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

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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 $\widehat K$ 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.
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ON THE INDUCED GEOMETRY ON SURFACES IN
3D CONTACT SUB-RIEMANNIAN MANIFOLDS
DAVIDE BARILARI, UGO BOSCAIN, AND DANIELE CANNARSA
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.
Keywords:
contact geometry, sub-Riemannian geometry, length space, Riemannian
approximation, Gaussian curvature, Heisenberg group.
Contents
1. Introduction 1
2. Preliminaries 5
3. Riemannian approximations and Gaussian curvature 6
4. Local study near a characteristic point 9
5. Global study of the characteristic foliation 13
6. Spheres in a tight contact distribution 16
7. Examples of surfaces in the Heisenberg structure 18
Appendix A. On the center manifold theorem 21
Appendix B. Tight and overtwisted distributions 22
References 23
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 Sby this metric tensor is not the restriction of the distance of R3
to 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 [Gro96, Sec. 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,
Date: September 24, 2020.
1
2 DAVIDE BARILARI, UGO BOSCAIN, AND DANIELE CANNARSA
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 [BBI01].) 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 [BBI01, Ch. 2] and [ABB19, Sec. 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 Lemma 2.4)
and, generically, it is composed of isolated points (see [Gei08, Par. 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
by adding a 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
S
connecting these two points.
Figure 1. The characteristic foliation deﬁned by the Heisenberg dis-
tribution (
R3,ker
(
dz
+
1
2
(
ydx xdy
)) 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 1.4.
INDUCED DISTANCE ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 3
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 gX0be the Riemannian extension of gfor which
hX0, DigX0= 0,|X0|gX0= 1.
The Riemannian metrics
gεX0
, for
ε >
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×DRdeﬁned by
BX0(X, Y ) = αif [X, Y ] = αX0mod D.
Since
D
is endowed with the metric
g
, the bilinear form
BX0
nant.
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
Din a neighbourhood of p. Then, in the notations deﬁned above, the limit
(1) b
Kp= lim
ε0
KεX0
p
det BεX0
p
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 Sin Uif, for all xin U,
(2) spanRX(x) = ({0},if xΣ(S),
TxSDx,otherwise,
and satisﬁes the condition
(3) div X(p)6= 0,pΣ(S)U.
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 (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 [Gei08, Par. 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.
Proposition 1.2.
Let
S
be a
C2
surface embedded in a 3D contact sub-Riemannian manifold.
Given a characteristic point
p
in Σ(
S
), let
X
be a characteristic vector ﬁeld
X
near
p
. Then,
tr DX(p)6= 0 and
(4) b
Kp=1 + det DX(p)
(tr DX(p))2.
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
(4)
shows that
b
Kp
is independent
on the sub-Riemannian metric, and depends only on the line ﬁeld deﬁned by Don S.
4 DAVIDE BARILARI, UGO BOSCAIN, AND DANIELE CANNARSA
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 xin and a characteristic vector ﬁeld Xof Ssuch that
(5) etX (x)pfor t+,
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)pfor 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
[ZZ95, Lem. 2.1] 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.
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
(6)
), then
S
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 1.4.
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 dSis ﬁnite.
We stress that having isolated characteristic points is a generic property for a surface in a
contact manifold. Example 7.4 and Example 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.
Previous literature.
Characteristic foliations of surfaces in 3D contact manifolds are
studied in numerous references; here we use notions contained in [Gir91,Gir00,Ben83] and
we refer to [Etn03,Gei08] for an introduction to the subject. Moreover, for an introduction
to sub-Riemannian geometry we refer to [Mon02,Rif14,Jea14,ABB19].
The use of the Riemannian approximation scheme to deﬁne sub-Riemannian geometric
invariants is a well-known technique. For example, it had already been used in [Pau04] to
study the horizontal mean curvature in relation to the minimal surfaces in the Heisenberg
group, whose integrability is discussed in [DGN12]. For a general description of the properties
of the Riemannian approximations in Heisenberg we also refer to [CDPT07].
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 [BTV17] the authors deﬁned the sub-Riemannian Gaussian curvature at a
point
xS
Σ(
S
)as
KS
(
x
) =
limε0KεX0
x
, and they proved that a Gauss-Bonnet type
theorem holds; here the authors worked in the setting of the Heisenberg group, and with
INDUCED DISTANCE ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 5
X0
equals to the Reeb vector ﬁeld of the Heisenberg group. This construction is extended
in [WW20] to the aﬃne group and to the group of rigid motions of the Minkowski plane,
and in [Vel20] to a general sub-Riemannian manifold. In the latter, the author linked
KS
with the curvature introduced in [DV16], 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 [ABS08]. We ﬁnally notice that the invariant
KS
also appears in [Lee13], where it is called
curvature of transversality. An expression for
KS
is provided also in [BBCH20], in relation
to a new notion of stochastic processes in this setting.
