Symplectic Applicability of Lagrangian Surfaces
We develop an approach to affine symplectic invariant geometry of Lagrangian surfaces by the method of moving frames. The fundamental invariants of elliptic Lagrangian immersions in affine symplectic four-space are derived together with their integrability equations. The invariant setup is applied t...
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irk-123456789-1491082019-02-20T01:26:56Z Symplectic Applicability of Lagrangian Surfaces Musso, E. Nicolodi, L. We develop an approach to affine symplectic invariant geometry of Lagrangian surfaces by the method of moving frames. The fundamental invariants of elliptic Lagrangian immersions in affine symplectic four-space are derived together with their integrability equations. The invariant setup is applied to discuss the question of symplectic applicability for elliptic Lagrangian immersions. Explicit examples are considered. 2009 Article Symplectic Applicability of Lagrangian Surfaces / E. Musso, L. Nicolodi // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 24 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 53A07; 53B99; 53D12; 53A15 http://dspace.nbuv.gov.ua/handle/123456789/149108 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We develop an approach to affine symplectic invariant geometry of Lagrangian surfaces by the method of moving frames. The fundamental invariants of elliptic Lagrangian immersions in affine symplectic four-space are derived together with their integrability equations. The invariant setup is applied to discuss the question of symplectic applicability for elliptic Lagrangian immersions. Explicit examples are considered. |
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Musso, E. Nicolodi, L. Symplectic Applicability of Lagrangian Surfaces Symmetry, Integrability and Geometry: Methods and Applications |
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Musso, E. Nicolodi, L. |
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Musso, E. |
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Symplectic Applicability of Lagrangian Surfaces |
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Symplectic Applicability of Lagrangian Surfaces |
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Symplectic Applicability of Lagrangian Surfaces |
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Symplectic Applicability of Lagrangian Surfaces |
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Symplectic Applicability of Lagrangian Surfaces |
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symplectic applicability of lagrangian surfaces |
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Інститут математики НАН України |
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2009 |
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Symplectic Applicability of Lagrangian Surfaces / E. Musso, L. Nicolodi // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 24 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 067, 18 pages
Symplectic Applicability of Lagrangian Surfaces?
Emilio MUSSO † and Lorenzo NICOLODI ‡
† Dipartimento di Matematica, Politecnico di Torino,
Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
E-mail: emilio.musso@polito.it
‡ Dipartimento di Matematica, Università degli Studi di Parma,
Viale G.P. Usberti 53/A, I-43100 Parma, Italy
E-mail: lorenzo.nicolodi@unipr.it
Received February 25, 2009, in final form June 15, 2009; Published online June 30, 2009
doi:10.3842/SIGMA.2009.067
Abstract. We develop an approach to affine symplectic invariant geometry of Lagrangian
surfaces by the method of moving frames. The fundamental invariants of elliptic Lagrangian
immersions in affine symplectic four-space are derived together with their integrability equa-
tions. The invariant setup is applied to discuss the question of symplectic applicability for
elliptic Lagrangian immersions. Explicit examples are considered.
Key words: Lagrangian surfaces; affine symplectic geometry; moving frames; differential
invariants; applicability
2000 Mathematics Subject Classification: 53A07; 53B99; 53D12; 53A15
1 Introduction
While the subject of metric invariants of Lagrangian submanifolds has received much attention
in the literature, the subject of affine symplectic invariants of Lagrangian submanifolds has
been less studied (cf. [21] for recent work in this direction). Early contributions to symplectic
invariant geometry of submanifolds go back to Chern and Wang [10], who studied submanifolds
in projective (2n + 1)-space under the linear symplectic group. Some recent works related to
the study of affine symplectic invariants, mainly for the case of curves and hypersurfaces in
Euclidean space, include [11, 19, 1]. The aim of this paper is to develop an approach to affine
symplectic invariant geometry of Lagrangian surfaces in standard affine symplectic four-space,
using the method of moving frames. The invariant setup is then applied to discuss the problem
of symplectic applicability for Lagrangian immersions.
Let M be a 2-dimensional manifold and let f : M → R4 be a Lagrangian immersion into
standard affine symplectic four-space (R4,Ω). The basic observation is that the natural Gauss
map of f , whose value at a point is determined by the 1-jet of f at that point, takes values
in the oriented Lagrangian Grassmannian of (R4,Ω), the manifold Λ+
2 of oriented Lagrangian
2-subspaces in R4. The oriented Lagrangian Grassmannian admits an alternative description as
the conformal compactification of Minkowski 3-space R2,1, the projectivization of the positive
nullcone of R3,2. The local isomorphisms between the group Sp(4, R) of linear symplectomor-
phisms and the conformal Lorentz group O(3, 2) establishes a close relationship between the
symplectic geometry of a Lagrangian surface in R4 and the conformal Lorentzian geometry of
its Gauss map. Accordingly, a Lagrangian immersion is of general type (resp., special type) if its
Gauss map is an immersion (resp., has constant rank one). Moreover, a Lagrangian immersion
?This paper is a contribution to the Special Issue “Élie Cartan and Differential Geometry”. The full collection
is available at http://www.emis.de/journals/SIGMA/Cartan.html
mailto:emilio.musso@polito.it
mailto:lorenzo.nicolodi@unipr.it
http://dx.doi.org/10.3842/SIGMA.2009.067
http://www.emis.de/journals/SIGMA/Cartan.html
2 E Musso and L. Nicolodi
of general type is elliptic (resp., hyperbolic, parabolic) if its Gauss map is a spacelike (resp.,
timelike, lightlike) immersion into Λ+
2 .
In this work we study elliptic Lagrangian immersions in (R4,Ω) under the group R4oSp(4, R)
of affine symplectic transformations. In Section 2, we collect the background material on affine
symplectic frames and their structure equations, and briefly describe the conformal geometry of
the Lagrangian Grassmannian Λ+
2 .
In Section 3, we develop the moving frame method for elliptic Lagrangian immersions in
affine symplectic geometry. If f : M → R4 is an elliptic Lagrangian immersion, we consider
on M the complex structure defined by the conformal structure induced by the Gauss map
of f and a choice of orientation on M . After the third frame reduction, we make a succes-
sive reduction with respect to a fixed local complex coordinate in the Riemann surface M ,
which yields a unique adapted frame field along f . From this we derive the three local affine
symplectic invariants for f , namely the complex-valued smooth functions t, h, p : M → C,
and establish their integrability equations, which give the existence and uniqueness theorem
for elliptic Lagrangian immersion in affine symplectic geometry. At this stage, a natural
question to ask is to what extent the invariants are actually needed to determine the ellip-
tic Lagrangian surface up to symmetry (cf. Bonnet’s problem in Euclidean geometry). We
will look at this problem from the classical point of view of applicable surfaces, a concept
generalized by É. Cartan to G-deformations of submanifolds of any homogeneous space G/G0
(cf. [12, 8, 9, 14, 17]).
