifolds
In this section we consider a particular type of submanifolds for which the ad-
missibility system reduces to a system of ODEs along the characteristic curves,
that rule these submanifolds by determining their degree since the other adapted
tangent vectors tangent to Mhave highest degree equal to s.
12 G. GIOVANNARDI
Definition 4.1. Let (N , H1,...,Hs) be an equiregular graded manifold and let ¯
M
am-dimensional manifold with m < n. We say that an immersion Φ : ¯
M→Nis
ruled if
(4.1) deg(M) = (m−1)s+ι0,
where 1 6ι06s−1 and M= Φ( ¯
M). In this case, we will call the image of the
immersion Ma ruled submanifold.
Let pbe a point of maximum degree in M. Let e1,...,embe a basis of TpM
adapted to the flag (2.7). Therefore deg(e1) = ι0and deg(ej) = sfor j= 2,...,m
and k=nι0−1. Then we follow the construction described in Section 3to provide
the metric gand the orthonormal basis E1,...,Em, Vm+1,...,Vnwhose sorting
is an adapted basis. Since deg(Ej)>deg(Vi) for each j= 2,...,m and i=
m+k+ 1,...,n, the only derivative that appears in (3.8) is E1. Therefore we
deduce that a vector field V⊥, given by equation (3.4), is admissible if and only if
it satisfies
(4.2) E1(fi) +
n
X
r=m+k+1
bi1rfr+
m+k
X
h=m+1
ai1hgh= 0,
for i=m+k+ 1,...,n and
ai1h(p) = hvi,[E1, Vh](p)i,
and
bi1r(p) = hvi,[E1, Vr](p)i.
Given pin Meach point qin a local neighborhood Uof pin Mcan be reached
using the exponential map as follows
q= exp(x1E1) exp
m
X
j=2
xjEj
(p).
On this open neighborhood Uwe consider the local coordinates x= (x1, x2,...,xm)
given by logarithmic map Ξ. We set ˆx:= (x2,...,xm). Given a relative compact
open subset Ω ⊂⊂ Ξ(U) we consider
(4.3) Σ0={x1= 0} ∩ Ω
be the (m−1)-dimensional leaf normal to E1. Then there exists ε > 0 so that the
closure of the cylinder
(4.4) Ωε={(x1,ˆx) : 0 < x1< ε, ˆx∈Σ0}
is contained in Ξ(U). Then Σε={(ε, ˆx) : ˆx∈Σ0}is the top of the cylinder. Since
dΞ(E1) = ∂x1in this logarithmic coordinates the admissibility system (4.2) is given
by
(4.5) ∂F (x)
∂x1
=−B(x)F(x)−A(x)G(x),
where we set
(4.6) F=
fm+k+1
.
.
.
fn
, G =
gm+1
.
.
.
gm+k
HIGHER DIMENSIONAL HOLONOMY MAP FOR RULED SUBMANIFOLDS 13
and we denote by Bthe (n−m−k) square matrix whose entries are bi1r, by Athe
(n−m−k)×kmatrix whose entries are ai1h.
5. The high dimensional holonomy map for ruled submanifolds
For ruled submanifolds the system (3.8) reduces to the system of ODEs (4.2)
along the characteristic curves. Therefore, a uniqueness and existence result for
the solution is given by the classical Cauchy-Peano Theorem, as in the case of
curves in [10, Section 5].
Let Φ : ¯
M→Nbe a ruled immersion in a graded manifold. Let Ωεbe the open
cylinder defined in (4.4) and TΣ0(f) = f(0,·) and TΣε(f) = f(ε, ·) be the operators
that evaluate functions at x1= 0 and at x1=ε, respectively. Then we consider
the following spaces:
(1) H0(Ωε) = (m+k
X
i=m+1
giVi:gi∈C0(Ωε)).
(2) V1(Ωε) = (n
X
i=m+k
fiVi:∂x1fi∈C(¯
Ωε), fi∈C(¯
Ωε), TΣ0(fi) = 0).
(3) V(Σε) is the set of compactly supported vertical vector fields in C0(Σε,Rn−m−k)
normal to M.
