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# On the induced geometry on surfaces in 3D contact sub-Riemannian manifolds

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## Abstract

Given a surface S in a 3D contact sub-Riemannian manifold M , we investigate the metric structure induced on S by M , in the sense of length spaces. First, we define a coefficient [[EQUATION]] at characteristic points that determines locally the characteristic foliation of S . Next, we identify some global conditions for the induced distance to be finite. In particular, we prove that the induced distance is finite for surfaces with the topology of a sphere embedded in a tight coorientable distribution, with isolated characteristic points.
ESAIM: COCV 28 (2022) 9 ESAIM: Control, Optimisation and Calculus of Variations
https://doi.org/10.1051/cocv/2021104 www.esaim-cocv.org
ON THE INDUCED GEOMETRY ON SURFACES IN 3D CONTACT
SUB-RIEMANNIAN MANIFOLDS
Davide Barilari1, Ugo Boscain2and Daniele Cannarsa3,*
Abstract.
Given a surface
S
in a 3D contact sub-Riemannian manifold
M
, we investigate the
metric structure induced on
S
by
M
, in the sense of length spaces. First, we deﬁne a coeﬃcient
b
K
at
characteristic points that determines locally the characteristic foliation of
S
. Next, we identify some
global conditions for the induced distance to be ﬁnite. In particular, we prove that the induced distance
is ﬁnite for surfaces with the topology of a sphere embedded in a tight coorientable distribution, with
isolated characteristic points.
Mathematics Sub ject Classiﬁcation. 53C17, 53A05, 57K33.
Received November 9, 2020. Accepted November 29, 2021.
1. Introduction
The study of the geometry of submanifolds
S
of an ambient manifold
M
with a given geometric structure is a
classical subject. A familiar example, whose study goes back to Gauss, is that of a surface
S
embedded in the
Euclidean space
R3
. In such case,
S
inherits its natural Riemannian structure by restricting the metric tensor to
the tangent space of
S
. The distance induced on
S
by this metric tensor is not the restriction of the distance of
R3to points on S, but rather the length space structure induced on Sby the ambient space.
Things are less straightforward for a smooth 3-manifold
M
endowed with a contact sub-Riemannian structure
(
D, g
); here
D
is a smooth contact distribution and
g
is a smooth metric on it. Indeed, for a two-dimensional
submanifold
S
, the intersection
TxSDx
is one-dimensional for most points
x
in
S
; thus,
T S D
is not a
bracket-generating distribution and there is no well-deﬁned sub-Riemannian distance induced by (
M, D, g
) on
S
.
This fact is indeed more general, as already observed in ([23], Sect. 0.6.B).
Nevertheless, one can still deﬁne a distance on
S
following the length space viewpoint: the sub-Riemannian
distance dsR deﬁnes the length of any continuous curve γ: [0,1] Mas
LsR(γ) = sup XN
i=1dsR (γ(ti), γ(ti+1 )) |0 = t0. . . tN= 1,
Keywords and phrases: Contact geometry, sub-Riemannian geometry, length space, Riemannian approximation, Gaussian
curvature, Heisenberg group.
1Dipartimento di Matematica “Tullio Levi-Civita”, Universit`a di Padova, via Trieste 63, Padova, Italy.
2
CNRS, Laboratoire Jacques-Louis Lions, team Inria CAGE, Universit´e de Paris, Sorbonne Universit´e boˆıte courrier 187, 75252
Paris Cedex 05 Paris France.
3Universit´e de Paris and Sorbonne Universit´e, CNRS, INRIA, IMJ-PRG, 75013 Paris, France.
*Corresponding author: daniele.cannarsa@imj-prg.fr
c
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
2D. BARILARI ET AL.
and one can deﬁne dS:S×S[0,+] with
dS(x, y) = inf{LsR(γ)|γ: [0,1] S, γ(0) = x, γ(1) = y}.
The space (
S, dS
) is called a length space, and
dS
the induced distance deﬁned by (
M, dsR
). (In the theory of
length metric spaces, the induced distance
dS
is called intrinsic distance, emphasising that it depends uniquely
on lengths of curves in
S
, see [
13
].) We stress that the induced distance
dS
is not the restriction
dsR|S×S
of the
sub-Riemannian distance to S.
This paper studies necessary and suﬃcient conditions on the surface
S
for which the induced distance
dS
is ﬁnite i.e.,
dS
(
x, y
)
<
+
for all points
x, y
in
S
; this is equivalent to (
S, dS
) being a metric space. In the
following lines we rephrase this property through the characteristic foliation of S.
Recall that a curve
¯γ
is horizontal with respect to
D
if it is Lipschitz, and its derivative
˙
¯γ
is in
D
whenever
deﬁned. Consider a continuous curve
γ
: [0
,
1]
S
. Its length is ﬁnite, i.e.,
LsR
(
γ
)
<
+
, if and only if
γ
is a
reparametrisation of a curve
¯γ
horizontal with respect to
D
; in such case, the length of
γ
coincides with the
sub-Riemannian length of
¯γ
,i.e., the integral of
|˙
¯γ|g
. We refer to ([
13
], Ch. 2) and ([
1
], Sect. 3.3) for more
details. Therefore, the distance
dS
(
x, y
) between two points
x
and
y
in
S
is ﬁnite if and only if there exists a
ﬁnite-length horizontal curve in Swith respect to Dconnecting the points xand y.
A point
p
in
S
is a characteristic point if the tangent space
TpS
coincides with the distribution
Dp
. The set
of characteristic points of
S
is the characteristic set, noted Σ(
S
). The characteristic set is closed due to the
lower semi-continuity of the rank, and it cannot contain open sets since
D
is bracket-generating. Moreover, since
the distribution
D
is contact and
S
is
C2
, the set Σ(
S
) is contained in a 1-dimensional submanifold of
S
(see
Lem. 2.4) and, generically, it is composed of isolated points (see [20], Sect. 4.6).
Outside of the characteristic set, the intersection
T S D
is a one-dimensional distribution and deﬁnes a
regular one-dimensional foliation on
S
Σ(
S
). This foliation extends to a singular foliation of
S
singleton at every characteristic point. The resulting foliation is the characteristic foliation of
S
. Note that any
horizontal curve contained in SΣ(S) stays inside a single one-dimensional leaf of the characteristic foliation.
In conclusion, the ﬁniteness of
dS
is equivalent to the existence, for any two points in
S
, of a ﬁnite-length
continuous concatenation of leaves of the characteristic foliation of Sconnecting these two points.
1.1. Main results
In this paper we prove two kind of results: local and global. On the local side, we are interested in the behaviour
of the characteristic foliation around the characteristic points. First, we use the Riemannian approximations
of the sub-Riemannian space to associate with each characteristic point a real number. Precisely, let
X0
be a
vector ﬁeld transverse to the distribution
D
in a neighbourhood of a characteristic point
p
Σ(
S
). Let
gX0
be
the Riemannian extension of gfor which
hX0, DigX0= 0,|X0|gX0= 1.
The Riemannian metrics
gX0
, for
>
0, are the Riemannian approximations of (
D, g
) with respect to
X0
. Let
KX0
be the Gaussian curvature of
S
with respect to
gX0
, and let
BX0
be the bilinear form
BX0
:
D×DR
deﬁned by
BX0(X, Y ) = αif [X, Y ] = αX0mod D.
Since Dis endowed with the metric g, the bilinear form BX0admits a well-deﬁned determinant.
Theorem 1.1.
Let
S
be a
C2
surface embedded in a 3D contact sub-Riemannian manifold. Let
p
be a characteristic
point of
S
, and let
X0
be a vector ﬁeld transverse to the distribution
D
in a neighbourhood of
p
. Then, in the
NDUCED GEOMETRY ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 3
Figure 1. The characteristic foliation deﬁned by the Heisenberg distribution (
R3,ker
(d
z
+
1
2
(
y
d
xx
d
y
)) on an Euclidean sphere centred at the origin: any horizontal curve connecting
points on diﬀerent spirals goes though one of the characteristic points, at the North or the
South pole. The sub-Riemannian length of the leaves spiralling around the characteristic points
is ﬁnite because of Proposition 1.3. Thus, the induced distance
dS
is ﬁnite: this is a particular
case of Theorem 1.5.
notations deﬁned above, the limit
b
Kp= lim
0
KX0
p
det BX0
p
(1.1)
is ﬁnite and independent on the vector ﬁeld X0.