In Section 4, we apply the invariant setup to discuss rigidity and applicability of elliptic
Lagrangian immersions. Already after the second reduction, we associate with any elliptic Lag-
rangian immersion f a cubic differential form F , the Fubini cubic form. Proposition 2 shows
that the position of a generic elliptic Lagrangian immersion is completely determined by F , up to
affine symplectic transformation. The elliptic Lagrangian immersions which are not determined
by the Fubini cubic form alone are called applicable. Theorem 2 relates applicability to the
complex structure of M , using the normalized Hopf differential H, a quadratic differential form
naturally associated to any elliptic Lagrangian immersion with respect to third order frames.
More precisely, it proves that an elliptic Lagrangian immersion is applicable if and only if there
exists on M a non-zero holomorphic quadratic differential Q such that H and Q are linearly
dependent over the reals at each point of M . An interesting consequence of this result is that
applicable elliptic Lagrangian immersions admit an associated family of noncongruent elliptic
Lagrangian immersions parametrized by R, which in turn suggests their integrable character.
This is not surprising when one considers that the Gauss map of an applicable elliptic La-
grangian immersion is an isothermic spacelike immersion of Λ+
2 . Indeed, it is well known that
a spacelike isothermic immersion in 3-dimensional conformal space comes in an associated fa-
mily of (second order) deformations which coincide with the classical T -transforms of Bianchi
and Calapso (cf. [22, 16, 24]). The equations governing elliptic Lagrangian immersions are
also derived. We then consider the class of elliptic Lagrangian surfaces with totally umbilic
Gauss map. They are defined by the vanishing of the normalized Hopf differential H. Propo-
sition 3 shows that, up to a biholomorphism of M , two noncongruent, totally umbilic, elliptic
Lagrangian immersions are applicable. Finally, Proposition 4 proves that a complex curve in C2
without flex points is a totally umbilic elliptic Lagrangian immersion and that, conversely, any
totally umbilic elliptic Lagrangian immersion is congruent to a complex curve in C2 without flex
points.
Section 5 is devoted to the analysis of explicit examples of applicable non-totally umbilic
elliptic Lagrangian immersions.
Symplectic Applicability of Lagrangian Surfaces 3
2 Preliminaries
2.1 Affine symplectic frames and structure equations
Let R4
Ω denote R4 with the standard symplectic form
Ω(X, Y ) = tXJY, X, Y ∈ R4,
where
J = (Jij) =
(
0 I2
−I2 0
)
,
I2 being the 2× 2 identity matrix. The linear symplectic group is
Sp(4, R) = {X ∈ GL(4, R) : tXJX = J}.
An element X ∈ Sp(4, R) is of the form
X =
(
A B
C D
)
,
where A, B, C and D are 2× 2 matrices such that
tAC − tCA = 0, tBD − tDB = 0, tAD − tCB = I2.
The Lie algebra of Sp(4, R) is
sp(4, R) =
{
x(a, b, c) =
(
a b
c −ta
)
: a ∈ gl(2, R), c, b ∈ S(2, R)
}
,
where S(2, R) is the vector space of 2 × 2 symmetric matrices endowed with the inner product
of signature (2, 1) defined by
(b, b) = −det b, b ∈ S(2, R).
The affine symplectic group R4 o Sp(4, R) is represented in GL(5, R) by
S(4, R) =
{
S(P,X) =
(
1 0
P X
)
: P ∈ R4, X ∈ Sp(4, R)
}
.
The Lie algebra of S(4, R) is given by
s(4, R) =
{
S(p,x) =
(
0 0
p x
)
: p ∈ R4,x ∈ s(4, R)
}
.
If S(P,X) ∈ S(4, R), we let Xj , j = 1, . . . , 4, denote the column vectors of X.
By an affine symplectic frame is meant a point P ∈ R4 and four vectors X1, . . . , X4 ∈ R4
such that X = (X1, X2, X3, X4) ∈ Sp(4, R). Upon choice of a reference frame, the manifold of
all affine symplectic frames of R4
Ω may be identified with the group S(4, R).
Consider the tautological projection maps
P : S(4, R) 3 S(P,X) 7→ P ∈ R4, (1)
Xj : S(4, R) 3 S(P,X) 7→ Xj ∈ R4,
4 E Musso and L. Nicolodi
for j = 1, . . . , 4. Then
dP = τ jXj , dXj = θi
jXi,
where the τ j , θi
j are the left-invariant Maurer–Cartan forms of S(4, R). Exterior differentiation
of
Ω(Xi,Xj) = Jij
yields that θ is sp(4, R)-valued. We then write
θ =
(
α β
γ −tα
)
,
where
α =
(
α1
1 α1
2
α2
1 α2
2
)
, β =
(
β1
1 β1
2
β1
2 β2
2
)
, γ =
(
γ1
1 γ1
2
γ1
2 γ2
2
)
.
Note that(
τ1, τ2, τ3, τ4, α1
1, α
2
2, α
1
2, α
2
1, γ
1
1 , γ2
2 , γ1
2 , β1
1 , β2
2 , β1
2
)
is a basis for the vector space of the left-invariant 1-forms of S(4, R). Moreover, for each
S(0,Y) ∈ S(4, R), we have
R∗
S(0,Y)
τ1
τ2
τ3
τ4
= Y−1
τ1
τ2
τ3
τ4
, R∗
S(0,Y)(θ) = Y−1θY + Y−1dY.