Therefore the Cauchy problem allows us to define the holonomy type map
(5.1) Hε
M:H0(Ωε)→ V(Σε),
in the following way: we consider a horizontal compactly supported continuous
vector
Yh=
m+k
X
l=m+1
glVl∈ H0(Ωε)
we fix the initial condition Yv(0,ˆx) = 0. Then there exists a unique solution
Yv=
n
X
r=m+k+1
frVr∈ V1(Ωε)
of the admissibility system (4.5) with initial condition Yv(0,ˆx) = 0. Letting
TΣε:V1(Ωε)→ V(Σε)
be the evaluating operator for vertical vectors fields at x1=εdefined by TΣε(V) =
V(ε, ·), we define Hε
M(Yh) = TΣε(Yv).
Definition 5.1. We say that Φ restricted to ¯
Ωεis regular if the holonomy map
Hε
Mis surjective.
The following result allows the integration of the differential system (4.5) to
explicitly compute the holonomy map.
Proposition 5.2. In the above conditions, there exists a square regular matrix
D(x1,ˆx)of order (n−k−m)such that
(5.2) F(ε, ˆx) = −D(ε, ˆx)−1Zε
0
(DA)(τ, ˆx)G(τ, ˆx)dτ,
for each ˆx∈Σ0.
14 G. GIOVANNARDI
Proof. Lemma 5.3 below allows us to find a regular matrix D(x1,ˆx) such that
∂x1D=DB. Then equation ∂x1F=−BF −AG is equivalent to ∂x1(DF ) = −DAG.
Integrating between 0 and ε, taking into account that F(0,ˆx) = 0 for each ˆx∈Σ0,
and multiplying by D(ε, ˆx)−1, we obtain (5.2).
Lemma 5.3. Let Ebe an open set of Rm−1. Let B(t, λ)be a continuous family of
square matrices on [0, ε]×E. Let D(t, λ)be the solution of the Cauchy problem
∂tD(t, λ) = D(t, λ)B(t, λ)on [0, ε]×E, D(0, λ) = Id,
for each λ∈E. Then det D(t, λ)6= 0 for each (t, λ)∈[0, ε]×E.
Definition 5.4. We say that the matrix ˜
A(x1,ˆx) := (DA)(x1,ˆx) on Ωεdefined in
Proposition 5.2 is linearly full Rn−m−kif and only if for each ˆx∈Σ0
dim span n˜
A1(x1,ˆx),..., ˜
Ak(x1,ˆx)∀x1∈[0, ε]o=n−m−k,
where ˜
Aifor i=m+ 1,...,m+kare the columns of ˜
A(x1,ˆx).
Lemma 5.5. Let L:X→Ybe a linear closed operator of Banach spaces. Then
Lis not surjective if and only if there exists µ∈Y∗,µ6≡ 0such that µ(y) = 0 for
each y∈Range(L).
Proof. Assume that Lis not surjective, namely the subspace Range(L) = Range(L)(
Y, then by [5, Corollary 1.8] we obtain the result. Conversely by contradiction as-
sume that Range(L) = Y, but by assumption there exists a a dual function µ6≡ 0
such that µ(y) = 0 for each y∈Y, which is absurd.
Proposition 5.6. The immersion Φrestricted to ¯
Ωεis regular if and only if
˜
A(x1,ˆx)is linearly full in Rn−m−k.
Proof. Assume that the holonomy map is not surjective. The representation for-
mula (5.2) allows us to deduce that the linear map Hε
Σis closed, since the limit
of integrals of an uniform sequence of continuos functions converges to the integral
of the uniform limit of the sequence. Since the dual of the space of compactly
supported continuous functions is the space of Radon measures (see [17, Chapter
7]), by Lemma 5.5 there exists a Radon measure µ6= 0 and a continuous row vector
Γ(ˆx) such that
0 = Γµ(F(ε, ·)) = −µΓ(ˆx)D(ε, ˆx)−1Zε
0
(DA)(τ, ˆx)G(τ, ˆx)dτ
=−Zε
0ZΣ0
˜
Γ(ˆx)(DA)(τ, ˆx)G(τ, ˆx)dµ(ˆx)dτ
where ˜
Γ = Γ(ˆx)D(ε, ˆx)−16= 0. As this formula holds for any G(t, ˆx), we have
Γ(ˆx)˜
A(t, ˆx) = 0 for all t∈[a, b] and µ-a.e. in ˆx. Since the supp(µ)6=∅there
exists ˆx0∈supp(µ) such that Γ(ˆx0)˜
A(t, ˆx0) = 0, then their columns are contained
in the hyperplane of Rn−m−kdetermined by Γ(ˆx0). Hence we deduce that ˜
Ais not
linearly full.