As we shall see, the coeﬃcient
b
Kp
determines the qualitative behaviour of the characteristic foliation near a
characteristic point
p
. Given an open set
U
in
S
, a vector ﬁeld
X
of class
C1
is a characteristic vector ﬁeld of
S
in Uif, for all xin U,
spanRX(x) = ({0},if xΣ(S),
TxSDx,otherwise,(1.2)
and satisﬁes the condition
div X(p)6= 0,pΣ(S)U. (1.3)
Notice that
div X
(
p
) is well-deﬁned since
X
(
p
) = 0, i.e.,
p
is a characteristic point, and it is independent on
the volume form; in particular
div X
(
p
) =
tr DX
(
p
). Due to Lemma 2.1, one can show that locally there always
exists a characteristic vector ﬁeld, and that two characteristic vector ﬁelds are multiples by an everywhere
non-zero function; in particular, if
X
is a characteristic vector ﬁeld, then also
X
it a characteristic vector ﬁeld.
Finally, condition (1.2) implies that the characteristic foliation of
S
in
U
is the set of orbits of the dynamical
system deﬁned by X, and that the characteristic points are precisely the zeros of X,i.e., equilibrium points.
Following the terminology of contact geometry (cf. for instance [
20
], Sect. 4.6), given a characteristic
point
p
Σ(
S
) and a characteristic vector ﬁeld
X
, the point
p
is elliptic if
det DX
(
p
)
>
0, and hyperbolic if
det DX(p)<0.
4D. BARILARI ET AL.
Proposition 1.2.
Let
S
be a
C2
surface embedded in a 3D contact sub-Riemannian manifold. Given a
characteristic point pin Σ(S), let Xbe a characteristic vector ﬁeld Xnear p. Then, tr DX(p)6= 0 and
b
Kp=1 + det DX(p)
(tr DX(p))2.(1.4)
Thus, pis hyperbolic if and only if b
Kp<1, and pit is elliptic if and only if b
Kp>1.
b
Kp
to the eigenvalues of
DX
(
p
), which determine the qualitative behaviour of the
characteristic foliation around the characteristic point
p
. This relation is made explicit in Corollary 4.5 for a non-
degenerate characteristic point, and in Corollary 4.7 for a degenerate characteristic point. Moreover, equation
(1.4)
shows that
b
Kp
is independent on the sub-Riemannian metric, and depends only on the line ﬁeld deﬁned by
D
on
S
.
Still about local properties, we prove that the one-dimensional leaves of the characteristic foliation of
S
which
converge to a characteristic point have ﬁnite length. Precisely, let
`
be a leaf of the characteristic foliation of
S
;
we say that a point
p
in
S
is a limit point of
`
if there exists a point
x
in
`
and a characteristic vector ﬁeld
X
of
Ssuch that
etX (x)pfor t+,(1.5)
where
etX
is the ﬂow of
X
. In such case, we denote the semi-leaf
`+
X
(
x
) =
{etX
(
x
)
|t
0
}
. With the above
deﬁnition, a leaf can have at most two limit points: one for each extremity. Finally, notice that a limit point of a
leaf must be a zero of the corresponding characteristic vector ﬁeld X,i.e., a characteristic point of S.
Proposition 1.3.
Let
S
be a
C2
surface embedded in a 3D contact sub-Riemannian manifold, and let
p
be a
limit point of a one-dimensional leaf
`
. Let
x`
, and
X
be a characteristic vector ﬁeld such that
etX
(
x
)
p
for
t+. Then, the length of `+
X(x)is ﬁnite.
This result is not surprising, and it is a consequence of the sub-Riemannian structure being contact. Indeed,
for a non-contact distribution this conclusion is false; for instance, in Lemma 2.1 of [
36
] the authors prove that
the length of the semi-leaves of the characteristic foliation of a Martinet surface converging to an elliptic point is
inﬁnite.
Remark 1.4.
This local analysis of the qualitative properties of the characteristic foliation around characteristic
points will be crucial to establish the global ﬁniteness of dSin Section 5.1.
On the global side, we determine some conditions for the induced metric
dS
to be ﬁnite under the assumption
that there exists a global characteristic vector ﬁeld of
S
. In such case, for a compact, connected surface
S
with isolated characteristic points, we show that
dS
is ﬁnite in the absence of the following classes of leaves in
the characteristic foliation of
S
: nontrivial recurrent trajectories, periodic trajectories, and sided contours; see
Proposition 5.1. Note that if
S
is orientable and the distribution
D
is coorientable,i.e., there exists a global
contact form
ω
deﬁning the distribution (cf. also
(2.1)
), then
S
admits a global characteristic vector ﬁeld; see
Lemma 2.1. Recall that a distribution is tight if it does not admit an overtwisted disk,i.e., an embedding of a
disk with horizontal boundary such that the distribution does not twists along the boundary.
Theorem 1.5.
Let (
M, D, g
)be a tight coorientable sub-Riemannian contact structure, and let
S
be a
C2
embedded surface with isolated characteristic points, homeomorphic to a sphere. Then the induced distance
dS
is
ﬁnite.
We stress that the property that any characteristic point is isolated is a generic property for surfaces in
contact manifolds. Here by a generic property we mean a property satisﬁed on an open and dense subset of
the set of all coorientable contact distributions for the
C
topology (once the surface is ﬁxed), and on an open
and dense subset of the set of all
C2
surfaces for the
C2
topology (once the contact distributions is ﬁxed).
NDUCED GEOMETRY ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 5
Examples 7.4 and 7.5 in the Heisenberg distribution show that, if
S
is not a topological sphere, then
S
presents
possibly nontrivial recurrent trajectories or periodic trajectories, cases in which
dS
is not ﬁnite. Moreover, if one
removes the hypothesis of the contact structure being tight, then a sphere
S
might present a periodic trajectory,
hence the induced distance
dS
would not be ﬁnite. The compactness hypothesis is also important, as one can see
in Example 7.1.
When considering hypersurfaces in contact structures, usually one needs to demand the absence of characteristic
points to obtain results that generalize from Riemannian ones. Consider, for instance, the result ([
33
], Thm. 1.1)
describing the induced distance on hypersurfaces of sub-Riemannian manifolds in dimension greater than three,
or the result in [
32
] about the small-time asymptotic of the heat content surfaces in the 3D case. On the contrary,
here the ﬁniteness of the distance in the 3D case needs the presence of characteristic points; this is mainly due
to the integrability of the characteristic vector ﬁeld and of the global nature of the question. Finally, we are left
to observe that for many surfaces the induced distance is not ﬁnite. For instance, the induced distance on a
surface without characteristic points is not ﬁnite, and the property of not having characteristic points do not
change with small perturbations of the surface or of the contact structure in the appropriate topologies.
1.2. Previous literature
Characteristic foliations of surfaces in 3D contact manifolds are studied in numerous references; here we use
notions contained in [
11
,
21
,
22
], and we refer to [
19
,
20
] for an introduction to the subject. For hypersurfaces
in Carnot groups we refer to [
16
] and the references therein. Moreover, for an introduction to sub-Riemannian
geometry we refer to [26,28,31] or [1].
The use of the Riemannian approximation scheme to deﬁne sub-Riemannian geometric invariants is a well-
29
] to study the horizontal mean curvature in
relation to the minimal surfaces in the Heisenberg group, whose integrability is discussed in [17]. For a general
description of the properties of the Riemannian approximations in Heisenberg we also refer to [14].
In this paper, we combine the Riemannian approximation scheme suitably normalised by the Lie bracket
structure on the distribution to deﬁne the metric coeﬃcient
b
K
at the characteristic points. Notice that usually in
the literature the Riemannian approximation is employed to deﬁne sub-Riemannian geometric invariants outside
of the characteristic set. For instance, in [
8
] the authors deﬁned the sub-Riemannian Gaussian curvature at
a point
xS
Σ(
S
) as
KS
(
x
) =
lim0KX0
x
, and they proved that a Gauss-Bonnet type theorem holds; here
the authors worked in the setting of the Heisenberg group, and with
X0
equals to the Reeb vector ﬁeld of the
Heisenberg group. This construction is extended in [
35
] to the aﬃne group and to the group of rigid motions of
the Minkowski plane, and in [
34
] to a general sub-Riemannian manifold. In the latter, the author linked
KS
with
the curvature introduced in [
18
], and, when Σ(
S
) =
, they proved a Gauss-Bonnet theorem by Stokes formula.
A Gauss-Bonnet theorem (in a diﬀerent setting) was also proven in [
2
]. We ﬁnally notice that the invariant
KS
also appears in [
27
], where it is called curvature of transversality. An expression for
KS
is provided also in [
10
],
in relation to a new notion of stochastic processes in this setting.
1.3. Structure of the paper
After some preliminaries contained in Section 2, in Section 3we prove Theorem 1.1, by introducing the metric
invariant
b
K
deﬁned at characteristic points. In Section 4, we write the metric invariant in terms of a characteristic
vector ﬁeld as in Proposition 1.2, and we study the length of the horizontal curves as in Proposition 1.3. In
Section 5, we use the topological decomposition of a 2D ﬂow to prove Proposition 5.1, from which we deduce
Theorem 1.5 in Section 6. Section 7is devoted some examples of induced distances on surfaces in the Heisenberg
group.