Exterior differentiation of
dP = τ jXj , dXi = θj
i Xj
yields the structure equations of the affine symplectic group:
dτ1 = −α1
1 ∧ τ1 − α1
2 ∧ τ2 − β1
1 ∧ τ3 − β1
2 ∧ τ4,
dτ2 = −α2
1 ∧ τ1 − α2
2 ∧ τ2 − β1
2 ∧ τ3 − β2
2 ∧ τ4,
dτ3 = −γ1
1 ∧ τ1 − γ1
2 ∧ τ2 + α1
1 ∧ τ3 + α2
1 ∧ τ4,
dτ4 = −γ1
2 ∧ τ1 − γ2
2 ∧ τ2 + α1
2 ∧ τ3 + α2
2 ∧ τ4,
dα1
1 = −α1
2 ∧ α2
1 − β1
1 ∧ γ1
1 − β1
2 ∧ γ1
2 ,
dα2
2 = −α2
1 ∧ α1
2 − β1
2 ∧ γ1
2 − β2
2 ∧ γ2
2 ,
dα2
1 = (α1
1 − α2
2) ∧ α2
1 − β1
2 ∧ γ1
1 − β2
2 ∧ γ1
2 ,
dα1
2 = (α2
2 − α1
1) ∧ α1
2 − β1
1 ∧ γ1
2 − β1
2 ∧ γ2
2 ,
(2)
dγ1
1 = 2α1
1 ∧ γ1
1 + 2α2
1 ∧ γ1
2 ,
dγ2
2 = 2α2
2 ∧ γ2
2 + 2α1
2 ∧ γ1
2 ,
dγ1
2 = (α1
1 + α2
2) ∧ γ1
2 + α2
1 ∧ γ2
2 + α1
2 ∧ γ1
1 ,
and
dβ1
1 = −2α1
1 ∧ β1
1 − 2α1
2 ∧ β1
2 ,
dβ2
2 = −2α2
2 ∧ β2
2 − 2α2
1 ∧ β1
2 ,
dβ1
2 = −(α1
1 + α2
2) ∧ β1
2 − α2
1 ∧ β1
1 − α1
2 ∧ β2
2 .
(3)
Symplectic Applicability of Lagrangian Surfaces 5
2.2 Oriented Lagrangian planes
Let Λ+
2 denote the set of oriented Lagrangian subspaces of R4, i.e. the set of oriented 2-
dimensional linear subspaces V ⊂ R4 such that Ω|V = 0. The space Λ+
2 is a smooth manifold
diffeomorphic to S2×S1. The standard action of Sp(4, R) on R4 induces an action on Λ+
2 which
is transitive. The projection map
πΛ : X ∈ Sp(4, R) 7→ [X1 ∧X2] ∈ Λ+
2
makes Sp(4, R) into a principal fiber bundle with structure group
Sp(4, R)1 =
{
X(A, b) =
(
A Ab
0 tA−1
)
: det A > 0, b ∈ S(2, R)
}
.
From this, it follows that the forms (γ1
1 , γ2
2 , γ2
1) span the semibasic forms of the projection πΛ.
Moreover, the symmetric quadratic form g = −γ1
1γ2
2 +(γ2
1)2 and the exterior 3-form γ1
1 ∧γ2
1 ∧γ2
2
are well defined on Λ+
2 , up to a positive multiple. They determine a conformal structure of
signature (2, 1) and an orientation, respectively.
Remark 1. The group Sp(4, R) is a covering group of the identity component of the conformal
Lorentz group O(3, 2). This fact implies that Λ+
2 can be identified with the quotient of the
nullcone of R3,2 by the action given by positive scalar multiplications. Therefore, Λ+
2 can be
seen as the conformal compactification of oriented, time-oriented affine Minkowski space R2,1 ∼=
S(2, R) (cf. [2, 15]). Any element of Λ+
2 can be represented by a 4× 2 matrix
X =
(
A1
A2
)
of rank 2 such that tA1A2 = tA2A1. Let U0 ⊂ Λ+
2 be the set of all Lagrangian subspaces [X1∧X2]
associated to ordered pairs of vectors
X1 = t
(
x1
1, . . . , x
4
1
)
, X2 = t
(
x1
2, . . . , x
4
2
)
such that
det
(
x1
1 x1
2
x2
1 x2
2
)
> 0.
The set U0 is a dense open subset of Λ+
2 . The map S : U0 → S(2, R) ∼= R2,1 given by
S([X1 ∧X2]) =
(
x3
1 x3
2
x4
1 x4
2
) (
x1
1 x1
2
x2
1 x2
2
)−1
is a conformal diffeomorphism, and (U0, S) is a local chart of Λ+
2 . Note that
U0 =
{
V ∈ Λ+
2 : V ∩W0 = {0}
}
,
the set of Lagrangian subspaces which are transversal to W0 = span {e3, e4}.1 Let
U1 =
{
V ∈ Λ+
2 : dim(V ∩W0) = 1
}
.
Then U1 ∪ {W0} is a closed subset of Λ+
2 , which can be interpreted as the ideal boundary of
R2,1 ∼= U0. Note that U0 and U1 are orbits of the closed subgroup of Sp(4, R) that preserves the
subspace W0.
1Here e1, e2, e3, e4 denote the standard basis of R4.
6 E Musso and L. Nicolodi
2.3 Lagrangian immersions
Let f : M → R4 be a smooth immersion of a connected 2-manifold, oriented by a volume
element A. For any q ∈ M , let Tf (q) be the 2-plane df(TqM) translated to the origin and
equipped with the orientation induced by −df(A|q).2
Definition 1. The immersion f : M → R4 is called Lagrangian if Tf (q) ∈ Λ+
2 , for each q ∈ M .
The resulting map
Tf : q ∈ M 7→ Tf (q) ∈ Λ+
2
is the (symplectic) Gauss map of the Lagrangian immersion f .
Definition 2. A Lagrangian immersion is said of general type (respectively, special type) if
its Gauss map is an immersion (respectively, has constant rank one). Moreover, a Lagrangian
immersion of general type is said elliptic (respectively, hyperbolic, parabolic) if its Gauss map is
a spacelike (respectively, timelike, lightlike) immersion into Λ+
2 .
3 Moving frames on elliptic Lagrangian immersions
3.1 Frame reductions and differential invariants
In this section, f : M → R4 will denote an elliptic Lagrangian immersion with Gauss map Tf .
Since we are working with immersions which are not necessarily one-to-one, it is not restrictive
to assume that M is simply connected.
Definition 3. A symplectic frame field along f is a smooth map
S = S(f,X) : U → S(4, R)
from an open connected subset U ⊂ M such that the projection (1) composed with S is f , i.e.
P ◦ S(f,X) = S(f,X) ·O = f .
Following the usual practice in the method of moving frames we will write φ instead of S∗(φ)
to denote forms on U which are pulled back from S(4, R) by the moving frame S. A symplectic
frame field along f gives a local representation of the derivative map,
df =
4∑
j=1
τ jXj .