Conversely, assume that ˜
Ais not linearly full. Then there exist a point ˆx0∈Σ0
and a row vector with (n−m−k) coordinates Γ 6= 0 such that Γ ˜
A(x1,ˆx0) = 0 for
all x1∈[0, ε]. Then, denoting by δˆx0(ϕ) = ϕ(ˆx0) the delta distribution, we have
Γδˆx0(D(ε, ·)F(ε, ·)) = −Zε
0
Γ(DA)(τ, ˆx0)G(τ, ˆx0)dτ = 0
HIGHER DIMENSIONAL HOLONOMY MAP FOR RULED SUBMANIFOLDS 15
Since the Radon measure δˆx0annihilates the image of the holonomy map by Lemma
5.5 we conclude that the holonomy map is not surjective.
The following result provides a useful characterization of non-regularity
Theorem 5.7. The immersion Φrestricted to ¯
Ωεis non-regular if and only if there
exist a point ˆx0∈Σ0and a row vector field Λ(x1,ˆx0)6= 0 for all x1∈[0, ε]that
solves the following system
(5.3) (∂x1Λ(x1,ˆx0) = Λ(x1,ˆx0)B(x1,ˆx0)
Λ(x1,ˆx0)A(x1,ˆx0) = 0.
Proof. Assume that Φ restricted to ¯
Ωεis non-regular, then by Proposition 5.6 there
exist a point ˆx0∈Σ0and a row vector Γ 6= 0 such that
ΓD(x1,ˆx0)A(x1,ˆx0) = 0
for all x1∈[0, ε], where D(x1,ˆx0) solves
(5.4) (∂x1D=DB
D(0,ˆx0) = In−m−k.
Since Γ is a constant vector and D(x1,ˆx0) is a regular matrix by Lemma 5.3 ,
Λ(x1,ˆx0) := ΓD(x1,ˆx0) solves the system (5.3) and Λ(x1,ˆx0)6= 0 for all x1∈[0, ε].
Conversely, any solution of the system (5.3) is given by
Λ(x1,ˆx0) = ΓD(x1,ˆx0),
where Γ = Λ(0,ˆx0)6= 0 and D(x1,ˆx0) solves the equation (5.4). Indeed, let us
consider a general solution Λ(t, ˆx0) of (5.3). If we set
Ψˆx0(t) = Λ(t, ˆx0)−ΓD(t, ˆx0),
where Γ = Λ(0,ˆx0)6= 0 and D(t, ˆx0) solves the equation (5.4), then we deduce
(∂tΨˆx0(t) = Ψˆx′(t)B(t, ˆx0)
Ψˆx0(0) = 0.
Clearly the unique solution of this system is Ψˆx0(t)≡0. Hence we conclude that
Γ˜
A(x1,ˆx0) = 0. Thus ˜
A(x1,ˆx0) is not fully linear and by Proposition 5.6 we are
done.
6. Integrability of admissible vector fields for a ruled regular
submanifold
In this section we deduce the main result Theorem 6.6. As we pointed out in
the Introduction we need that the space of simple m-vectors of degree grater than
deg(M) is quite simple. Therefore we give the following definition.
Definition 6.1. We say that a m-dimensional ruled immersion, see Definition 4.1,
Φ : ¯
M→Ninto an equiregular graded manifold (N, H1,...,Hs)
(i) fills the grading from the top if ns−ns−1=m−1, where ns= dim(Hs)
and ns−1= dim(Hs−1);
(ii) is foliated by curves of degree grater than or equal to s−3 if ι0>s−3.
16 G. GIOVANNARDI
A ruled submanifold verifying (i) and (ii) will be called a FGT-(s−3) ruled sub-
manifold and in this case (ii) is equivalent to
(6.1) s−36ι06s−1.
Remark 6.2. Since ns−ns−1=m−1 and the condition (6.1) holds we have that
the only simple m-vectors of degree strictly grater than deg(M) are
Vi∧E2∧ · · · ∧ Em
for i=m+k+ 1,...,n. When ι0=s−1 the submanifold has maximum degree
therefore all vector fields are admissible, thus there are no singular submanifold.