2. Preliminaries
In this paper,
M
is a smooth 3-dimensional manifold, (
D, g
) a smooth contact sub-Riemannian structure on
M
, and
S
an embedded surface of class
C2
. The contact distribution is, locally, the kernel of a contact form
6D. BARILARI ET AL.
ω1(M), which can be normalised to satisfy
D= ker ω, ω 6= 0, |D= volg.(2.1)
Recall that a point
p
in
S
is a characteristic point of
S
if
TpS
=
Dp
, and that the characteristic points of
S
form the characteristic set Σ(S). For xSΣ(S), the intersection
lx=DxTxS(2.2)
is one-dimensional, and we can think of
(2.2)
as deﬁning a generalised distribution
l
in
S
whose rank increases
at characteristic points. Sometimes in the literature the (generalised) distribution
l
is called the trace of
D
on
S
. The distribution
l
is not smooth at the characteristic points, hence it is more convenient to work with a
characteristic vector ﬁeld, that is a C1vector ﬁeld of Ssatisfying (1.2) and (1.3).
Lemma 2.1.
Assume that
S
is orientable and that
D
is coorientable. Then,
S
vector ﬁeld; moreover, the characteristic vector ﬁelds of
S
are the vector ﬁelds
X
for which there exists a volume
form of Ssuch that
Ω(X, Y ) = ω(Y)for all YT S. (2.3)
Indeed, formula
(2.3)
is the deﬁnition of characteristic vector ﬁeld as given in Section 4.6 of [
20
], meaning that
the characteristic vector ﬁelds are dual to the contact form
ω|S
with respect to the volume forms of
S
. In the
previous reference it is shown that if a vector ﬁeld satisﬁes
(2.3)
, then it satisﬁes
(1.2)
and
(1.3)
. Reciprocally,
a vector ﬁeld
¯
X
satisfying
(1.2)
is a multiple of any vector ﬁeld
X
satisfying
(2.3)
for some function
φ
with
φ|SΣ(S)6= 0; additionally, if (1.3) holds, then φ|Σ(S)6= 0; thus, ¯
Xsatisﬁes (2.3) with 1
φ as volume form of S.
Remark 2.2.
Since the volume forms of
S
are proportional by nowhere-zero functions, the same holds for the
characteristic vector ﬁelds.
Therefore, if the orientability hypotheses hold, an equivalent deﬁnition of the characteristic foliation is the
partition of
S
into the orbits of a global characteristic vector ﬁeld. This is a generalised foliation, as the dimension
of the leaves is not constant since the characteristic set is partitioned in singletons.
Let us provide another way to ﬁnd, locally, an explicit expression for a local characteristic vector ﬁeld. Any
point in
S
U
in
M
in which there exists an oriented orthonormal frame (
X1, X2
) for
D|U
, and a submersion
f
of class
C2
for which
S
is a level set, i.e.,
SU
=
f1
(0) and
df|U6
= 0. In such case,
a vector VT M |Uis in T S if and only if V f = 0; thus, for a point pUS,
pΣ(S) X1f(p) = X2f(p) = 0.(2.4)
Moreover, since [X2, X1]p6∈ Dp=TpSat a characteristic point p, then [X2, X1]f(p)6= 0.
Remark 2.3. In the previous notation, the vector ﬁeld Xfdeﬁned by
Xf= (X1f)X2(X2f)X1,(2.5)
is a characteristic vector ﬁeld of
S
. Indeed, it follows from the deﬁnition that, for all
x
in
S
, the vector
Xf
(
x
) is
in
TxSDx
, and that, due to
(2.4)
,
Xf
(
p
) = 0 if and only if
p
Σ(
S
); thus,
Xf
satisﬁes
(1.2)
. Moreover, for all
pΣ(S),
div Xf(p) = X2X1f(p)X1X2f(p)=[X2, X1]f(p),
NDUCED GEOMETRY ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 7
which is nonzero due to the contact condition; thus, Xfsatisﬁes (1.3).
Lemma 2.4.
The characteristic set Σ(
S
)of a surface
S
of class
C2
is contained in a 1-dimensional submanifold
of S of class C1.
Proof.
It suﬃces to show that for every point
p
in Σ(
S
) there exists a neighbourhood
V
of
p
such that
V
Σ(
S
)
is contained in an embedded
C1
curve. Let us ﬁx a point
p
in Σ(
S
), and a neighbourhood
U
of
p
in
M
equipped
with a frame (
X1, X2
) and a function
f
with the properties described above. Because of
(2.4)
, the characteristic
points in V=USare the solutions of the system f=X1f=X2f= 0.
Due to the implicit function theorem, it suﬃces to show that
dp
(
X1f
)
6
= 0 or
dp
(
X2f
)
6
= 0. Thanks to the
contact condition, we have that [
X2, X1
]
f
(
p
)
6
= 0. As a consequence, since
X2X1f
(
p
) = [
X2, X1
]
f
(
p
) +
X1X2f
(
p
),
at least one of the following is true:
X2X1f
(
p
)
6
= 0, or
X1X2f
(
p
)
6
= 0. Assume that the ﬁrst is true; then
dp(X1f)(X2) = X2X1f(p)6= 0. The other case being similar, the lemma is proved.
For a more general discussion on the size of the characteristic set, we refer to [7] and references therein.
3. Riemannian approximations and Gaussian curvature
In this section we discuss the Riemannian approximations of a sub-Riemannian structure, and we prove
Theorem 1.1 by using the asymptotic expansion of the Gaussian curvature KX0
pat a characteristic point p.
In order to deﬁne the metric coeﬃcient
b
Kp
, one needs to ﬁx a vector ﬁeld
X0
transverse to the distribution in
a neighbourhood of
p
. If the distribution is coorientable, it is possible to make this choice globally. As described
in the introduction, once this choice has been made, one can extend the sub-Riemannian metric
g
to a family of
Riemannian metrics
gX0
such that, for every
>
0, one has
hD, X0igX0
= 0 and
|X0|gX0
= 1
/
. To simplify
the notation, we drop the dependance from X0in the superscript, writing g=gX0.
Let
be the Levi-Civita connection of (
M, g
). Since we study local properties, we can restrict to a domain
equipped with an orthonormal oriented frame (
X1, X2
) of
D
; thus, (
X0, X1, X2
) is an orthonormal basis of
g
.
Due to the Koszul formula, one has
XiXj, Xkg=1
2 hXi,[Xj, Xk]ig+hXk,[Xi, Xj]ig+hXj,[Xk, Xi]ig,
for all
i, j, k
= 0
,
1
,
2. This identity enables us to describe
using the frame (
X0, X1, X2
), which is independent
from . This is done using the Lie bracket structure of the frame, i.e., the Cfunctions ck
ij such that
[Xj, Xi] = c1
ij X1+c2
ij X2+c0
ij X0for i, j = 0,1,2.(3.1)
The functions ck
ij are the structure constants of the frame.
Thus, for every > 0, we have that
XiXi=ci
i02X0+ci
i1
2X1+ci
i2
2X2i= 0,1,2 (3.2)
XjXi=1
2cj
0i2ci
0j2+c0
ij X0+cj
ij Xji6=j= 1,2
X0X1=c0
01X0+1
2c1
02 c2
01 +c0
12
2X2
X0X2=c0
02X0+1
2c2
01 c1
02 c0
12
2X1,
and the remaining derivatives
X1X0and
X2X0are computed using that the connection is torsion-free.
8D. BARILARI ET AL.
Given the surface
S
, the second fundamental form
II
of
S
is the projection of the Levi-Civita connection
on the orthogonal to the tangent space of the surface. The Gaussian curvature
K
=
KX0
of
S
in (
M, g
) is
deﬁned by the Gauss formula
K=K
ext + det(II),(3.3)
where, given a frame (X, Y ) of T S , the extrinsic curvature K
ext is
K
ext =
X
YY
Y
XY
[X,Y ]Y, X g
|X|2
g|Y|2
g hX, Y i2
g
,(3.4)
and the determinant det IIof the second fundamental form is
det II=II(X, X),II(Y , Y )gII(X, Y ),II(X, Y )g
|X|2
g|Y|2
g hX, Y i2
g
.(3.5)
Both these quantities are independent on the frame (X, Y ) of T S chosen to compute them.
3.1. Proof of Theorem 1.1
To prove the theorem, we explicitly compute the asymptotic of the quantities in limit (1.1). Let us ﬁx a
characteristic point
p
, and, in a neighbourhood of
p
, let us ﬁx an oriented orthonormal frame (
X1, X2
) of
D
and
a submersion fdeﬁning S.
The determinant of the bilinear form BX0
pis homogeneous in , and satisﬁes
det BX0
p=det BX0
p
2=BX0
p(X1, X2)2
2=(c0
12(p))2
2,(3.6)
where
c0
12
is deﬁned in (3.1). Therefore, in order to prove the convergence of the limit in (1.1), it suﬃces to show
that the Gaussian curvature KX0
pat pdiverges at most as 1/2.