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 1.4 in Section 6. Section 7is devoted some examples of induced
distances on surfaces in the Heisenberg group.
Acknowledgements.
We would like to thank Daniel Bennequin and Nicola Garofalo for
stimulating discussions. This work was supported by the Grant ANR-15-CE40-0018 SRGI of
the French ANR. The third author is supported by the DIM Math Innov grant from Région
Île-de-France.
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 ω1(M), which can be normalised to satisfy
(6) D= ker ω, ω 6= 0, dω|D= volg.
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
xS
Σ(
S
), the intersection
(7) lx=DxTxS
is one-dimensional, and we can think of
(7)
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 (2) and (3).
Lemma 2.1.
Assume that
S
is orientable and that
D
is coorientable. Then,
S
global characteristic vector ﬁeld; moreover, the characteristic vector ﬁelds of
S
are the vector
ﬁelds Xfor which there exists a volume form of Ssuch that
(8) Ω(X, Y ) = ω(Y)for all YT S.
Indeed, formula
(8)
is the deﬁnition of characteristic vector ﬁeld as given in [Gei08, Par. 4.6],
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
(8)
,
then it satisﬁes
(2)
and
(3)
. Reciprocally, a vector ﬁeld
¯
X
satisfying
(2)
is a multiple of any
vector ﬁeld
X
satisfying
(8)
for some function
φ
with
φ|SΣ(S)6
(3)
holds,
then φ|Σ(S)6= 0; thus, ¯
Xsatisﬁes (8) with 1
φas 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.
6 DAVIDE BARILARI, UGO BOSCAIN, AND DANIELE CANNARSA
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 |U
is in
T S
if and only
if V f = 0; thus, for a point pUS,
(9) pΣ(S)X1f(p) = X2f(p)=0.
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
(10) Xf= (X1f)X2(X2f)X1,
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
(9)
,
Xf
(
p
) = 0 if and only if
p
Σ(
S
);
thus, Xfsatisﬁes (2). Moreover, for all pΣ(S),
div Xf(p) = X2X1f(p)X1X2f(p)=[X2, X1]f(p),
which is nonzero due to the contact condition; thus, Xfsatisﬁes (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
(9)
, the characteristic points in
V
=
US
are the
solutions of the system 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 [Bal03] 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
KεX0
pat a characteristic point p.
In order to deﬁne the metric coeﬃcient b
Kp, one needs to ﬁx a vector ﬁeld X0transverse
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
gεX0
such that, for every
ε >
0, one has
hD, X0igεX0
= 0 and
|X0|gεX0
= 1
. To simplify the
notation, we drop the dependance from X0in the superscript, writing gε=gεX0.
Let
ε
be the Levi-Civita connection of (
M, gε
). Since we study local properties, we
can restrict to a domain equipped with an orthonormal oriented frame (
X1, X2
)of
D
; thus,
(εX0, X1, X2)is an orthonormal basis of gε. Due to the Koszul formula, one has
ε
XiXj, Xkgε=1
2− hXi,[Xj, Xk]igε+hXk,[Xi, Xj]igε+hXj,[Xk, Xi]igε,
for all
i, j, k
= 0
,
1
,
2. This identity enables us to describe
ε
using the frame (
X0, X1, X2
),
which is independent from
ε
. This is done using the Lie bracket structure of the frame, i.e.,
the Cfunctions ck
ij such that
(11) [Xj, Xi] = c1
ij X1+c2
ij X2+c0
ij X0for i, j = 0,1,2.
The functions ck
ij are the structure constants of the frame.
INDUCED DISTANCE ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 7
Thus, for every ε > 0, we have that
ε
XiXi=ci
i0ε2X0+ci
i1
ε2X1+ci
i2
ε2X2i= 0,1,2(12)
ε
XjXi=1
2cj
0iε2ci
0jε2+c0
ij X0+cj
ij Xji6=j= 1,2
ε
X0X1=c0
01X0+1
2c1
02 c2
01 +c0
12
ε2X2
ε
X0X2=c0
02X0+1
2c2
01 c1
02 c0
12
ε2X1,
and the remaining derivatives
ε
X1X0
and
ε
X2X0
are computed using that the connection
is torsion-free.
Given the surface
S
, the second fundamental form
IIε
of
S
is the projection of the Levi-
Civita connection on the orthogonal to the tangent space of the surface. The Gaussian
curvature Kε=KεX0of Sin (M, gε)is deﬁned by the Gauss formula
(13) Kε=Kε
ext + det(IIε),
where, given a frame (X, Y )of T S , the extrinsic curvature Kε
ext is
(14) Kε
ext =ε
Xε
YY− ∇ε
Yε
XY− ∇ε
[X,Y ]Y, X gε
|X|2
gε|Y|2
gε− hX, Y i2
gε
,
and the determinant det IIεof the second fundamental form is
(15) det IIε=IIε(X, X),IIε(Y, Y )gεIIε(X, Y ),IIε(X, Y )gε
|X|2
gε|Y|2
gε− hX, Y i2
gε
.