Keeping the previous notation we now consider the following spaces
(1) H(Σ0) = (YH=
m+k
X
i=m+1
giVi:gi∈C(¯
Ωε), TΣ0(gi) = 0)where the norm
is given by
kYhk∞:= max
i=m+1,...,m+ksup
x∈¯
Ωε
|gi|
(2) V1(Σ0) = (Yv=
n
X
i=m+k
fiVi:∂x1fi∈C(¯
Ωε), fi∈C(¯
Ωε), TΣ0(fi) = 0),
where the norm is given by
kYvk1:= max
i=m+k,...,n( sup
x∈¯
Ωε
|fi|+ sup
x∈¯
Ωε
|∂x1fi|)
(3) Λ(Σ0) is the set of elements given by
n
X
i=m+k+1
zi(x1,...,xm)Vi∧E2∧ · · · ∧ Em
where zi∈C(¯
Ωε) vanishing on Σ0.
We denote by Πdthe orthogonal projection over the space Λ(Σ0), that is the bundle
over the vector space of simple m-vectors of degree strictly grater than d, thanks
to Remark 6.2. Then we set
(6.2) G:H(Σ0)× V1(Σ0)→ H(Σ0)×Λ(Σ0),
defined by
G(Y1, Y2) = (Y1,F(Y1+Y2)),
where
F(Y) = Πd(dΓ(Y)(e1)∧...∧dΓ(Y)(em)) ,
and Γ(Y)(p) = expΦ(¯p)(Yp). Observe that now F(Y) = 0 implies that the degree
of the variation Γ(Y) is less than or equal to d. Then
DG(0,0)(Y1, Y2) = (Y1, D F (0)(Y1+Y2)),
where DF (0)Yis given by
DF (0)Y=
n
X
i=m+k+1 ∂fi(x)
∂x1
+
n
X
r=m+k
bi1rfr+
m+k
X
h=m+1
ai1hghVi∧E2∧ · · · ∧ Em.
Observe that DF (0)Y= 0 if and only if Yis an admissible vector field, namely Y
solves (4.5).
HIGHER DIMENSIONAL HOLONOMY MAP FOR RULED SUBMANIFOLDS 17
Our objective now is to prove that the map DG(0,0) is an isomorphism of Banach
spaces. To show this, we shall need the following result.
Proposition 6.3. The differential DG(0,0) is an isomorphism of Banach spaces.
Proof. We first observe that DG(0,0) is injective, since DG(0,0)(Y1, Y2) = (0,0)
implies that Y1= 0 and that the vertical vector field Y2satisfies the compatibility
equations with initial condition Y2(0,ˆx) = 0 for each ˆx∈Σ0. Hence Y2= 0. The
map DG(0,0) is continuous. Indeed, if for instance we consider the 1-norm on the
product space we have
kDG(0,0)(Y1, Y2)k=k(Y1, D F (0)(Y1+Y2))k
6kY1k∞+kDF (0)(Y1+Y2))k∞
6(1 + k(ahij )k∞)kY1k∞+ (1 + k(brij )k∞)kY2k1.
To show that DG(0,0) is surjective, we take (Y1, Y2) in the image, and we find
a vector field Yon Ωεsuch that YH=Y1,DF(0)(Y) = Y2and Yv(0,ˆx) = 0. The
map DG(0,0) is open because of the estimate (6.3) given in Lemma 6.4 below.
Lemma 6.4. In the above conditions, assume that DF(0)(Y) = Y2and Yh=Y1
and Y(a) = 0. Then there exists a constant Ksuch that
(6.3) kYvk16K(kY2k∞+kY1k∞)
Proof. We write
Y1=
m+k
X
h=m+1
ghVh, Y2=
n
X
i=k+1
ziVi∧E2∧ · · · ∧ Emand Yv=
n
X
r=k+1
frVr.
Then Yvis a solution of the ODE (4.5) given by
(6.4) ∂x1F(x1,ˆx) = −B(x)F(x1,ˆx) + Z(x1,ˆx)−A(x)G(x1,ˆx)
where B(x), A(x) are defined after (4.6), F,Gare defined in (4.6) and we set
Z=
zm+k+1
.
.
.
zn
.