Let us start with the computation of the determinant (3.5) of the second fundamental form at a characteristic
point. It is convenient to write the second fundamental form as
II(X, Y ) =
XY, N N.
where Nis the Riemannian unitary gradient of f,i.e.,
N=(X1f)X1+ (X2f)X2+(X0f)X0
p(X1f)2+ (X2f)2+(X0f)2.
At the characteristic point p, the gradient N(p) simpliﬁes to
N(p) = sign(X0f)X0(p).(3.7)
To compute (3.5) one needs to choose a frame of T S ; we will use the frame (F1, F2) with
Fi= (X0f)Xi(Xif)X0for i= 1,2.(3.8)
NDUCED GEOMETRY ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 9
This frame is well-deﬁned for
X0f6
= 0; in particular, it is suited to calculate the Gaussian curvature at the
characteristic points. Recall that the horizontal Hessian of fis
HessH(f) = X1X1f X1X2f
X2X1f X2X2f.
Lemma 3.1.
Let
pS
be a characteristic point. Then, in the previous notations, for every
>
0, the
determinant (3.5) of the second fundamental form in pis
det II(p) = 1
2det HessHf(p)
(X0f(p))2(c0
12(p))2
4+O(1).
Proof. Let pbe a characteristic point. Because X1f(p) = X2f(p) = 0, one can show that,
FiFj(p) = (X0f)2
XiXj+ (X0f)(XiX0f)Xj(X0f)(XiXjf)X0p,
for
i, j
= 1
,
2. Using formula
(3.7)
for
N
, one ﬁnds that only the component along
X0
plays a role in the second
fundamental form in p. Thus, using the covariant derivatives in (3.2),
h∇
FiFj, N ip=|X0f(p)|
XiXjf+ (X0f)c0
ij
2+ (X0f)2cj
0i+ci
0j
2p
,
for i, j = 1,2. This, together with |F1|2|F2|2 hF1, F2i2p= (X0f(p))4, gives the result.
Next, the extrinsic curvature (3.4) is the sectional curvature of the plane
TpS
in
M
, which is known when
X0
is the Reeb vector ﬁeld and
= 1; this can be found for instance in ([
9
], Prop. 14). In our setting, the resulting
expression for 0 is the following.
Lemma 3.2. Let pSbe a characteristic point. Then, for every > 0,
K
ext(p) = 3
42(c0
12(p))2+O(1).
Proof.
To compute the extrinsic curvature we use the frame (
X1, X2
) of
T M
, which coincides with
TpS
=
Dp
at
the characteristic point p. Then, to compute
K
ext(p) = h∇
X1
X2X2
X2
X1X2
[X1,X2]X2, X1ip
it suﬃces to use the expressions (3.2).
Remark 3.3.
Following the proof of Lemma 3.1 and Lemma 3.2, the exact expressions for
det II
(
p
) and
K
ext
(
p
)
at a characteristic point pare, for all > 0,
det II(p) = + 1
2det HessHf
(X0f)2(c0
12)2
4p+2c1
01c2
02 c1
02 +c2
012
4p
+1
X0f(p)c2
02X1X1f+c1
01X2X2fc2
01 +c1
02
2(X2X1f+X1X2f)p,
K
ext(p) = 3
4
(c0
12(p))2
22c1
01c2
02 (c2
01 +c1
02)2
4p
10 D. BARILARI ET AL.
+X2(c1
12)X1(c2
12)(c1
12)2(c2
12)2+c0
12
c2
01 c1
02
2p.
If one chooses as transversal vector ﬁeld the Reeb vector ﬁeld of the contact sub-Riemannian manifold, then one
recognises the ﬁrst and the second functional invariants of the sub-Riemannian structure, deﬁned in Chapter 17
of [1]. Finally, notice that these expressions are still valid for non-contact distributions.
Proof of Theorem 1.1.
In the previous notations, due to the Gauss formula (3.3), Lemma 3.1 and Lemma 3.2,
the Gaussian curvature at a characteristic point psatisﬁes
K
p=KX0
p=(c0
12(p))2
21 + det HessHf(p)
[X2, X1]f(p)2+O(1).
Here we have used that
c0
12
(
p
)
X0f
(
p
)=[
X2, X1
]
f
(
p
) at
p
, which holds due to deﬁnition
(3.1)
and
X1f
(
p
) =
X2f(p) = 0. Using formula (3.6) for the determinant of BX0
p, one ﬁnds that
KX0
p
det BX0
p
=1 + det HessHf(p)
[X2, X1]f(p)2+O(2),(3.9)
which shows that the limit (1.1) is ﬁnite. Moreover,
b
Kp
is independent of
X0
because the transversal vector ﬁeld
X0is absent in the constant term of equation (3.9).
Formula (3.9) is useful to compute
b
Kp
explicitly, as it contains only derivatives of the submersion
f
; thus, let
us enclose it with the following corollary.
Corollary 3.4.
Let
p
be a characteristic point of
S
. Let
f
be a local submersion of class
C2
describing
S
, and
let (X1, X2)be a local oriented orthonormal frame of D. Then,
b
Kp=1 + det HessHf(p)
[X2, X1]f(p)2.(3.10)
Note that both
det HessHf
(
p
) and [
X2, X1
]
f
(
p
) calculated at the characteristic point
p
are invariant with
respect to the frame (
X1, X2
). Moreover, we emphasise that their ratio, which appears in
(3.10)
, is independent
on the choice of f.
4. Local study near a characteristic point
In this section, we prove Proposition 1.2, and we discuss the local qualitative behaviour of the characteristic
foliation near Σ(
S
) in relation to the metric coeﬃcient
b
K
; next, we estimate the length of a semi-leaf converging
to a point, proving Proposition 1.3.
Let us ﬁx a characteristic point
p
in Σ(
S
), and a characteristic vector ﬁeld
X
. Since
X
(
p
) = 0, there exists a
well-deﬁned linear map
DX
(
p
) :
TpSTpS
. Indeed, let
etX
be the ﬂow of
X
. The pushforward of the ﬂow gives,
for every
x
in
S
, a family of linear maps
etX
:
TxSTetX (x)S
. Since
etX
(
p
) =
p
for all
t
, then the preceding
gives the linear ﬂow etX
:TpSTpS, whose inﬁnitesimal generator is the diﬀerential DX(p).
Deﬁnition 4.1.
A characteristic point
p
Σ(
S
) is non-degenerate if, given a characteristic vector ﬁeld
X
of
S
,
the diﬀerential DX(p) is invertible. Otherwise, pis called degenerate.
Remark 4.2.
Condition (1.3) in the deﬁnition of characteristic vector ﬁeld ensures that the degeneracy of a
characteristic point is independent on the choice of characteristic vector ﬁeld.
NDUCED GEOMETRY ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 11
Since
TpS
coincides with
Dp
at the characteristic point
p
, we can endow
TpS
with a metric; thus,
DX
(
p
)
admits a well-deﬁned determinant and trace. Now, let
X
be the vector ﬁeld
X
=
a1X1
+
a2X2
, where (
X1, X2
) is
an orthonormal oriented frame of
D
and
aiC1
(
S
), for
i
= 1
,
2. Then, in the frame deﬁned by (
X1, X2
) one has
DX =X1a1X2a1
X1a2X2a2,(4.1)
and the formulas for the determinant and the trace are
det DX = (X1a1)(X2a2)(X1a2)(X2a1),(4.2)
tr DX = div X= (X1a1)+(X2a2).(4.3)
4.1. Proof of Proposition 1.2
Let us ﬁx a characteristic point
p
in Σ(
S
). We claim that the right-hand side of (1.4) is independent on the
choice of the characteristic vector ﬁeld
X
. Indeed, due to Remark 2.2 any two characteristic vector ﬁelds are
multiples by nonzero functions, thus, at characteristic point
p
, their diﬀerentials are multiples by nonzero scalars;
precisely, if
Y
=
φX
, for
φ
in
C1
(
S
), then one has
DY
(
p
) =
φ
(
p
)
DX
(
p
). Thus, the claim follows because both
determinant and trace-squared are homogenous of the degree two.
Thus, we ﬁx a local submersion
f
deﬁning
S
near
p
, and the characteristic vector ﬁeld
Xf
= (
X1f
)
X2
(X2f)X1deﬁned in (2.5). Using expression (4.1) for the diﬀerential of a vector ﬁeld, we get
DXf(p) = X1X2f(p)X2X2f(p)
X1X1f(p)X2X1f(p).
Thus, using expressions (4.2) and (4.3) for the determinant and the trace, we ﬁnd that
det DXf
(
p
) =
det HessHf(p), and tr DXf(p)=[X2, X1]f(p). In conclusion,
det DXf(p)
tr DXf(p)2=det HessHf(p)
[X2, X1]f(p)2,
which, together with Corollary 3.4, gives the desired result.