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). 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
f
deﬁning
S
.
The determinant of the bilinear form BεX0
pis homogeneous in ε, and satisﬁes
(16) det BεX0
p=det BX0
p
ε2=BX0
p(X1, X2)2
ε2=(c0
12(p))2
ε2,
where
c0
12
is deﬁned in (11). Therefore, in order to prove the convergence of the limit in (1),
it suﬃces to show that the Gaussian curvature KεX0
pat pdiverges at most as 12.
Let us start with the computation of the determinant (15) of the second fundamental
form at a characteristic point. It is convenient to write the second fundamental form as
IIε(X, Y ) = ε
XY, N εNε.
where Nεis the Riemannian unitary gradient of f, i.e.,
Nε=(X1f)X1+ (X2f)X2+ε(X0f)X0
p(X1f)2+ (X2f)2+ε(X0f)2.
At the characteristic point p, the gradient Nε(p)simpliﬁes to
(17) Nε(p) = εsign(X0f)X0(p).
To compute (15) one needs to choose a frame of T S ; we will use the frame (F1, F2)with
(18) Fi= (X0f)Xi(Xif)X0for i= 1,2.
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.
8 DAVIDE BARILARI, UGO BOSCAIN, AND DANIELE CANNARSA
Lemma 3.1.
Let
pS
be a characteristic point. Then, in the previous notations, for every
ε > 0, the determinant (15) of the second fundamental form in pis
det IIε(p) = 1
ε2det 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,
ε
FiFj(p) = (X0f)2ε
XiXj+ (X0f)(XiX0f)Xj(X0f)(XiXjf)X0p,
for
i, j
= 1
,
2. Using formula
(17)
for
Nε
, one ﬁnds that only the component along
X0
plays
a role in the second fundamental form in p. Thus, using the covariant derivatives in (12),
h∇ε
FiFj, N εip=|X0f(p)|
εXiXjf+ (X0f)c0
ij
2+ (X0f)ε2cj
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 (14) is the sectional curvature of the plane
TpS
in
M
, which
is known when
X0
is the Reeb vector ﬁeld and
ε
= 1; this can be found for instance in
[BBL20, Prop. 14]. In our setting, the resulting expression for ε0is the following.
Lemma 3.2. Let pSbe a characteristic point. Then, for every ε > 0,
Kε
ext(p) = 3
4ε2(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=Dpat the characteristic point p. Then, to compute
Kε
ext(p) = h∇ε
X1ε
X2X2− ∇ε
X2ε
X1X2− ∇ε
[X1,X2]X2, X1ip
it suﬃces to use the expressions (12).
Remark 3.3.Following the proof of Lemma 3.1 and Lemma 3.2, the exact expressions for
det IIε(p)and Kε
ext(p)at a characteristic point pare, for all ε > 0,
det IIε(p) = + 1
ε2det HessHf
(X0f)2(c0
12)2
4p+ε2c1
01c2
02 c1
02 +c2
012
4p
+1
X0f(p)c2
02X1X1f+c1
01X2X2fc2
01 +c1
02
2(X2X1f+X1X2f)p,
Kε
ext(p) = 3
4
(c0
12(p))2
ε2ε2c1
01c2
02 (c2
01 +c1
02)2
4p
+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 [ABB19, Ch. 17]. 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 (13), Lemma 3.1
and Lemma 3.2, the Gaussian curvature at a characteristic point psatisﬁes
Kε
p=KεX0
p=(c0
12(p))2
ε21 + 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
(11)
and X1f(p) = X2f(p)=0. Using formula (16) for the determinant of BεX0
p, one ﬁnds that
(19) KεX0
p
det BεX0
p
=1 + det HessHf(p)
[X2, X1]f(p)2+O(ε2),
which shows that the limit (1) is ﬁnite. Moreover,
b
Kp
is independent of
X0
because the
transversal vector ﬁeld X0is absent in the constant term of equation (19).
INDUCED DISTANCE ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 9
Formula (19) 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,
(20) b
Kp=1 + det HessHf(p)
[X2, X1]f(p)2.
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 (20), 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 Xof S, the diﬀerential DX(p)is invertible. Otherwise, pis called degenerate.
Remark 4.2.Condition (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.