Since Yv(0,ˆx) = 0 an Yvsolves (6.4) in (0, ε), by Lemma 6.5 there exists a constant
Ksuch that
(6.5) kYvk1=kFk16KkZ(x)−A(x)G(x)k∞
6˜
K(kY2k∞+kY1k∞).
where ˜
K=Kmax{1,kA(x)k∞}.
Lemma 6.5. Let Ebe an open set of Rm−1. Let u: [0, ε]×E→Rdbe the solution
of the inhomogeneous problem
(6.6) (u′(t, λ) = A(t, λ)u(t, λ) + c(t, λ),
u(0, λ) = u0(λ)
18 G. GIOVANNARDI
where A(t, λ)is a d×dcontinuos matrix on [0, ε]×Eand c(t, λ)a continuos vector
field on [0, ε]×E. We denote by u′the partial derivative ∂tu. Then, there exists
a constant Ksuch that
(6.7) kuk1:= kuk∞+ku′k∞6K(kck∞+|u0|∞).
Proof. We start from the case r= 1. By [21, Lemma 4.1] it follows
u(t, λ)6|u0(λ)|+Zt
0
|c(s, λ)|dse|Rt
0kA(s,λ)kds|,
for each λ∈Eand where the norm of Ais given by sup|x|=1 |A x|. Therefore we
have
(6.8) sup
t∈[0,ε]
sup
λ∈E
|u(t, λ)|6C1( sup
t∈[0,ε]
sup
λ∈E
|c(t, λ)|+ sup
λ∈E
|u0(λ)|),
where we set
C1=εeεsupt∈[0,ε]supλ∈EkA(t,λ)k.
Since uis a solution of (6.6) it follows
(6.9)
sup
t∈[0,ε]
sup
λ∈E
|u′(t, λ)|6sup
t∈[0,ε]
sup
λ∈E
kA(t, λ)ksup
t∈[0,ε]
sup
λ∈E
|u(t, λ)|+ sup
t∈[0,ε]
sup
λ∈E
|c(t, λ)|
6(C2+ 1) sup
t∈[0,ε]
sup
λ∈E
|c(t, λ)|.
Hence by (6.8) and (6.9) we obtain
kuk16K(kck∞+ku0k∞).
Finally, we use the previous constructions to give a criterion for the integrability
of admissible vector fields along a horizontal curve.
Theorem 6.6. Let Φ : ¯
M→Nbe a ruled FGT-(s−3) immersion into an equireg-
ular graded manifold (N , H1,...,Hs)such that deg(M) = (m−1)s+ι0, where
m= dim( ¯
M), and (i)and (ii)in 6.1 hold. Let Ωε={(x1,ˆx) : 0 < x1< ε, ˆx∈Σ0}
with Σ0defined in (4.3). Assume that Φis regular on the compact ¯
Ωε. Then every
admissible vector field with compact support in Ωεis integrable.
Proof. If ι0=s−1 all vector fields are admissible, then all immersions are auto-
matically regular. Each vector field is integrable for instance by the exponential
map.
Let now s−36ι06s−2. Let us take Vvector field on Ωεand {Vi}∞
i=1 vector
fields equi-bounded in the supremum norm on ¯
Ωε. Let l1(R) the space of summable
sequences. We consider the map
˜
G:(−ε, ε)×l1(R)× H(Σ0)× V1(Σ0)→ H(Σ0)×Λ(Σ0),
given by
˜
G((τ, (τi), Y1, Y2)) = (Y1, F (τ V +
∞
X
i=1
τiVi+Y1+Y2)).
The map ˜
Gis continuous with respect to the product norms (on each factor we put
the natural norm, the Euclidean one on the interval, the l1norm and || · ||∞and
|| · ||1in the spaces of vectors on Ω). Moreover
˜
G(0,0,0,0) = (0,0),
HIGHER DIMENSIONAL HOLONOMY MAP FOR RULED SUBMANIFOLDS 19
since the curve γis horizontal. Denoting by DYthe differential with respect to the
last two variables of ˜
Gwe have that
DY˜
G(0,0,0,0)(Y1, Y2) = DG(0,0)(Y1, Y2)
is a linear isomorphism. We can apply the Implicit Function Theorem to obtain
maps
Y1: (−ε, ε)×l1(ε)→ H(Σ0), Y2: (−ε, ε)×l1(ε)→ V1(Σ0),
such that ˜
G(τ, (τi),(Y1)(τ, τi),(Y2)(τ, τi)) = (0,0). We denote by l1(ε) the ball of
radio εin Banach space l1(R). This implies that (Y1)(τ, (τi)) = 0 and that
F(τV +
∞
X
i=1
τiVi+Y2(τ, τi)) = 0.