Let S : U → S(4, R) be a symplectic frame field along f . Any other symplectic frame field
along f on U is given by
S̃ = SY,
where Y : U → Sp(4, R) is a smooth map3. If Θ and Θ̃ are the pull-backs of the Maurer–Cartan
form of S(4, R) by S and S̃, respectively, then
Θ̃ = Y−1ΘY + Y−1dY. (4)
2One reason for choosing the opposite orientation is that totally umbilic elliptic Lagrangian immersions are
then described in terms of holomorphic functions instead of anti-holomorphic ones (cf. Sections 4.4 and 4.5).
3Here we are identifying an element S(0,Y) ∈ S(4, R) with Y ∈ Sp(4, R), and hence an element S(0,x) ∈ s(4, R)
with x ∈ sp(4, R).
Symplectic Applicability of Lagrangian Surfaces 7
3.1.1 First order frame fields
A symplectic frame field S = S(f,X) : U → S(4, R) along f is of first order if
Tf (q) = [X1|q ∧X2|q], for each q ∈ U.
It is clear that first order frame fields exist locally. With respect to a first order frame, the
1-forms τ3 and τ4 vanish identically on U and
df = τ1X1 + τ2X2,
where τ1 ∧ τ2 < 0. If S is a first order frame field along f on U , then any other is given by
S̃ = SY(A, b), where Y(A, b) : U → Sp(4, R)1 is a smooth map into the subgroup Sp(4, R)1
introduced in Section 2.2.
Calculated with respect to a first order frame field, the quadratic form g = −γ1
1γ2
2 + (γ2
1)2 is
positive definite. Under a change S̃ = SY(A, b) of first order frames, it transforms by
−γ̃1
1 γ̃2
2 + (γ̃2
1)2 = det(A)2
(
− γ1
1γ2
2 + (γ2
1)2
)
,
and hence defines a conformal structure on M . On M , we will consider the unique complex
structure compatible with the given orientation and the conformal structure defined by g. The
1-forms
ω1 := 1
2
(
γ1
1 − γ2
2
)
, ω2 := γ2
1
are everywhere linearly independent and
ω = ω1 + iω2
is complex-valued of bidegree (1, 0). Moreover, there exist smooth functions `1, `2 : U → R such
that
1
2
(
γ1
1 + γ2
2
)
= `1ω
1 + `2ω
2, `2
1 + `2
2 < 1.
3.1.2 Second order frame fields
A first order frame field along f is of second order if it satisfies
1
2
(
γ1
1 + γ2
2
)
= 0.
Lemma 1. About any point of M there exists a second order frame field along f .
Proof. Let S : U → S(4, R) be any first order frame field along f . Let φ : U → (−π/2, π/2)
be the smooth function defined by
sin(φ) = − `2√
1− `2
1
and let A : U → GL+(2, R) be given by
A =
√
1−`1
2 cos(φ)
√
1−`1
2 sin(φ)
0
√
1+`1
2
.
Then Y(A, 0) : U → Sp(4, R)1 is a smooth map. Let S̃ = SY(A, 0). The transformation rule (4)
gives γ̃ = tAγA, from which we compute γ̃1
1 + γ̃2
2 = 0. This means that S̃ is a second order frame
field along f . �
8 E Musso and L. Nicolodi
If S and S̃ are second order frame fields on U , then S̃ = SY(A, b), where A is a 2× 2 matrix
such that tAA = r2I2 and det(A) > 0. The group of all such A identifies with the multiplicative
group C∗ of nonzero complex numbers by
C∗ 3 reis 7→ r
(
cos(s) − sin(s)
sin(s) cos(s)
)
.
Let S be a second order frame field along f . We know that the complex-valued 1-form ω =
ω1 + iω2 is of bidegree (1, 0) and never zero.
Differentiating the equations τ3 = τ4 = 0 and using the structure equations, we find that{
τ1 ∧ ω1 + τ2 ∧ ω2 = 0,
τ1 ∧ ω2 − τ2 ∧ ω1 = 0,
which, by Cartan’s Lemma, implies that
τ := τ1 − iτ2 = tω,
for some smooth function t = t1 + it2 : U → C. The 1-form τ is of bidegree (1, 0) and never
zero.
Under a change of second order frame fields S̃ = SY(reis, b), the forms ω and τ transform by
ω̃ = r2e−2isω, τ̃ = r−1eisτ. (5)
This implies that
τ̃2ω̃ = τ2ω.
Thus τ2ω is independent of the choice of second order frame field and is never zero.
Definition 4. The cubic differential form F := τ2ω, globally defined on M , is called the Fubini
cubic form of the Lagrangian immersion f .
Next, consider the complex-valued 1-form
η := η1 − iη2 = 1
2
(
α1
1 − α2
2
)
− i
2
(
α2
1 + α1
2
)
.
Differentiation of γ1
1 + γ2
2 = 0, combined with the structure equations, gives
η1 ∧ ω1 + η2 ∧ ω2 = 0,
which, by Cartan’s Lemma, implies that
η = hω + `ω̄
for some smooth functions h = h1 + ih2 : U → C and ` : U → R.
3.1.3 Third order frame fields
Under a change S̃ = SY(reis, b) of second order frame fields, the 1-form η transforms by
η̃ = e2is
(
η − r2
2
tr (b) ω̄
)
.
This implies that locally there exist second order frame fields satisfying ` = 0. Such frame fields
are said of third order.
If S̃ and S are third order frame fields on U , then S̃ = SY(reis, b) where tr (b) = 0. With
respect to a third order frame field, the 1-form η is of bidegree (1, 0) and transforms by
η̃ = e2isη. (6)
Symplectic Applicability of Lagrangian Surfaces 9
Definition 5. According to (5) and (6), the differential forms
B = η̄η, H = |t|2/3ηω,
are globally defined on M , independent of the choice of third order frames. We call B the
Thomsen quadratic form and H the normalized Hopf differential of f .
Remark 2. The invariant form B amounts to the metric induced on M by the conformal Gauss
map [5] (or central sphere congruence [4]) of the spacelike conformal immersion Tf : M → Λ+
2 .
The invariant form H is instead a normalization of the Hopf differential of Tf .
3.2 Adapted frame fields
Being simply connected, M is either the Riemann sphere S2, the complex plane C, or the unit
disk ∆. By a complex parameter on M is meant a complex coordinate chart (U, z) in M defined
on a maximal contractible open subset U of M : if M = C, or ∆, then U = M ; if M = S2, then
U = S2 \ {q}.