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
(21) DX =X1a1X2a1
X1a2X2a2,
and the formulas for the determinant and the trace are
det DX = (X1a1)(X2a2)(X1a2)(X2a1),(22)
tr DX = div X= (X1a1)+(X2a2).(23)
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 (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
)
X1
deﬁned in
(10)
. Using expression (21) for the diﬀerential of a
vector ﬁeld, we get
DXf(p) = X1X2f(p)X2X2f(p)
X1X1f(p)X2X1f(p).
Thus, using expressions (22) and (23) 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.
10 DAVIDE BARILARI, UGO BOSCAIN, AND DANIELE CANNARSA
The eigenvalues of the linearisation
DX
(
p
)of a characteristic vector ﬁeld
X
can be written
as a function of b
Kpby rearranging equation (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
(24) λ±(X, p) = tr DX(p)1
2±r3
4b
Kp.
Proof. Let us note λ±=λ±(X, p), and α= tr DX(p). Equation (4) reads
b
Kp=1 + λ+λ
α2.
Using that
λ+
+
λ
=
α
, equation
(4)
implies that the eigenvalues satisfy the equation
z2αz +α2(b
Kp+ 1) = 0, which implies (24).
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
(25) XS=(X1f)X2(X2f)X1
Zf ,
where
Z
is the Reeb vector ﬁeld of the contact form
ω
of
D
deﬁned in
(6)
, i.e., the unique
vector ﬁeld satisfying
ω
(
Z
)=1and
(
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 [Lee13]; in the case of the
Heisenberg group, pScoincides with the imaginary curvature introduced in [AF07,AF08].
Expression (24) for the eigenvalues of the linearisation
DX
(
p
)implies the following
relations between the eigenvalues and the metric coeﬃcient b
Kp:
(i) b
Kp<1if and only if λ±Rwith diﬀerent signs;
(ii) b
Kp=1if and only if λ= 0 and λ+R;
(iii) 1<b
Kp≤ −3/4if 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
p
is 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 [Har60]. For
this theorem to hold, one needs the characteristic vector ﬁeld Xto 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
S
in 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 [Per12, Par. 2.8]. Finally,
for a Csurface some informations can be found in [GHR03].
INDUCED DISTANCE ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 11
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.
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 charac-
teristic point in Σ(
S
). Then,
b
Kp
=
1, and the characteristic foliation in a neighbourhood
centred at pis C0-conjugate at the origin to the orbits of a system of the form
(26) ˙u=φ(u)
˙v=v,
for a function
φ
with
φ
(0) =
φ0
(0) = 0. If
p
is isolated, then the characteristic foliation
described in
(26)
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 (26). The line
{v
= 0
}
, parametrised by
u
, is a center manifold of (26), and
the function
φ
determines the dynamic of (26); this illustrates the fact that the nonlinear
terms on a center manifold determine the dynamic near a degenerate characteristic point.
The equilibria of (26) 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+>0and φ<0, then the origin is a topological node.
- If φ+<0and 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 saddle-node.
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.
12 DAVIDE BARILARI, UGO BOSCAIN, AND DANIELE CANNARSA
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 > 0such that
(27) 1
C|V|R2≤ |V|gC|V|R2VDT S|U,
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
(27)
follows from the local equivalence of
˜g
with the pullback by
Φof the Euclidean metric of R2. Now, inequality (27) implies that
(28) LsR(+
X(y)) = Z+
0|X(etX (y))|gdt CZ+
0|X(etX (y))|R2dt.
At this point the proof of the ﬁniteness of the sub-Riemannian length of
+
X
(
y
)diﬀers de-
pending on whether pis 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 [Per12, Par. 2.8], 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>0such that
(29) |etX (y)p|R2Ceαt t>t0.
Since X(p)=0, for all t > 0one has
X(etX (y))R2=X(etX (y)) X(p)R2sup
B||DX(x)|| |etX (y)p|R2.
Due to the inequality (28) and (29), 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>
0and a trajectory
etX (z)contained in C, such that
(30) |etX (y)etX (z)|R2Ceαt tt0.
The triangle inequality implies that
(31) |X(etX (y))|R2≤ |X(etX (y)) X(etX (z))|R2+|X(etX (z))|R2.
Due to inequality
(28)
, to prove that
LsR
(
+
X
(
y
)) is ﬁnite, it suﬃces to show that the two
terms on the right-hand side of (31) are integrable for t0. Thanks to (30) and
|X(etX (y)) X(etX (z))|R2sup
B||DX|| |etX (y)etX (z)|R2,
then the ﬁrst term in
(31)
is integrable. Next, because
etX
(
z
)is a regular parametrisation of
a bounded interval inside a
C1
embedded curve (the center manifold
C
), then its derivative
|X(etX (z))|R2is integrable.
INDUCED DISTANCE ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 13
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
dSis not ﬁnite. For a discussion on non-characteristic domains we refer to [DGN06, Ch. 3].