Hence the submanifolds
Γ(τV +X
i
τiVi+Y2(τ, τi))
have degree equal to or less than d.
Now we assume that Vis an admissible vector field compactly supported on Ωε,
and that Viare admissible vector fields such that Vi
vvanishing on Σ0. Then the
vector field ∂Y2
∂τ (0,0),∂ Y2
∂τi
(0,0)
on Ωεare vertical and admissible. Since they vanish at (0,ˆx), they are identically
0.
Since the holonomy map is surjective we choose {Vi}∞
i=1 on Ωεsuch that {TΣε(Vi
v)}i∈N
is a normalized Schauder basis for V(Σε). Then we consider the map
P: (−ε, ε)×l1(ε)→ C0(Σε, N )
given by
(τ, (τi)) 7→ Γ(τ V +
∞
X
i=1
τiVi+Y2(τ, τi))|Σε,
where C0(Σε, N ) is the Banach manifold based on C0(Σε,Rn). Notice that
∂P(0,0)
∂τi
=TΣε(Vi) = TΣε(Vi
v),
which is invertible since the holonomy map is surjective and
∂P(0,0)
∂τ =TΣε(V) = 0,
since Vis compactly supported in Ωε. Hence we can apply the Implicit Function
Theorem to conclude that there exist ε′< ε and a family of smooth functions τi(τ),
with Pi|τi(τ)|< ε for all τ∈(−ε′, ε′), so that
Γ(τV +X
i
τi(τ)Vi+Y2(τ, τi(τ)))
take the value Φ(¯p) for almost each ¯p∈Σε. Since the vector fields {Vi}∞
i=1 are
equi-bounded in the supremum norm on ¯
Ωε, the series Piτi(τ)Viis absolutely
convergent on ¯
Ωε.
Clearly, we have
P(τ, (τi(τ)))(¯p) = Φ(¯p),
20 G. GIOVANNARDI
for almost each ¯p∈Σε. Differentiating with respect to τat τ= 0 we obtain
∂P(0,0)
∂τ +X
i
∂P(0,0)
∂τi
τ′
i(0) = 0.
Therefore τ′
i(0) = 0 for each i∈N. Thus, the variational vector field to Γ is
(6.10) Γ(τ)
∂τ τ=0
=V+X
i
τ′
i(0)Vi+∂Y2
∂τ (0,0) + X
i
∂Y2
∂τi
(0,0) = V.
Here we show an unexpected application of Theorem 6.6.
Example 6.7. An Engel structure (E, H) is 4-dimensional Carnot manifold where
His a two dimensional distribution of step 3. A representation of the Engel group
E, which is the tangent cone to each Engel structure, is given by R4endowed with
the distribution Hgenerated by
X1=∂x1and X2=∂x2+x1∂x3+x2
1
2∂x4.
The second layer is generated by
X3= [X1, X2] = ∂x3+x1∂x4
and the third layer by X4= [X1, X3] = ∂x4. A well-known example of horizontal
singular curve, first discovered by Engel, is given by γ:R→R4,γ(t) = (0, t, 0,0).
R. Bryant and L. Hsu proved in [8] that γis rigid in the C1topology therefore this
curve γdoes not satisfy any geodesic equation. However H. Sussman [36] proved
that γis the minimizer among all the curves whose endpoints belongs to the x2-axis.
Let Ω be an open set in R2and Φ : Ω →R4be the ruled immersion parametrized
by Φ(u, v) = (0, u, 0, v) whose tangent vectors are (X2)Φ(u,v)and (X4)Φ(u,v). Then
we have that the degree deg(Φ(Ω)) is equal to four. Fix the left invariant metric g
that makes X1, . . . , X4an orthonormal basis. Taking into account equation (4.2),
we have that a normal vector field V=f3X3+g1X1is admissible if and only if
∂f3
∂u =g1,
since b313 =hX3,[X2, X3]i= 0 and a311 =hX3,[X2, X1]i=−1. Therefore
A(u, v) = (−1) for all (u, v )∈Ω, then Ais linearly full in R. Thus, by Propo-
sition 5.6 we gain that ruled immersion Φ is regular.