The eigenvalues of the linearisation DX (p) of a characteristic vector ﬁeld Xcan be written as a function of
b
Kpby rearranging equation (1.4), as in the following corollary.
Corollary 4.3.
In the hypothesis of Proposition 1.2, let
λ+
(
X, p
)and
λ
(
X, p
)be the two eigenvalues of
DX
(
p
).
Then
λ±(X, p) = tr DX(p)1
2±r3
4b
Kp.(4.4)
Proof. Let us note λ±=λ±(X, p), and α= tr DX (p). Equation (1.4) reads
b
Kp=1 + λ+λ
α2.
Using that
λ+
+
λ
=
α
, equation
(1.4)
implies that the eigenvalues satisfy the equation
z2αz
+
α2
(
b
Kp
+ 1) = 0,
which implies (4.4).
12 D. BARILARI ET AL.
Remark 4.4.
It is possible to choose canonically a characteristic vector ﬁeld with trace 1. Indeed, in the
notations used in Remark 2.3, let us deﬁne the characteristic vector ﬁeld
XS=(X1f)X2(X2f)X1
Zf ,(4.5)
where
Z
is the Reeb vector ﬁeld of the contact form
ω
of
D
deﬁned in
(2.1)
,i.e., the unique vector ﬁeld satisfying
ω
(
Z
) = 1 and
(
Z, ·
) = 0. The vector ﬁeld
XS
is a characteristic vector ﬁeld in a neighbourhood of
p
because it
is a nonzero multiple of
Xf
near Σ(
S
), since
Zf
(
p
) = [
X2, X1
]
f
(
p
)
6
= 0. Using the latter, one can verify that
div XS(p) = tr DXS(p) = 1.
It is worth mentioning that the vector ﬁeld
XS
is independent on
f
and on the frame (
X1, X2
), i.e., it
depends uniquely on
S
and (
M, D, g
). Moreover, the norm of
XS
satisﬁes
|XS|1
g
=
|pS|
, where
pS
is the degree
of transversality deﬁned in [
27
]; in the case of the Heisenberg group,
pS
coincides with the imaginary curvature
introduced in [5,6].
Expression (4.4) for the eigenvalues of the linearisation
DX
(
p
) implies the following relations between the
eigenvalues and the metric coeﬃcient b
Kp:
(i) b
Kp<1 if and only if λ±Rwith diﬀerent signs;
(ii) b
Kp=1 if and only if λ= 0 and λ+R;
(iii) 1<b
Kp 3/4 if and only if λ±Rwith same sign;
(iv) 3/4<b
Kpif and only if <(λ±)6= 0 6==(λ±) and λ=λ+.
Notice that the characteristic point pis degenerate if and only if b
Kp=1, which is case (ii).
Assume that
p
is a non-degenerate characteristic point. Then, the linear dynamical system deﬁned by
DX
(
p
)
is a saddle, a node, and a focus respectively in case (i),(iii) and (iv). In these cases there exists a local
C1
-diﬀeomorphism near
p
which sends the ﬂow of
X
to the ﬂow of
DX
(
p
) in
R2
,i.e., the ﬂows are
C1
-conjugate,
as proven by Hartman in [
25
]. For this theorem to hold, one needs the characteristic vector ﬁeld
X
to be of class
C2. For this reason, in the following corollary we assume the surface Sto be of class C3.
Corollary 4.5.
Assume that the surface
S
is of class
C3
, and let
p
be a non-degenerate characteristic point in
Σ(S). Then, b
Kp6=1, and the characteristic foliation of Sin a neighbourhood of pis C1-conjugate to
a saddle if and only if b
Kp<1;
a node if and only if 1<b
Kp 3/4;
a focus if and only if 3/4<b
Kp.
Those chases are depicted, respectively, in the ﬁrst, third and fourth image in Figure 2.
Remark 4.6.
For surfaces of class
C2
,i.e., with characteristic vector ﬁelds of class
C1
, one can use the
Hartman-Grobman theorem, by which one recovers a
C0
-conjugation to the corresponding linearisation. However,
under this hypothesis, a node and a focus become indistinguishable. For the Hartman-Grobman theorem we
refer to Section 2.8 of [30]. Finally, for a Csurface some informations can be found in [24].
Next, if
p
is a degenerate characteristic point, then we are in case (ii). Thus,
b
Kp
=
1, and the diﬀerential
DX
(
p
) has a zero eigenvalue with multiplicity one. In this situation, the qualitative behaviour of the characteristic
foliation does not depend uniquely on the linearisation, but also on the nonlinear dynamic along a center manifold,
i.e., an embedded curve
C S
with the same regularity as
X
, invariant with respect to the ﬂow of
X
, and
tangent to the zero eigenvector of DX (p). The analogue of Corollary 4.5 is the following.
Corollary 4.7.
Assume that the surface
S
is of class
C2
, and let
p
be a degenerate characteristic point in Σ(
S
).
Then,
b
Kp
=
1, and the characteristic foliation in a neighbourhood centred at
p
is
C0
-conjugate at the origin to
NDUCED GEOMETRY ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 13
Figure 2. The qualitative picture for the characteristic foliation at an isolated characteristic point,
with the corresponding values for
b
K
. From left to right, we recognise a saddle, a saddle-node, a node,
and a focus.
the orbits of a system of the form
˙u=φ(u)
˙v=v,(4.6)
for a function
φ
with
φ
(0) =
φ0
(0) = 0. If
p
is isolated, then the characteristic foliation described in
(4.6)
at the
origin is either a saddle, a saddle-node, or a node; those cases are depicted, respectively, in the ﬁrst, second, and
third image in Figure 2.
The proof of Corollary 4.7 follows from considerations on the center manifold of the dynamical system deﬁned
by X, which we recall in Appendix A.
Remark 4.8.
A node and a focus are not distinguishable by a conjugation
C0
. However, the center manifold of
the characteristic point
p
is an embedded curve of class
C1
, thus it does not spiral around
p
. Therefore, the
existence of a center manifold gives further properties then what is expressed in Corollary 4.7.
To justify the last sentence of Corollary 4.7 let us get a sense of the qualitative properties of a system as (4.6).
The line
{v
= 0
}
, parametrised by
u
, is a center manifold of (4.6), and the function
φ
determines the dynamic
of (4.6); this illustrates the fact that the nonlinear terms on a center manifold determine the dynamic near a
degenerate characteristic point.
The equilibria of (4.6) occur only in
{v
= 0
}
,i.e., on a center manifold, and a point (
u,
0) is an equilibrium if
and only if
φ
(
u
) = 0. Thus, if the characteristic point
p
is isolated, then
u0
= 0 is an isolated zero of
φ
. In such
case, let us note
φ+
=
φ|u>0
and
φ
=
φ|u<0
, and without loss of generality let us suppose that the signs of
φ+
and φare constant.
If φ+>0 and φ<0, then the origin is a topological node.
If φ+<0 and φ>0, then the origin is a a topological saddle.
If
φ+
and
φ
have the same sign, then the two half spaces
{u >
0
}
and
{u <
0
}
have two diﬀerent
behaviours: one is a node, and the other one is a saddle. This gives the characteristic foliation called
Remark 4.9.
For an isolated characteristic point, combining Corollary 4.5 and Corollary 4.7, we obtain the
four characteristic foliations depicted in Figure 2.
14 D. BARILARI ET AL.
4.2. Proof of Proposition 1.3
In this section we prove the ﬁniteness of the sub-Riemannian length of a semi-leaf converging to a point. Since
we are interested in a local property, it is not restrictive to assume the existence of a global characteristic vector
ﬁeld Xof S.
Let
`
be a one-dimensional leaf of the characteristic foliation of
S
, and
x`
such that
etX
(
x
)
p
as
t
+
.
The limit point
p
has to be an equilibrium of
X
,i.e.,
X
(
p
) = 0, hence
p
is a characteristic point of
S
. Let
U
be
a small open neighbourhood of
p
in
S
for which we have a coordinate chart Φ :
UBR2
with Φ(
p
) = 0,
where
B
is the open unit ball. Let
y
be the point of last intersection between
`+
X
(
x
) and the boundary
∂U
. Since
LsR
(
`+
X
(
x
)) =
LsR
(
`|[x,y]
) +
LsR
(
`+
X
(
y
)) and
LsR
(
`|[x,y]
) is ﬁnite, it suﬃces to show that
LsR
(
`+
X
(
y
)) is ﬁnite.