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 yin 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 charac-
teristic foliation of
S
is described by a global characteristic vector ﬁeld of
S
which 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, . . . , ps
in Σ(
S
)and trajectories
1, . . . , s
such that:
- for all j= 1, . . . , s, we have ω(j, X ) = pj=α(j+1, X)(where s+1 =1);
-
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
xiiU
, for
i
= 1
,
2, there exists a curve
Ti
going through
xi
and transversal to the ﬂow of
X
such 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 [ABZ96, Par. 2.3.5].
14 DAVIDE BARILARI, UGO BOSCAIN, AND DANIELE CANNARSA
Figure 3. The illustration of a right-sided hyperbolic sector
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 [ABZ96, Par. 3.4].
The singular trajectories of the characteristic foliation of Sare precisely the following:
- characteristic points;
- separatrices of characteristic points (see [ABZ96, Par. 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
S ST
(
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
([ABZ96, Par. 3.4.3])
.
Assume that the ﬂow of
X
has a ﬁnite number of
singular trajectories. Let Rbe a cell ﬁl led 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;
(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 Rof 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 Γbe one of the connected
components of the boundary
∂R
(of which there are either one or two, due to Proposition 5.3).
The curve Γis the union of characteristic points and separatrices. If all characteristic points
have a hyperbolic sector towards
R
(right-sided or left-sided), then Γwould be a sided contour,
which is excluded. Therefore, there exists a characteristic point
p
Γwithout a hyperbolic
sector towards
R
. As shown in [ALGM73, Par. 8.18], 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 phas a parabolic sector towards R.
Due to Proposition 5.3, the point pis 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.
INDUCED DISTANCE ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 15
Figure 4. The sectors of an isolated equilibrium of a dynamical system.
Figure 5. How to connect the points of a cell with the points in the boundary.
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.
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.
Lemma 5.5.
Let
S
be surface satisfying the hypothesis of Proposition 5.1. Then, for every
xin 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 xbelongs 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 Γin two connected components. Those two connected components are
contained in some cell
R1
and
R2
, which contain Γin 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 Γitself 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
Sis connected, there is only one class.
16 DAVIDE BARILARI, UGO BOSCAIN, AND DANIELE CANNARSA
Figure 6. An embedded polygon which bounds a right-sided contour
6. Spheres in a tight contact distribution
In this section we prove Theorem 1.4, 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.
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
S
does 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
S
does not contain sided
contours.
Proof.
Assume that the characteristic foliation presents a sided contour Γ. Its complementary
S
Γhas two connected components, which are topologically half-spheres. Let us call the
component on the same side of Γ, i.e., if Γis right-sided (resp. left-sided) then is on the
right (resp. left). For instance, if Γis 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
xiiU
, 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.
INDUCED DISTANCE ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 17
Figure 7. The characteristic foliation of the perturbed surface.
Figure 8. The lift to an horizontal curve connecting diﬀerent leaves.
Consider the Heisenberg distribution (
R3,ker
(
dz
+
1
2
(
ydx xdy
)). 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 Pis 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
P
. A generalisation of a theorem of Giroux
[Gei08, 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 qin 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
qto 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)ds t[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 (Figure 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.
18 DAVIDE BARILARI, UGO BOSCAIN, AND DANIELE CANNARSA
We can ﬁnally prove Theorem 1.4.
Proof of Theorem 1.4.
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 [ABZ96, Lem. 2.4]. 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, dSis ﬁnite.
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
R3
for 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 Xin coordinates u, v becomes
(32) X=zv+xv
y
2yv
x
2
∂u +zu+xu
y
2yu
x
2
∂v ,
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
P
does 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
P
has exactly one characteristic point
p
= (
2
b/c,
2
a/c,
0). One
has that
b
Kp=3
4.
Thus, because of formula (24), there is one eigenvalue of multiplicity two. Due to Corollary 4.5,
the characteristic foliation of
P
has a node at
p
. An explicit computation of
XS
shows 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
dP
induced by the Heisenberg group on
P
satisﬁes 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,
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).
INDUCED DISTANCE ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 19
Figure 9. The qualitative picture of the characteristic foliation of a
vertical plane (left), and of a non-vertical plane (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 1.4.
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.
This is the torus obtained by revolving a circle of radius
r >
0in the
xz
-plane around a
R > r
surrounding the
z
-axis. Using formula (32), a characteristic vector
ﬁeld Xin the coordinates (u, v)is
(33) X=(R+rcos(u))2
2
∂u rcos(u)
2
∂v .
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 (33), in the coordinates u, v the trajectories of Xsatisfy
(34) ˙u= (R+rcos(u))2/2
˙v=rcos(u)/2.
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 (34)
are v-translations of the solution γ0(t)=(u(t), v(t)) with initial condition γ0(0) = (0,0).