Despite the immersion Φ is foliated by singular curves that are also rigid in the
C1topology, Φ is a regular ruled immersion. Thus, by Theorem 6.6 we obtain that
each admissible vector field is integrable. Therefore it possible to compute the first
variation formula [11, Eq. (8.7), Section 8] and verify that Φ is a critical point
for the area functional with respect to the left invariant metric gsince its mean
curvature vector H4of degree 4 vanishes. Hence this plane foliated by abnormal
geodesics, that do not verify any geodesic equations, satisfies the mean curvature
equations for surface of degree 4.
Here we show some applications of Theorem 6.6 to lifted surfaces immersed
of codimension 2 in an Engel structure that model the visual cortex, taking into
account orientation and curvature.
HIGHER DIMENSIONAL HOLONOMY MAP FOR RULED SUBMANIFOLDS 21
Example 6.8. Let E=R2×S1×Rbe a smooth manifold with coordinates
p= (x, y, θ, k). We set H= span{X1, X2}, where
(6.11) X1= cos(θ)∂x+ sin(θ)∂y+k∂θand X2=∂k.
The second layer is generated by
X3= [X1, X2] = −∂θ
and X1, X2. The third layer by adding X4= [X1,[X1, X2]] = −sin(θ)∂x+ cos(θ)∂y
to X1,...,X3. Notice that the Carnot manifold (E, H) is a Engel structure. Let Ω
be an open set of R2endowed with the Lebesgue measure. Then we consider the
immersion Φ : Ω →E, Φ(x, y) = (x, y, θ(x, y), κ(x, y)) where we set Σ = Φ(Ω).The
tangent vectors to Σ are
(6.12) Φx= (1,0, θx, kx),Φy= (0,1, θy, κy).
Following the computation in [11, Section 4.3] the 2-vector tangent to Σ is given
(6.13)
Φx∧Φy=(cos(θ)κy−sin(θ)κx)X1∧X2−(cos(θ)θy−sin(θ)θx)X1∧X3
+X1∧X4+ (θxκy−θyκx−κ(cos(θ)κy−sin(θ)κx))X2∧X3
+ (sin(θ)κy+ cos(θ)κx)X2∧X4
+ (κ−X1(θ))X3∧X4.
Since the curvature is the derivative of orientation we gain that κ(x, y) = X1(θ(x, y ))
and therefore the degree of these immersion is always equal to four. Then a tangent
basis of TpΣ adapted to 2.7 is given by
(6.14) E1= cos(θ)Φx+ sin(θ)Φy=X1+X1(κ)X2,
E2=−sin(θ)Φx+ cos(θ)Φy=X4−X4(θ)X3+X4(κ)X2.
Therefore Σ is a FGT-(s−3) ruled submanifoldruled manifold foliated by horizontal
curves. Adding V3=X2−X1(κ)X1and V4=X3we obtain a basis of T E. Choosing
the metric gthat makes E1, E2, V3, V4an orthonormal basis we gain that
a413 =hV4,[E1, V3]i= 1 + X1(κ)2,
b414 =hV4,[E1, V4]i=X4(θ).
Therefore the admissibility system (4.2) on the chart Ω is given by
¯
X1(f4) = −¯
X4(θ)f4−(1 + ¯
X1(θ)2)g3,
where V⊥=g3V3+f4V4and the projection of the vector field X1and X4onto Ω
is given by
¯
X1= cos(θ(x, y))∂x+ sin(θ(x, y))∂y
¯
X4=−sin(θ(x, y))∂x+ cos(θ(x, y))∂y.
Notice that the matrix A(x, y) = ((1 + ¯
X1(θ(x, y))2)) never vanishes for all (x, y)∈
Ω, then also the matrix ˜
A=DA defined in Proposition 5.2 never vanishes since
D(x, y)6= 0 for all (x, y )∈Ω. Therefore by Proposition 5.6 the surface Σ is regular,
then by Theorem 6.6 they are deformable.
22 G. GIOVANNARDI
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Dipartimento di Matematica, Piazza di Porta S. Donato 5, 401 26 Bologna, Italy
E-mail address:gianmarc.giovannard2@unibo.it