Definition 6. Let (U, z) be a complex parameter on M . A third order frame field S : U →
S(4, R) along f is adapted to (U, z) if
ω = ω1 + iω2 = dz, α1
1 + α2
2 = α2
1 − α1
2 = 0. (7)
Proposition 1. Let (U, z) be a complex parameter on M . There exist third order frame fields
along f adapted to (U, z). If S and S′ are two such frame fields, then S′ = ±S.
Proof. Let q0 ∈ U and let S′ : V → S(4, R) be any third order frame field defined on a simply
connected open neighborhood V ⊂ U of q0. On V there exist real-valued functions r, ϑ : V → R
such that ω′ = r2e−2iϑdz. Define S′′ = S′Y(r−2e2iϑ, 0). Then, S′′ is a third order frame field
such that ω′′ = dz. Two such frame fields are related by
S̃′′ = S′′Y(±I2, b),
where b : V → S(2, R) is a smooth map such that tr (b) = 0. From the structure equations we
find that{ (
α1
1 + α2
2
)
∧ dx +
(
α2
1 − α1
2
)
∧ dy = 0,(
α1
1 + α2
2
)
∧ dy −
(
α2
1 − α1
2
)
∧ dx = 0,
which implies
(α1
1 + α2
2)− i(α2
1 − α1
2) = w dz,
for some smooth function w : V → C. If S̃′′ = S′′Y(±I2, b) then(
α̃1
1 + α̃2
2
)
− i
(
α̃2
1 − α̃1
2
)
=
(
α1
1 + α2
2
)
− i
(
α2
1 − α1
2
)
+ 2
(
b1
1 − ib2
1
)
dz.
If we choose b such that 2(b1
1 − ib2
1) = −w, then S̃′′ satisfies
ω′′ = dz,
(
α̃1
1 + α̃2
2
)
− i
(
α̃2
1 − α̃1
2
)
= 0.
This shows that adapted frame fields do exist locally near any point of U . Moreover, two such
frames S and S′ on V are related by S′ = ±S. The existence of an adapted frame on U follows
from the fact that U is simply connected. �
10 E Musso and L. Nicolodi
3.2.1 Invariant functions and integrability equations
Let S : U → S(4, R) be a frame field adapted to (U, z). Then
τ = t dz, η = h dz, (8)
for smooth functions t = t1 + it2 : U → C and h = h1 + ih2 : U → C. Define υ and ρ by
υ := 1
2
(
β1
1 + β2
2
)
, ρ := 1
2
(
β1
1 − β2
2
)
− iβ2
1 .
The structure equations (2) imply
υ = hz̄dz + h̄zdz̄, (9)
and
ρ = pdz +
(
h2
1 + h2
2
)
dz̄ = pdz + |h|2dz̄, (10)
for some smooth function p = p1 + ip2 : U → C.
Definition 7. The functions t, h and p defined by (8) and (10) are called the invariant functions
of the frame field along f adapted to (U, z). These functions are subject to the equations
tz̄ = t̄ h,
pz̄ = 2hh̄z − i|h|2z,
p̄h− ph̄ = hz̄z̄ − h̄zz.
(11)
Remark 3. From (11) it follows that
∂̄(F) = 2N , (12)
where N := τ̄ τηω is an invariant form. The form B, H and N are completely determined by
the Fubini cubic form.
Theorem 1 (Existence). Let (U, z) be a complex parameter on M . Let t : U → C, h :
U → C and p : U → C be smooth functions satisfying the equations (11). Then there exist an
elliptic Lagrangian immersion f : U → R4, unique up to affine symplectic transformation, and
a unique frame field S : U → S(4, R) along f adapted to (U, z) whose invariant functions are t, h
and p, i.e.
τ = tdz, η = hdz, ρ(1,0) = pdz.
Proof. Let(
τ1
τ2
)
=
(
t1dx− t2dy
−t2dx− t1dy
)
, γ =
(
dx dy
dy −dx
)
,
α =
(
h1dx− h2dy −h2dx− h1dy
−h2dx− h1dy −h1dx + h2dy
)
, β =
(
β1
1 β2
1
β2
1 β2
2
)
,
where
1
2
(
β1
1 + β2
2
)
= hz̄dz + h̄zdz̄
and
1
2
(
β1
1 − β2
2
)
− iβ2
1 = pdz + |h|2dz̄.
Symplectic Applicability of Lagrangian Surfaces 11
Then
Θ =
0 0 0(
τ1
τ2
)
α β
0 γ −tα
is a smooth 1-form on U with values in s(4, R). By (11),
dΘ = −Θ ∧Θ.
Therefore, by the Cartan–Darboux theorem, there exists a smooth map S : U → S(4, R) such
that S−1dS = Θ, unique up to left multiplication by an element of S(4, R). The elliptic Lagran-
gian immersion f : U → R4 defined by f = S ·O has the required properties. �
Remark 4 (Uniqueness). If f , f̃ : U → R4 are two elliptic Lagrangian immersions inducing
the same invariant functions, then, by the Cartan–Darboux uniqueness theorem, there exists
a symplectic motion S : R4 → R4 such that f̃ = Sf .
4 Symplectic applicability
In this section we investigate the extent to which the invariants are really needed to determine an
elliptic Lagrangian immersion up to symmetry. We take the point of view of surface applicability
as in classical projective differential geometry and Lie sphere geometry [12, 8, 9, 4, 23, 18].
4.1 Generic Lagrangian immersions
Suppose that H, the normalized Hopf differential, is never zero. Then the Hermitian form
P =
(
(h̄zh̄
−1)z̄ − (hz̄h
−1)z
)
dzdz̄
is well defined, independent of the choice of the complex parameter (U, z) and of the adapted
frame (the reason for considering this form will be clear in the proof of the next proposition).
Definition 8. An elliptic Lagrangian immersion f : M → R4 is called generic if H(q) 6= 0 and
P(q) 6= 0, for each q ∈ M .
Proposition 2. A generic elliptic Lagrangian immersion is uniquely determined, up to affine
symplectic transformation, by the Fubini cubic form.
Proof. Fix a complex parameter (U, z) and choose an adapted frame field S : U → S(4, R). If
h 6= 0, then (11) implies the existence of a unique real-valued function s : U → R such that
p = hs +D2(h),
where
D2(h) =
1
2h̄
(
h̄zz − hz̄z̄
)
.