20 DAVIDE BARILARI, UGO BOSCAIN, AND DANIELE CANNARSA
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 .
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 [ABZ96, 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.
This is the torus obtained by turning a circle of radius
r
in the
xy
-plane around a circle
R
surrounding the
x
-axis. Due to formula (32), a characteristic vector ﬁeld
X
in
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
(35) tan v=2
rsin u,cos u=4 + r2
rR .
System
(35)
has solutions if and only if
R >
4and
|
2
rR| ≤ R216
, in which case we
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±=3
41
r(Rr).
Note that
b
KF±>
3
/
4, thus, due to Corollary 4.5,
F±
is a focus for all value of
r
and
R
.
On the other hand, b
KV±can attain any value between −∞ and 3/4; precisely:
- if R < 4or |2rR|>R216, then b
-
if
|
2
rR|
=
R216
, then
b
KV±
=
1and
V±
is a degenerate characteristic point;
due to the Poincaré Index theorem, the points V±are saddles.
- if |2rR|<R216, then 1<b
KV±<3/4and V±are nodes.
INDUCED DISTANCE ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 21
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.
The values for which
|
2
rR|
=
R216
are a bifurcation of the dynamical system
X
,
because the number of characteristic point changes from 4 to 8. The characteristic points
Si
which appears at this bifurcation are saddles, due to the Poincaré Index theorem. The
bifurcation which takes place is the one presented in [Per12, E.g. 4.2.6].
Appendix A. On the center manifold theorem
In the language of dynamical systems, a non-degenerate characteristic point
p
is a hy-
perbolic equilibrium for any characteristic vector ﬁeld
X
, i.e., an equilibrium for which the
real parts of the eigenvalues of
DX
(
p
)are non-zero. For a hyperbolic equilibrium
p
, the
Hartman-Grobman theorem and the Hartman theorem give a conjugation between the ﬂow
of Xand the ﬂow of DX(p), see [Per12, Par. 2.8] and [Har60].
Let us discuss here the case of a non-hyperbolic equilibrium, i.e., of a degenerate charac-
teristic point. Let
E
be an open set of
Rn
containing the origin, and let
X
be a vector ﬁeld
in
C1
(
E, Rn
)with
X
(0) = 0. Due to the Jordan decomposition theorem, we can assume that
the linearisation of Xat the origin is
DX(0) =
C
P
Q
,
where
C
is a square
c×c
matrix with
c
complex (generalised) eigenvalues with zero real part,
P
with
p
complex (generalised) eigenvalues with positive real part, and
Q
with
q
complex
(generalised) eigenvalues with negative real part. Thus, the dynamical system
˙γ
=
X
(
γ
)can
be rewritten as
˙x=Cx +F(x, y, z)
˙y=P y +G(x, y, z)
˙z=Qz +H(x, y, z)
for (
x, y, z
)
Rc×Rp×Rq
=
Rn
, and for suitable functions
F
,
G
and
H
with
F
(0) =
G
(0) =
H(0) = 0 and D F (0) = DG(0) = DH(0) = 0.
The origin is a non-hyperbolic characteristic point if and only if
c
1. Under these
hypotheses, the following theorem shows that there exists an embedded submanifold
C
of
dimension
c
, tangent to
Rc
, and invariant for the ﬂow of
X
. Such manifold is called a central
manifold of Xat the origin.
22 DAVIDE BARILARI, UGO BOSCAIN, AND DANIELE CANNARSA
Proposition A.1
([Per12, Par. 2.12])
.
Under the previous notations, there exists an open
set
URc
containing the origin, and two functions
h1
:
URp
and
h2
:
URq
of class
C1
with
h1
(0) =
h2
(0) = 0 and
Dh1
(0) =
Dh2
(0) = 0, and such that the map
x7→
(
x, h1
(
x
)
, h2
(
x
)) parametrises a submanifold invariant for the ﬂow of
X
. Moreover, the
ﬂow of Xis C0-conjugate to the ﬂow of
(36)
˙x=Cx +F(x, h1(x), h2(x))
˙y=P y
˙z=Qz.
In general, the central manifold
C
is non-unique. Note that the dynamic of the
x
-variable
in equation (36) is simply the restriction of
X
to the center manifold
C
. One can show that
the trajectory converging to the origin approaches
C
exponentially fast: this is the asymptotic
approximation property we used in (30).
Proposition A.2
([Bre07, p. 330])
.
Under the previous assumptions, let us denote
C
a
center manifold of the ﬂow of
X
at the origin. Then, for every trajectory
l
(
t
)such that
l
(
t
)
0as
t
+
, there exists
η >
0and a trajectory
ζ
(
t
)in the center manifold
C
, such
that
eηt|l(t)ζ(t)|Rn0, as t +.