We claim that there exists a constant C > 0 such that
1
C|V|R2 |V|gC|V|R2VDT S|U,(4.7)
where we have dropped Φ
in the notation. Indeed, let
˜g
be any Riemannian extension of
g
on the surface
S
(for
example
˜g
=
gX0|S
). Since
˜g
is an extension, one has
|v|g
=
|v|˜g
for all
v
in
DT S
. Equivalence
(4.7)
follows
from the local equivalence of
˜g
with the pullback by Φ of the Euclidean metric of
R2
. Now, inequality
(4.7)
implies that
LsR(`+
X(y)) = Z+
0|X(etX (y))|gdtCZ+
0|X(etX (y))|R2dt. (4.8)
At this point the proof of the ﬁniteness of the sub-Riemannian length of
`+
X
(
y
) diﬀers depending on whether
p
is
a non-degenerate or a degenerate characteristic point.
First, assume that
p
is a non-degenerate characteristic point. Since
p
is non-degenerate, then the set of
point
w
with
etX
(
w
)
p
for
t
+
form a manifold, called the stable manifold at
p
for the dynamical
system deﬁned by
X
. In our case, since
etX
(
y
)
p
for
t
+
, the semi-leaf
`+
X
(
y
) is contained in the stable
manifold at
p
. Moreover, the stable manifold convergence property, precisely stated in Section 2.8 of [
30
], shows
that each trajectory inside the stable manifold converges to
p
sub-exponentially in
t
. Precisely, if
α
satisﬁes
|<(λ±(p, X))|> α, then there exists constants C, t0>0 such that
|etX (y)p|R2Ceαt t>t0.(4.9)
Since X(p) = 0, for all t > 0 one has
X(etX(y))R2=X(etX(y)) X(p)R2sup
B||DX(x)|| |etX (y)p|R2.
Due to the inequality (4.8) and (4.9), this shows that LsR(`+
X(y)) is ﬁnite.
Next, assume that
p
is a degenerate characteristic point. As we said in the introduction of Corollary 4.7,
there exists a center manifold
C
at
p
for the dynamical system deﬁned by
X
. The asymptotic approximation
property of the center manifold, recalled in Proposition A.2, shows that if a trajectory converges to
p
, then it
approximates any center manifold exponentially fast. Precisely, since
etX
(
y
)
p
, then there exist constants
C, α, t0>0 and a trajectory etX (z) contained in C, such that
|etX (y)etX (z)|R2Ceαt tt0.(4.10)
The triangle inequality implies that
|X(etX (y))|R2 |X(etX (y)) X(etX (z))|R2+|X(etX (z))|R2.(4.11)
NDUCED GEOMETRY ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 15
Due to inequality
(4.8)
, to prove that
LsR
(
`+
X
(
y
)) is ﬁnite, it suﬃces to show that the two terms on the right-hand
side of (4.11) are integrable for t0. Thanks to (4.10) and
|X(etX (y)) X(etX (z))|R2sup
B||DX|| |etX (y)etX (z)|R2,
then the ﬁrst term in
(4.11)
is integrable. Next, because
etX
(
z
) is a regular parametrisation of a bounded interval
inside a C1embedded curve (the center manifold C), then its derivative |X(etX (z))|R2is integrable.
Remark 4.10.
Let
X
be a characteristic vector ﬁeld of a compact surface
S
. If the
ω
-limit set with respect to
X
of a non-periodic leaf
`
contains more then one point, then
LsR
(
`+
X
) = +
. Therefore, if a leaf
`
does not
converge to a point in any of its extremities, then the points in `have inﬁnite distance from the points in S`.
In particular, if the characteristic set of a surface
S
is empty, then the induced distance
dS
is not ﬁnite. For a
discussion on non-characteristic domains we refer to Chapter 3 of [15].
5. Global study of the characteristic foliation
The main goal of this section is to identify a suﬃcient condition for the induced distance
dS
to be ﬁnite.
As explained in the introduction, this is done by excluding the existence of certain leaves in the characteristic
foliation of
S
, as in Proposition 5.1. In this section we assume the existence of a global characteristic vector ﬁeld
Xof S.
The leaves the characteristic foliation of
S
are precisely the orbits of the dynamical system deﬁned by
X
,
therefore we are going to call them trajectories, stressing that they are parametrised by the ﬂow of
X
. Moreover,
the vector ﬁeld
X
enables us to use the notions of
ω
-limit set and
α
-limit set of a point
y
in
S
, which are,
respectively,
ω(y, X ) = nqStn+such that etnX(y)qo, α(y, X ) = ω(y, X).
The points yin a leaf `have the same limit sets, thus one can deﬁne ω(`, X ) and α(`, X).
Proposition 5.1.
Let
S
be a compact, connected surface
C2
embedded in a contact sub-Riemannian structure.
Assume that
S
has isolated characteristic points, and that the characteristic foliation of
S
is described by a global
characteristic vector ﬁeld of Swhich does not contain any of the following trajectories:
nontrivial recurrent trajectories,
periodic trajectories,
sided contours.
Then, dSis ﬁnite.
Let us give a formal deﬁnition of these objects. A periodic trajectory is a leaf of the characteristic foliation
homeomorphic to a circle. A periodic trajectory has inﬁnite distance from its complementary, hence it is necessary
to exclude its presence for dSto be ﬁnite.
Next, a leaf
`
is recurrent if
`ω
(
`, X
) and
`α
(
`, X
). A nontrivial recurrent trajectory is a recurrent
trajectory which is not an equilibrium nor a periodic trajectory. Because the
ω
-limit and the
α
-limit set of a
nontrivial recurrent trajectory contains more then one point, then, due to Remark 4.10, those trajectories have
inﬁnite distance from their complementary.
Lastly, a sided contour is either a left-sided or right-sided contour. A right-sided contour (resp. left-sided) is a
family of points p1, . . . , psin Σ(S) and trajectories `1, . . . , `ssuch that:
for all j= 1, . . . , s, we have ω(`j, X) = pj=α(`j+1 , X) (where `s+1 =`1);
16 D. BARILARI ET AL.
Figure 3. The illustration of a right-sided hyperbolic sector.
for every
j
= 1
, . . . , s
, there exists a neighbourhood
Uj
of
pj
such that
Uj
is a right-sided hyperbolic sector
(resp. left-sided) for pjwith respect to `jand `j+1.
Let us give a precise deﬁnition of a hyperbolic sector. Note that, given a non-characteristic point
xS
, and a
curve
T
going through
x
and transversal to the ﬂow of
X
, the orientation deﬁned by
X
deﬁnes the right-hand
and the left-hand connected component of T{x}, denoted Trand Tlrespectively.
Deﬁnition 5.2.
Let
p
be a characteristic point, and
`1
and
`2
be two trajectories such that
ω
(
`1, X
) =
p
=
α
(
`2, X
). A neighbourhood
U
of
p
homeomorphic to a disk is a right-sided hyperbolic sector (resp. left-sided)
with respect to
`1
and
`2
if, for every point
xi`iU
, for
i
= 1
,
2, there exists a curve
Ti
going through
xi
and
transversal to the ﬂow of Xsuch that:
for every point
yTr
1
(resp.
Tl
1
) the positive semi-trajectory
`+
X
(
y
) starting from
y
intersects
Tr
2
(resp.
Tl
2
)
before leaving U;
the point of ﬁrst intersection of `+
X(y) and Tr
2(resp. Tl
2) converges to x2, for yx1.
Note that a right-sided hyperbolic sector for
X
is a left-sided hyperbolic sector for
X
. An illustration of
hyperbolic sector can be found in Figure 3, an example of sided contours can be found in Figure 6, and for the
general theory we refer to Section 2.3.5 of [4].
5.1. Topological structure of the characteristic foliation
Now, assume that
S
does not contain any nontrivial recurrent trajectories. To prove Proposition 5.1 we are
going to use the topological structure of a ﬂow. We resume here the relevant theory, following the exposition in
Section 3.4 of [4].
The singular trajectories of the characteristic foliation of Sare precisely the following:
characteristic points;
separatrices of characteristic points (see [4], Sect. 2.3.3);
isolated periodic trajectories;
periodic trajectories which contain in every neighbourhood both periodic and non-periodic trajectories.
The union of the singular trajectories is noted
ST
(
S
), and it is closed. The open connected components of
SST
(
S
) are called cells. The leaves of the characteristic foliation of
S
contained in the same cell have the same
behaviour, as shown in the following proposition.
Proposition 5.3
([
4
], Sect. 3.4.3)
.
Assume that the ﬂow of
X
has a ﬁnite number of singular trajectories. Let
Rbe a cell ﬁlled by non-periodic trajectories; then:
(i) Ris homeomorphic to a disk, or to an annulus;
(ii) the trajectories contained in Rhave all the same ω-limit and α-limit sets;
(iii) the limit sets of any trajectory in Rbelongs to ∂R;
NDUCED GEOMETRY ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 17
Figure 4. The sectors of an isolated equilibrium of a dynamical system.
(iv) each connected component of ∂R contains points of the ω-limit or α-limit sets.
Using this proposition, we show the following lemma.
Lemma 5.4.
Let
S
be surface satisfying the hypothesis of Proposition 5.1. Then, for every cell
R
of the
characteristic foliation of S, we have that
dS(x, y)<+ x, y RR.