Differentiating this identity and using
pz̄ = 2hh̄z − i|h|2z,
12 E Musso and L. Nicolodi
we get
ds = −s
(
1
h
hz̄dz̄ +
1
h̄
h̄zdz
)
−
(
D3(h)dz̄ +D3(h)dz
)
, (13)
where
D3(h) =
1
h
[
(D2(h))z̄ − 2hh̄z + i|h|2z
]
.
Differentiation of (13) gives
P2(h)s +D4(h) = 0,
where
P2(h) = (h̄zh̄
−1)z̄ − (hz̄h
−1)z
and
D4(h) = D3(h)z̄ −D3(h)z − h̄−1h̄zD3(h) + h−1hz̄D3(h).
If P2(h) 6= 0, then
s = −D4(h)
P2(h)
.
Therefore, for a generic elliptic Lagrangian immersion (i.e. h 6= 0 and P2(h) 6= 0), the functions h
and p are determined by the Fubini cubic form. Thus, a generic f is uniquely determined, up
to symplectic congruence, by its Fubini cubic form. �
4.2 Applicable elliptic Lagrangian immersions
In this section we investigate the special class of surfaces which are not determined by the Fubini
cubic form alone.
Definition 9. Two noncongruent elliptic Lagrangian immersions f , f̃ : M → R4 are applicable if
they induce the same Fubini cubic forms, F = F̃ . An elliptic Lagrangian immersion f : M → R4
is applicable if there exists an elliptic Lagrangian immersion f̃ : M → R4 such that f and f̃
are applicable. Moreover, an elliptic Lagrangian immersion f is rigid if any elliptic Lagrangian
immersion inducing the same Fubini cubic form as that of f is congruent to f .
Remark 5. According to Proposition 2, any generic elliptic Lagrangian immersion is rigid.
Theorem 2. An elliptic Lagrangian immersion f : M → R4 is applicable if and only if there
exists a non-zero holomorphic quadratic differential Q on M such that H ∧R Q = 0.4
Proof. First, we consider the case H ≡ 0. Then, F is a never vanishing holomorphic cubic
differential. Since M is simply connected, M is either the complex plane or the unit disk. In
both cases, there exist a global complex coordinate z : M → C and a global adapted frame
field S : M → S(4, R) with invariant functions t : M → C, h = 0 and p : M → C. Consider a
non-zero holomorphic differential Q = λ dz2 and set p̃ = p − λ. The functions t, h = 0 and p̃
satisfy equations (11). The corresponding f̃ has F̃ = F , but is not congruent to f .
We now examine the case H 6= 0. Let f , f̃ be two noncongruent elliptic Lagrangian im-
mersions inducing the same Fubini cubic forms. Let S, S̃ : U → S(4, R) be the corresponding
4H ∧R Q = 0 means that H and Q are linearly dependent over the reals.
Symplectic Applicability of Lagrangian Surfaces 13
adapted frame fields with respect to a fixed complex parameter (U, z). Since t = t̃, equations (11)
imply that h = h̃ and that p− p̃ is a non-zero holomorphic function satisfying
(p1 − p̃1)h2 − (p2 − p̃2)h1 = 0.
Then, QU = (p−p̃)dz2 is a non-zero holomorphic quadratic differential on U such thatH∧RQU =
0. Let (Û , ẑ) be another complex parameter and let Q̂Û be the corresponding holomorphic
quadratic differential. The equations H ∧R QU = 0 and H ∧R Q̂Û = 0 imply that Q = λUÛ Q̂
on U ∩ Û 6= ∅, where λUÛ is a real-valued, locally constant function. The functions {expλUÛ},
defined on each ordered pair (U, Û) when U ∩ Û 6= ∅, generate an R+-valued 1-cocycle on M ,
which is trivial because M is simply connected. Therefore, there exists a globally defined non-
zero holomorphic quadratic differential Q′ such that QU = Q′|U , for every (U, z).
Conversely, suppose there exists a non-zero holomorphic quadratic differential Q such that
H ∧R Q = 0. This implies that M is either C or the unit disk. Let z be a global complex
coordinate on M and let S : M → S(4, R) be the associated adapted frame field. Set Q = w dz2,
where w = w1 + iw2 is a holomorphic non-zero function. Now, the invariant functions t, h and p,
calculated with respect to S, satisfy (11). Since H∧R Q = 0, that is w1h2 − w2h1 = 0, also t, h
and p̃ = p − w satisfy (11). The corresponding elliptic Lagrangian immersions f , f̃ : M → R4
are applicable. �
Remark 6. If H 6= 0, then equation H ∧R Q = 0 determines Q up to a real constant multiple.
For each λ ∈ R, define the invariant functions tλ = t, hλ = h and pλ = p − λ. It is clear
that tλ, hλ and pλ satisfy the integrability equations (11). Therefore, up to affine symplectic
transformation, there exists a unique fλ with Fλ = F . As pλ are distinct for distinct values
of λ, the immersions fλ are noncongruent for distinct values of λ. This shows that applicable
elliptic Lagrangian immersions come in 1-parameter families.
Interestingly enough, away from the points where h vanishes (umbilics), the Gauss map of
an applicable elliptic Lagrangian immersion f is an isothermic spacelike immersion. Moreover,
the Gauss maps of the 1-parameter family of associates to f are the T -transforms of Bianchi
and Calapso of γf (cf. [7, 3, 16]).
Remark 7. The 1-parameter family of associates to an applicable elliptic Lagrangian immersion
are related to its second order affine symplectic deformations. We will not discuss this issue
here. We just recall that two elliptic Lagrangian immersions f, f̃ : M → R4 are second order
affine symplectic deformations of each other if there exists a smooth map D : M → S(4, R)
such that, for each q ∈ M , the Taylor expansions about q of D(q)f and f̃ agree through second
order terms. For the general notion of applicability and deformation in homogeneous spaces we
refer the reader to [14, 17].
4.3 The differential equations of applicable Lagrangian immersions
Let f : M → R4 be an applicable elliptic Lagrangian immersion and let Q be a non-zero
holomorphic quadratic differential on M such that H ∧R Q = 0. If we suppose that Q never
vanishes, we can choose a complex coordinate (U, z) such that Q = dz2. Let S : U → S(4, R) be
a frame field along f adapted to (U, z) with invariant functions t, h and p. Now, the equation
H ∧R Q = 0 implies that h is real valued.