Appendix B. Tight and overtwisted distributions
In this section we brieﬂy recall the theory of tight distributions. For a more comprehensive
presentation, we refer to [Gei08, Par. 4.5]. In what follows
M
is a 3-dimensional contact
manifold, whose distribution is D.
To deﬁne an overtwisted disk, let us ﬁrst consider an embedding of ∆ =
{xR2
:
|x| ≤
1
}
in
M
, and denote Γ =
. Let Γbe horizontal with respect to the contact distribution
D
,
i.e.,
T
Γ
D
. Then, the normal bundle
N
Γ =
T M |Γ/T
Γcan be decomposed in two ways:
the ﬁrst with respect to the tangent space of , i.e.,
(37) NΓ
=T M
TT
TΓ,
and the second with respect to the contact distribution D, i.e.,
(38) NΓ
=T M
DD
TΓ.
A frame (
Y1, Y2
)of
N
Γis called a surface frame if it respects the splitting (37), i.e.,
Y1T M T
and
Y2T
T
Γ; similarly, it is called a contact frame if it respects the
splitting (38). Since the contact distribution is cooriented near , both bundles (37) and
(38) are trivial, thus one can always ﬁnd contact and surface frames.
The Thurston–Bennequin invariant of Γ, noted
tb
(Γ), is the number of twists of a contact
frame of Γwith respect to a surface frame: the right-handed twists are counted positively,
and the left-handed twists negatively (cf. for instance [Gei08, Def. 3.5.4]). Note that
tb
(Γ) is
independent of the orientation of Γ. The requirement that the distribution
D
does not twist
along the boundary of is equivalent to
tb
(
∆) = 0, i.e., the Thurston–Bennequin invariant
of being zero.
Deﬁnition B.1.
An embedded disk in a cooriented contact manifold (
M, D
)with smooth
boundary
is an overtwisted disk if
is a horizontal curve of
D
,
tb
(
∆) = 0, and there
is exactly one characteristic point in the interior of the disk.
Note that the elimination lemma of Giroux allows to remove the condition that there is
only one characteristic point in the interior of the owertwisted disk, as discussed for instance
in [Gei08, Def. 4.5.2].
Deﬁnition B.2.
A contact structure (
M, D
)is called overtwisted if it admits an overtwisted
disk, and tight otherwise.
INDUCED DISTANCE ON SURFACES IN 3D CONTACT SUB-RIEMANNIAN MANIFOLDS 23
References
[ABB19]
A. Agrachev, D. Barilari, and U. Boscain, A Comprehensive Introduction to Sub-
Riemannian Geometry, Cambridge Studies in Advanced Mathematics, Cambridge Uni-
versity Press, 2019.
[ABS08]
A. Agrachev, U. Boscain, and M. Sigalotti, A Gauss-Bonnet-like formula on two-
dimensional almost-Riemannian manifolds, Discrete &Continuous Dynamical Systems -
A20 (2008), no. 4, 801–822.
[ABZ96]
S. Kh. Aranson, G. R. Belitsky, and E. V. Zhuzhoma, Introduction to the qualitative
theory of dynamical systems on surfaces, Translations of Mathematical Monographs,
vol. 153, American Mathematical Society, 1996.
[AF07]
N. Arcozzi and F. Ferrari, Metric normal and distance function in the Heisenberg group,
Mathematische Zeitschrift 256 (2007), no. 3, 661–684.
[AF08]
,The Hessian of the distance from a surface in the Heisenberg group, Annales
Academiae Scientiarum Fennicae. Mathematica 33 (2008), no. 1, 35–63.
[ALGM73]
A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, Qualitative theory of
second order dynamic systems, Israel Program for Scientiﬁc Trasnlations Ltd., 1973.
[Bal03]
Z. M. Balogh, Size of characteristic sets and functions with prescribed gradient, Journal für
die Reine und Angewandte Mathematik. [Crelle’s Journal]
564
(2003), 63–83. MR2021034
[BBCH20]
D. Barilari, U. Boscain, D. Cannarsa, and K. Habermann, Stochastic processes on surfaces
in three-dimensional contact sub-Riemannian manifolds, arXiv e-prints (2020).
[BBI01]
D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry, Graduate Studies in
Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR1835418
[BBL20]
D. Barilari, I. Beschastnyi, and A. Lerario, Volume of small balls and sub-Riemannian
curvature in 3D contact manifolds, Journal of Symplectic Geometry 18 (2020), no. 2.
[Ben83]
D. Bennequin, Entrelacements et équations de Pfaﬀ., IIIe rencontre de géométrie du
Schnepfenried, 10-15 mai 1982. Vol. 1: Feuilletages. Géométrie symplectique et de contact.
Analyse non standard et applications, 1983, pp. 87–161 (French).