Proof.
Since the surface
S
is compact and the characteristic points in Σ(
S
) are isolated, there is a ﬁnite number
of characteristic points. Moreover, there are no periodic trajectories. This implies that there is a ﬁnite number of
singular trajectories, hence we can apply Proposition 5.3.
Let
R
be a cell of the characteristic foliation of
S
, and let Γ be one of the connected components of the
boundary
∂R
(of which there are either one or two, due to Prop. 5.3). The curve Γ is the union of characteristic
points and separatrices. If all characteristic points have a hyperbolic sector towards
R
(right-sided or left-sided),
then Γ would be a sided contour, which is excluded. Therefore, there exists a characteristic point
p
Γ without
a hyperbolic sector towards
R
. As shown in Section 8.18 of [
3
], around an isolated equilibrium there are only the
three kinds of sectors depicted in Figure 4. Since there is no elliptic sector due to Remark 4.9, the point
p
has a
parabolic sector towards R.
Due to Proposition 5.3, the point
p
is the
ω
-limit or the
α
-limit of every trajectory in
R
. Then, for every point
xR
, there exists a semi-leaf
`+
X
(
x
) or
`+
X
(
x
) starting from
x
and converging to
p
. Due to Proposition 1.3,
this semi-leaf has ﬁnite sub-Riemannian length, hence dS(x, p) is ﬁnite.
Next, for every point yΓ, note that
dS(x, y)dS(x, p) + dS(p, y).
dS
(
x, p
) is ﬁnite, and the same holds for
dS
(
p, y
). Indeed, one can ﬁnd a horizontal
curve of ﬁnite length connecting
p
and
y
using a concatenation of the separatrices contained in Γ. See Figure 5
for a graphical representation.
If the boundary of
R
has a second connected component, then the above argument holds also for the other
connected component because it suﬃces to repeat the above argument for it. Thus, we have shown that
dS(x, y)<+ xR, y R,
which implies the statement of the lemma.
18 D. BARILARI ET AL.
Figure 5. How to connect the points of a cell with the points in the boundary.
Figure 6. An embedded polygon which bounds a right-sided contour.
Lemma 5.5.
Let
S
be surface satisfying the hypothesis of Proposition 5.1. Then, for every
x
in
S
, there exists
an open neighbourhood Uof xsuch that, for all yin U,
dS(x, y)<+.
Proof.
Let
x
be a point of
S
. If
x
does not belong to the union of the singular trajectories, then it is in the
interior of a cell
R
. Thus, due to Lemma 5.4, one can choose
U
=
R
. Otherwise, the point
x
belongs to a
separatrix, or it is a characteristic point of S.
Assume that
x
belongs to a separatrix Γ. Then, there exists a neighbourhood
U
of
x
which is divided by Γ
in two connected components. Those two connected components are contained in some cell
R1
and
R2
, which
contain Γ in their boundary. For every
yU
, then either
yRi
, for
i
= 1
,
2, or
y
Γ. If
yRi
, then it suﬃces
to apply Lemma 5.4. Otherwise, if yΓ, the separatrix Γ itself connects xand y.
Finally, assume that
x
is a characteristic point. Due to Corollary 4.5, Remark 4.6, and Corollary 4.7, there
exists a neighbourhood
U
of
x
in which the characteristic foliation of
S
is topologically conjugate to a saddle, a
node or a saddle-node. Thus, one can repeat the same argument as before: for every
yU
, if
y
belongs to a cell
then one applies Lemma 5.4; otherwise, if ybelongs to a separatrix one can connect xand ydirectly.
The proof of Proposition 5.1 is an immediate corollary of Lemma 5.5.
Proof of Proposition 5.1.
The property of having ﬁnite distance is an equivalence relation on the points of
S
.
Because of Lemma 5.5, the equivalence classes are open. Thus, because
S
is connected, there is only one class.
6. Spheres in a tight contact distribution
In this section we prove Theorem 1.5,i.e., in a tight coorientable contact distribution the topological spheres
have ﬁnite induced distance. This is done by showing that the hypothesis of Proposition 5.1 are satisﬁed in this
setting.
NDUCED GEOMETRY ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 19
Figure 7. The characteristic foliation of the perturbed surface.
An overtwisted disk, precisely deﬁned in Deﬁnition B.1, is en embedding of a disk with horizontal boundary
such that the distribution does not twist along the boundary. A contact distribution is called overtwisted if it
admits a overtwisted disk, and it is called tight if it is non-overtwisted.
Remark 6.1.
Note that if the boundary of a disk is a periodic trajectory of its characteristic foliation, then the
disk is overtwisted. Indeed, since a periodic trajectory does not contain characteristic points, then the plane
distribution never coincides with the tangent space of the disk, thus the distribution can’t perform any twists.
Lemma 6.2.
Let (
M, D
)be a tight contact 3-manifold, and
S
an embedded surface with the topology of a sphere.
Then, the characteristic foliation of Sdoes not contain periodic trajectories.
Proof.
Assume that the characteristic foliation of
S
has a periodic trajectory
`
. Then, because
`
does not have
self-intersections, the leaf
`
divides
S
in two topological half-spheres
1
and
2
. The disks
i
, for
i
= 1
,
2, are
overtwisted, which contradicts the hypothesis that the distribution is tight because Remark 6.1.
Now, let us discuss the sided contours.
Lemma 6.3.
Let (
M, D
)be a tight contact 3-manifold, and
SM
an embedded surface with the topology of a
sphere. Then the characteristic foliation of Sdoes not contain sided contours.
Proof.
Assume that the characteristic foliation presents a sided contour Γ. Its complementary
S
Γ has two
connected components, which are topologically half-spheres. Let us call the component on the same side of Γ,
i.e., if Γ is right-sided (resp. left-sided) then is on the right (resp. left). For instance, if Γ is right-sided, then
the characteristic foliation of looks like that of the polygon in Figure 6.
Let
p
be one of the vertices of ∆, let
`1
and
`2
p
, and let
U
be a neighbourhood
of
p
such that we are in the condition of Deﬁnition 5.2. Let us ﬁx two points
xi`iU
, for
i
= 1
,
2. Due to the
deﬁnition of hyperbolic sector, in a neighbourhood of x1the leaves pass arbitrarily close to x2.
We are going to give the idea of how to perturb the surface near
x1
and
x2
so that the separatrices
`1
and
`2
are diverted to the same nearby leaf, therefore bypassing
p
. In other words, via a
C
-small perturbation of
S
supported in a neighbourhood of
x1
and
x2
, we obtain a sphere which contains a sided contour with one less
vertex, see Figure 7. By repeating such perturbation for every vertex, one obtains a new surface with a periodic
trajectory in its characteristic foliation, which is excluded due to Lemma 6.2.
Consider the Heisenberg distribution (
R3,ker
(d
z
+
1
2
(
y
d
xx
d
y
)). Let
P
be the vertical plane
P
=
{x
= 0
}
,
and
q
a point in
P
contained in the
y
-axis. As one can see in Example 7.1, the characteristic foliation of
P
is
made up of parallel horizontal lines.
Locally, it is possible to rectify the surface
S
into the plane
P
using a contactomorphism of the respective
ambient spaces, as explained in the following lines. Due to the rectiﬁcation theorem of dynamical systems,
the characteristic foliation of
S
in a neighbourhood of
x1
is diﬀeomorphic to that of a neighbourhood of
q
in
20 D. BARILARI ET AL.
Figure 8. The lift to an horizontal curve connecting diﬀerent leaves.
P
. A generalisation of a theorem of Giroux ([
20
], Thm. 2.5.23) implies that the
C1
-conjugation between the
characteristic foliations of the two surfaces can be extended, in a smaller neighbourhood, to a contactomorphism.
Precisely, there exists a contactomorphism
ψ
from a neighbourhood
VM
of
x1
to a neighbourhood of
q
in
R3
,
with ψ(S) P.
For what it has been said above, the image of
`1
by
ψ
is contained in the
y
-axis. By creating a small bump in
P
after the point
q
, we will be able to divert the leaf going through
q
to any other parallel line. Precisely, for any
curve γ(t)=(x(t), y(t)), deﬁning
z(t) = 1
2Zt
t1
x(s)y0(s)y(s)x0(s)dst[t1, t2],
we obtain a horizontal curve (
x
(
t
)
, y
(
t
)
, z
(
t
)). Now, let
γ
be a smooth curve which joins smoothly to the
y
-axis
at its end points
γ
(
t1
) =
q
and
γ
(
t2
), and let be the set between
γ
and the
y
-axis. One can verify that
z
(
t2
) =
Area
, where the area is a signed area. By choosing an appropriate curve
γ
, we can connect the
y
-axis from
q
to any other parallel line in
P
via a horizontal curve (Fig. 8). Next, by creating a small bump
in
P
in order to include this horizontal curve one has successfully diverted the leaf. This procedure can be
done
C
-small, provided one wants to connect to parallel lines suﬃciently close to the
y
-axis. Thus, one can
make sure that no new characteristic points are created. Finally, this perturbation has to be transposed to a
perturbation of Susing ψ.