Next, in analogy with [6, 24] we deduce the differential equations satisfied by the invariant
functions of an elliptic Lagrangian immersion. With respect to the adapted frame field S, we
have
α1
1 = −α2
2 = hdx, α2
1 = α1
2 = −hdy,
14 E Musso and L. Nicolodi
where h is a real-valued function. From (2), we compute
β1
1 = (p1 + hx + h2)dx− (p2 + hy)dy,
β2
2 = −(p1 − hx + h2)dx + (p2 − hy)dy,
β2
1 = −p2dx− (p1 − h2)dy.
Using (3), we find
hxy = −2hp2,
(p1)y = −(p2)x − 2(h2)y,
(p1)x = (p2)y + 2(h2)x.
Thus, calculated with respect to a complex parameter z such that Q = dz2, the integrability
equations of an applicable elliptic Lagrangian immersion are
tz̄ = ht̄,
hxy = −2hp2,
∆p2 = −4(h2)xy.
(14)
If (t, h, p2) is a solution of this system, the 1-form[
(p2)y + 2(h2)x
]
dx−
[
(p2)x + 2(h2)y
]
dy
is closed. If p1 : M → R is so that
dp1 =
[
(p2)y + 2(h2)x
]
dx−
[
(p2)x + 2(h2)y
]
dy,
then t, h and p = p1 + ip2 give a solution of (11). Thus, up to congruence, there exists a unique
elliptic Lagrangian immersion f : M → R4 with invariant functions t, h and p. Different choices
of the primitive p1 yield the family of associate immersions applicable to f .
4.4 Totally umbilic Lagrangian immersions
Definition 10. An elliptic Lagrangian immersion f is called totally umbilic if its Hopf differen-
tial H vanishes identically, or, equivalently, h vanishes identically. This terminology is based on
the observation that H ≡ 0 if and only if the Gauss map Tf of f is a totally umbilic spacelike
immersion (cf. [5]).
Remark 8. Note that f is totally umbilic if and only if its Fubini cubic form F is holomorphic
(cf. (12)). In this case, M is either the complex plane or the unit disk. Therefore, there exists
a global parameter z on M and a global adapted frame field S : M → S(4, R) along f .
Proposition 3. Let f, f̃ : M → R4 be two noncongruent, totally umbilic elliptic Lagrangian
immersions. Then there exists a biholomorphism Φ : M → M such that f and f̃ ◦ Φ are
applicable.
Proof. On M there are complex parameters z, w : M → W , W = C or ∆, such that F = (dz)3
and F̃ = (dw)3. Then Φ := w−1 ◦ z : M → M is a biholomorphic map such that f and f̃ ◦ Φ
have the same Fubini cubic differential. This yields the required result. �
Symplectic Applicability of Lagrangian Surfaces 15
4.5 Complex curves and Lagrangian immersions
On R4, consider the complex structure defined by the identification of R4 with C2 given by
I : t
(
x1, x2, x3, x4
)
∈ R4 7→ t
(
x1 − ix2, x3 + ix4
)
∈ C2.
Then, any complex line generates a Lagrangian plane, which implies that a complex immersion
f : M → R4 ' C2 is automatically Lagrangian.
Let ISL(2, C) ∼= C2 o SL(2, C) be the inhomogeneous group associated with the unimodular
complex group SL(2, C). The group ISL(2, C) can be realized as a closed subgroup of S(4, R) by(
1 0
v A
)
∈ ISL(2, C) 7→
(
1 0
I−1v [I−1AI]
)
∈ S(4, R),
where, if we write A ∈ SL(2, C) in the form A = (vi
j) + i(wi
j), for vi
j , wi
j ∈ R,
[I−1AI] =
v1
1 w1
1 v1
2 −w1
2
−w1
1 v1
1 −w1
2 −v1
2
v2
1 w2
1 v2
2 −w2
2
w2
1 −v2
1 w2
2 v2
2
.
The Lie algebra isl(2, C) identifies with the Lie subalgebra of s(4, R) consisting of all S(p,x) ∈
s(4, R) such that
x =
(
a b
c −ta
)
,
where
b, c ∈ S(2, R), tr (b) = tr (c) = 0, a1
1 − a2
2 = 0, a2
1 + a1
2 = 0. (15)
The precise relation between complex curves and elliptic Lagrangian immersions is summa-
rized in the following statement.
Proposition 4. If f : M → C2 is a complex curve without flex points5, then f is a totally
umbilic elliptic Lagrangian immersion. Conversely, let f : M → R4 be a totally umbilic elliptic
Lagrangian immersion. Then there exists an affine symplectic transformation D ∈ S(4, R) such
that D · f is a complex curve without flex points.
Proof. Let f be a complex curve without flex points. We already observed that f is Lagrangian.
Moreover, since f has no flex points, it is also elliptic. To prove that f is totally umbilic, we
have just to observe that the Maurer–Cartan form of an adapted frame along f with respect to
a given complex parameter takes values in isl(2, C). The defining conditions (7) for an adapted
frame, combined with (15), yield that the invariant h vanishes identically, and hence f is totally
umbilic.
Conversely, let f be a totally umbilic elliptic Lagrangian immersion. For a fixed complex
parameter z : M → C, let S : M → S(4, R) be an adapted frame field along f . Since f is totally
umbilic, using (8), (9) and (10), we find that
τ1 = t1dx− t2dy, τ2 = −t2dx− t1dy
and
γ =
(
dx dy
dy −dx
)
, α = 0, β =
(
p1dx− p2dy −p2dx− p1dy
−p2dx− p1dy −p1dx + p2dy
)
.
5A point q ∈ M is a flex point if fz|q ∧ fzz|q = 0, z complex coordinate on M .
16 E Musso and L. Nicolodi
This implies that S−1dS is an isl(2, C)-valued 1-form of bidegree (1, 0), which implies that there
exists an element D ∈ S(4, R) such that DS is holomorphic and takes values in ISL(2, C).
Therefore, Df is a complex curve without flex points. �
Remark 9. Let f : M → C2 be a complex curve. Possibly replacing f with Df , for some
D ∈ S(4, R), the homogeneous part X : M → SL(2, C) of the frame field adapted to a complex
coordinate z satisfies
X−1dX =
(
0 −ip
1 0
)
dz.
In particular, f can be obtained by integrating the first column vector of X, i.e.
f =
∫
X1dz.