[Bre07]
A. Bressan, Tutorial on the center manifold theorem, Lecture Notes in Mathematics,
Springer Verlag, Germany, 2007 (English (US)).
[BTV17]
Z. M. Balogh, J. T. Tyson, and E. Vecchi, Intrinsic curvature of curves and surfaces
and a Gauss–Bonnet theorem in the Heisenberg group, Mathematische Zeitschrift
287
(2017), no. 1, 1–38.
[CDPT07]
L. Capogna, D. Danielli, Scott D. Pauls, and Jeremy T. Tyson, An introduction to the
Heisenberg group and the sub-Riemannian isoperimetric problem, Progress in Mathemat-
ics, vol. 259, Birkhäuser Verlag, Basel, 2007 (English (US)).
[DGN06]
D. Danielli, N. Garofalo, and D.-M. Nhieu, Non-doubling Ahlfors measures, perimeter
measures, and the characterization of the trace spaces of Sobolev functions in Carnot-
Carathéodory spaces, Memoirs of the American Mathematical Society
182
(2006), no. 857.
[DGN12]
,Integrability of the sub-Riemannian mean curvature of surfaces in the Heisenberg
group, Proceedings of the American Mathematical Society 140 (2012).
[DV16]
M. M. Diniz and J. M. M. Veloso, Gauss-Bonnet theorem in sub-Riemannian Heisenberg
space H1, Journal of Dynamical and Control Systems 22 (2016), no. 4, 807–820.
[Etn03]
J. B. Etnyre, Introductory lectures on contact geometry, Topology and Geometry of
Manifolds (2003), 81–107.
[Gei08]
H. Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathe-
matics, Cambridge University Press, 2008.
[GHR03]
M. Guysinsky, B. Hasselblatt, and V. Rayskin, Diﬀerentiability of the Hartman–Grobman
linearization, Discrete and Continuous Dynamical Systems 9(2003), no. 4, 979–984.
[Gir00]
E. Giroux, Structures de contact en dimension trois et bifurcations des feuilletages de
surfaces, Inventiones mathematicae 141 (2000), no. 3, 615–689.
[Gir91]
E. Giroux, Convexité en topologie de contact., Commentarii mathematici Helvetici
66
(1991), no. 4, 637–677.
[Gro96]
M. Gromov, Carnot-Carathéodory spaces seen from within (A. Bellaïche and J.-J.
Risler, eds.), Progress in Mathematics, vol. 144, Birkhäuser Basel, Basel, 1996.
[Har60]
P. Hartman, On local homeomorphisms of Euclidean space, Boletin de la Sociedad
Matematica Mexicana (2) 5(1960), 220–241.
[Jea14]
F. Jean, Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion
Planning, SpringerBriefs in Mathematics, Springer International Publishing, 2014.
[Lee13]
P. W. Y. Lee, On surfaces in three dimensional contact manifolds, Canadian Journal of
Mathematics 65 (2013Jun), no. 3, 621–633.
24 DAVIDE BARILARI, UGO BOSCAIN, AND DANIELE CANNARSA
[Mon02]
R. Montgomery, A tour of sub-Riemannian geometries, their geodesics and applications,
Mathematical surveys and monographs, American Mathematical Society, 2002.
[Pau04]
S. D. Pauls, Minimal surfaces in the Heisenberg group, Geometriae Dedicata
104
(2004),
no. 1, 201–231.
[Per12]
L. Perko, Diﬀerential equations and dynamical systems, Texts in Applied Mathematics,
Springer New York, 2012.
[Rif14]
L. Riﬀord, Sub-Riemannian Geometry and Optimal Transport, SpringerBriefs in Mathe-
matics, Springer International Publishing, 2014.
[Vel20]
J. Veloso, Limit of Gaussian and normal curvatures of surfaces in Riemannian approx-
imation scheme for sub-Riemannian three dimensional manifolds and Gauss-Bonnet
theorem, 2020.
[WW20]
Y. Wang and S. Wei, Gauss-Bonnet theorems in the aﬃne group and the group of rigid
motions of the Minkowski plane, Science China Mathematics (2020Apr).
[ZZ95]
I. Zelenko and M. Zhitomirskii, Rigid paths of generic 2-distributions on 3-manifolds,
Duke Mathematical Journal 79 (1995).
Davide Barilari, Dipartimento di Matematica "Tullio Levi-Civita", Università di
Ugo Boscain, CNRS, Laboratoire Jacques-Louis Lions, team Inria CAGE, Université
de Paris, Sorbonne Université boîte courrier 187, 75252 Paris Cedex 05 Paris France
Daniele Cannarsa, Université de Paris, Sorbonne Université, CNRS, Inria, Institut
de Mathématiques de Jussieu-Paris Rive Gauche, F-75013 Paris, France
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