The same argument has to be repeated mutatis mutandis in a neighbourhood of
x2
, ensuring that one connects
x2
exactly to the leaf coming from
x1
. This is possible due to the continuity property of a hyperbolic sector,
which ensures that the leaf coming from
x1
intersects the domain of the rectifying contactomorphism of
x2
.
We can ﬁnally prove Theorem 1.5.
Proof of Theorem 1.5.
The surface
S
admits a global characteristic vector ﬁeld, due to Lemma 2.1. Next, a
surface with the topology of a sphere doesn’t allow ﬂows with nontrivial recurrent trajectories, see Lemma 2.4 of
[
4
]. Indeed, from a nontrivial recurrent trajectory one can construct a closed curve transversal to the ﬂow which
does not separate the surface, which contradicts the Jordan curve theorem.
Then, Lemma 6.2 and Lemma 6.3 imply that the ﬂow of a characteristic vector ﬁeld of
S
does not contain
periodic trajectories and sided contours, thus the hypothesis of Proposition 5.1 are satisﬁed. Consequently,
dS
is
ﬁnite.
NDUCED GEOMETRY ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 21
7. Examples of surfaces in the Heisenberg structure
In this section we present some examples of surfaces in the Heisenberg sub-Riemannian structure, that is the
contact, tight, sub-Riemannian structure of R3for which (X1, X2) is a global orthonormal frame, where
X1=xy/2z, X2=y+x/2z.
If (
u, v
)
7→
(
x
(
u, v
)
, y
(
u, v
)
, z
(
u, v
)) is a parametrisation of a surface
S
, then the characteristic vector ﬁeld
X
in
coordinates u, v becomes
X=zv+xv
y
2yv
x
2
∂u +zu+xu
y
2yu
x
2
∂v ,(7.1)
where have used the subscripts to denote a partial derivative. When the surface is the graph of a function
S={z=h(x, y)}, then in the graph coordinates
X=x
2yh
∂x +xh+y
2
∂y ,
and, at a characteristic point p= (x, y, z), the metric coeﬃcient b
Kpis computed by
b
Kp=3/4 + 2
xxh(x, y)2
yy h(x, y)2
xyh(x, y )2
yxh(x, y ).
7.1. Planes
Let us consider aﬃne planes in Heisenberg. Thanks to the left-invariance, it is not restrictive to consider a
plane Pgoing throughout the origin. Thus,
P={(x, y, z)R3|ax +by +cz = 0}with (a, b, c)6= (0,0,0).
If c= 0, i.e., the plane is vertical, then Pdoes not contain characteristic points. Every characteristic vector
ﬁeld is parallel to the vector (
b, a,
0), therefore the characteristic foliation of
P
consists of lines that are parallel
to the
xy
-plane. This implies that points with diﬀerent
z
-coordinate are not at ﬁnite distance from each other,
see Figure 9(left).
Otherwise, if c6= 0, then Phas exactly one characteristic point p= (2b/c, 2a/c, 0). One has that
b
Kp=3
4.
Thus, because of formula (4.4), there is one eigenvalue of multiplicity two. Due to Corollary 4.5, the characteristic
foliation of Phas a node at p. An explicit computation of XSshows that
XS(q) = qp
2q P,
which shows that the characteristic foliation of
P
is composed of Euclidean half-lines radiating out of
p
. The
metric dPinduced by the Heisenberg group on Psatisﬁes the following relation: for all q, q0 P, one has
dPq, q0=(|(x, y )(x0, y0)|R2,if (qp)//(q0p)
dP(q, p) + dP(q0, p),otherwise,
22 D. BARILARI ET AL.
Figure 9. The qualitative picture of the characteristic foliation of a vertical plane (left), and of a
non-vertical plane (right).
where we have written
q
= (
x, y, z
) and
q0
= (
x0, y0, z0
). This distance is sometimes called British Rail metric.
See Figure 9(right).
7.2. Ellipsoids
Fix a, b, c > 0, and consider the surface E=Ea,b,c deﬁned by
Ea,b,c =(x, y, z)R3x2
a2+y2
b2+z2
c21=0.
This surface has exactly two characteristic points
p1
= (0
,
0
, c
) and
p2
= (0
,
0
,c
), respectively at the North and
the South pole. For both points, one has
b
Kpi=3
4+c2
a2b2, i = 1,2.
Because of Corollary 4.5, the characteristic foliation of
E
spirals around the two poles, as in Figure 1. Due to
Proposition 1.3, the spirals converging to the poles have ﬁnite sub-Riemannian length, thus the length distance
dS
is ﬁnite. Indeed,
dS
is realised by the length of the curves joining the points with either the North, or the
South pole. Here, the ﬁniteness of dSis also a particular case of Theorem 1.5.
7.3. Symmetric paraboloids
Let aR, and consider the paraboloid Pawith
Pa=(x, y, z)R3|z=ax2+y2.
The origin pis the unique characteristic point of Pa. Note that
b
Kp=3
4+ 4a4,
therefore the characteristic foliation is a focus.
7.4. Horizontal torus
Fix R > r > 0, and consider the torus parametrised by
Φ(u, v) = (R+rcos u) cos v, (R+rcos u) sin v , r sin u.
NDUCED GEOMETRY ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 23
Figure 10. A leaf of the characteristic foliation of two Horizontal tori. On the left-hand side the leaf
is periodic, and on the right-hand side there is a portion of an everywhere dense leaf.
This is the torus obtained by revolving a circle of radius
r >
0 in the
xz
-plane around a circle of radius
R > r
surrounding the z-axis. Using formula (7.1), a characteristic vector ﬁeld Xin the coordinates (u, v ) is
X=(R+rcos(u))2
2
∂u rcos(u)
2
∂v .(7.2)
It is immediate to see that the characteristic set is empty. Thus, no point can be a limit point of any leaves of
the characteristic foliation; due to Remark 4.10, this implies that the length distance is inﬁnite.
Lemma 7.1.
The characteristic foliation of a horizontal torus is ﬁlled either with periodic trajectories, or with
everywhere dense trajectories.
Proof. Using expression (7.2), in the coordinates u, v the trajectories of Xsatisfy
˙u= (R+rcos(u))2/2
˙v=rcos(u)/2.(7.3)
Because the Heisenberg distribution and the horizontal torus are invariant under rotations around the
z
-axis,
the same applies to the characteristic foliation. Thus, the solutions of (7.3) are
v
-translations of the solution
γ0(t)=(u(t), v(t)) with initial condition γ0(0) = (0,0).
Note that (
r
+
R
)
2/
2
˙u
(
t
)
(
Rr
)
2/
2. Thus, there exists a time
t0
in which the trajectory
γ0
(
t
), satisﬁes
u
(
t0
)=2
π
. Deﬁne
αr,R
=
v
(
t0
). If
αr,R/
(2
π
) =
m/n
is rational, then
γ0
(
nt0
) = 0 (
mod
2
π
). This shows that
γ0
(
t
) is periodic, as every other trajectory. On the other hand, if
αr,R/
(2
π
) is irrational, then a classical argument
shows that γ(t) is dense in the torus, see for instance ([4], E.g .2.3.1).
See Figure 10 for a picture of a leaf in these two cases.
7.5. Vertical torus
Fix R > r > 0, and consider the torus T=Tr,R parametrised by
Φ(u, v) = rsin u, (R+rcos u) cos v, (R+rcos u) sin v.
24 D. BARILARI ET AL.
Figure 11. The topological skeleton, i.e., the singular trajectories, of the characteristic foliations
of two vertical tori: the torus on the left-hand side has four characteristic points, and the torus
on the right-hand side has eight.
This is the torus obtained by turning a circle of radius
r
in the
xy
-plane around a circle of radius
R
surrounding
the x-axis. Due to formula (7.1), a characteristic vector ﬁeld Xin coordinates u, v is
X=(R+rcos u)cos v+r
2sin vsin u
∂u
+r
22 sin usin vRcos ucos vrcos v
∂v .
The characteristic points are critical points of the vector ﬁeld
X
. If
cos v
=
sin u
= 0, then (
u, v
) corresponds to
a solution; this gives 4 characteristic points
F±=0,0,±(R+r), V±=0,0,±(Rr).
The other critical points of Xoccur if and only if
tan v=2
rsin u,cos u=4 + r2
rR .(7.4)
System
(7.4)
has solutions if and only if
R >
4 and
|
2
rR| R216
, in which case we have 4 additional
characteristic points
Si
(
r, R
), for
i
= 1
,
2
,
3
,
4. Now, the metric coeﬃcient at the characteristic points
F±
and
V±
is
b
KF±=3
4+1
r(R+r),
b
KV±=