Note that X : M → SL(2, C) is a contact complex curve in SL(2, C). This shows that totally
umbilic Lagrangian surfaces are strictly related to the geometry of flat fronts in hyperbolic
3-space [13, 20]. For a fixed choice of the complex parameter, let Xλ : M → SL(2, C) be so that
X−1
λ dXλ =
(
0 −i(p− λ)
1 0
)
dz.
Then, the 1-parameter family of complex curves fλ obtained by integrating the first column
vector of Xλ are not congruent to each other and are all applicable over f .
5 Examples
The simplest non-umbilic solutions of (14) are given by taking h and p real constants. Possibly
rescaling the complex parameter z by a real constant, we may assume h = 1. The only equation
to be solved is then
tz̄ = t̄.
The homogeneous part of the Maurer–Cartan form of the adapted frame S is
θ = Adx + Bdy,
where A, B ∈ sp(4, R) are given by
A =
1 0 p + 1 0
0 −1 0 −(p + 1)
1 0 −1 0
0 −1 0 1
, B =
0 −1 0 1− p
−1 0 1− p 0
0 1 0 1
1 0 1 0
.
Since A and B commute, the general solution of X−1dX is
X(x, y) = YExp(xA+ yB),
where Y ∈ Sp(4, R). Without loss of generality, we may assume Y = I4. Then the first two
column vectors of X are
X1 =
cosh(
√
2− py)
(
cosh(
√
2 + px) + sinh(
√
2+px)√
2+p
)
− (
√
p+2 cosh(
√
p+2x)+p sinh(
√
p+2x)) sinh(
√
2−py)√
4−p2
cosh(
√
2−py) sinh(
√
p+2x)√
p+2
(
√
p+2 cosh(
√
p+2x)+2 sinh(
√
p+2x)) sinh(
√
2−py)√
4−p2
Symplectic Applicability of Lagrangian Surfaces 17
and
X2 =
(−
√
p+2 cosh(
√
p+2x)+p sinh(
√
p+2x)) sinh(
√
2−py)√
4−p2
cosh(
√
2− py)
(
cosh(
√
2 + px)− sinh(
√
2+px)√
2+p
)
(
√
p+2 cosh(
√
p+2x)−2 sinh(
√
p+2x)) sinh(
√
2−py)√
4−p2
− cosh(
√
2−py) sinh(
√
p+2x)√
p+2
.
If t = t1 + it2 is a solution of tz̄ = t̄, that is
(t1)x − (t2)y = t1, (t1)y + (t1)x = −t2,
then the R4-valued 1-form
Y1dx + Y2dy = (t1X1 − t2X2)dx− (t2X1 + t1X2)dy
is closed and the corresponding Lagrangian immersion f can be computed by solving
df = Y1dx + Y2dy = (t1X1 − t2X2)dx− (t2X1 + t1X2)dy.
This is an elliptic Lagrangian immersion with Fubini’s cubic form F = t2 dz3. For a fixed t and
p 6= 2, we have the 1-parameter family of applicable immersions. Explicit solutions of tz̄ = t̄ can
be obtained with the ansatz
t1(x, y) = v1(x) + w1(y), t2(x, y) = v2(x) + w2(y).
We then have
v1(x) = c1e
2x − a1, v2(x) = c2e
−2x − a2,
w1(y) = m1e
2y + m2e
−2y − a1, w2(y) = −m1e
2y + m2e
−2y + a2,
where a1, a2, c1, c2, m1 and m2 are real constants. In this case, the integration involves
elementary functions and can be performed explicitly. The general formulae are rather involved
expressions and can be obtained with any standard software of scientific computation such as
Mathematica 6.
For instance, if a1 = a2 = 0 and m1 = m2 = 0, the components of the immersions f are given
by
fj(x, y) =
e−2x
(p− 2)
√
(p + 2)(4− p2)
(
cosh(
√
p + 2x)Aj(x, y) + sinh(
√
p + 2x)Bj(x, y)
)
,
j = 1, . . . , 4, where the functions Aj(y) and Bj(y) are given by
A1(x, y) = −c1e
4x
√
(p + 2)(4− p2) cosh(
√
2− py)− c2(p2 − 4) sinh(
√
2− py),
A2(x, y) = −c1e
4x(p2 − 4) sinh(
√
2− py)− c2
√
(p + 2)(4− p2) cosh(
√
2− py)),
A3(x, y) = c1e
4x
√
(p + 2)(4− p2) cosh(
√
2− py),
A4(x, y) = c2
√
(p + 2)(4− p2) cosh(
√
2− py),
and by
B1(x, y) = c1e
4xp
√
4− p2 cosh(
√
2− py)− c2(p− 2)
√
p + 2 sinh(
√
2− py),
B2(x, y) = c1e
4x(p− 2)
√
p + 2 sinh(
√
2− py)− c2p
√
4− p2 cosh(
√
2− py),
B3(x, y) = −2c1e
4x
√
4− p2 cosh(
√
2− py)− c2(p− 2)
√
p + 2 sinh(
√
2− py),
B4(x, y) = c1e
4x(p− 2)
√
p + 2 sinh(
√
2− py) + 2c2
√
4− p2 cosh(
√
2− py).
These provide explicit examples of elliptic Lagrangian immersions with the same Gauss map but
which are not applicable and of applicable non-umbilic elliptic Lagrangian immersions. Note
that for p > 2 some of the hyperbolic functions transform into trigonometric functions.
18 E Musso and L. Nicolodi
Acknowledgements
The work was partially supported by MIUR projects: Metriche riemanniane e varietà dif-
ferenziabili (E.M.); Proprietà geometriche delle varietà reali e complesse (L.N.); and by the
GNSAGA of INDAM. The authors would like to thank the referees for their useful comments
and suggestions.
References
[1] Álvarez Paiva J.C., Durán C.E., Geometric invariants of fanning curves, math.SG/0502481.
[2] Barbot T., Charette V., Drumm T., Goldman W.M., Melnik K., A primer on the (2+1) Einstein universe,
arXiv:0706.3055.
[3] Bianchi L., Complementi alle ricerche sulle superficie isoterme, Ann. Mat. Pura Appl. 12 (1905), 20–54.
[4] Blaschke W., Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Rela-
tivitätstheorie, Bd. 3, bearbeitet von G. Thomsen, J. Springer, Berlin, 1929.
[5] Bryant R.L., A duality theorem for Willmore surfaces, J. Differential Geom. 20 (1984), 23–53.
[6] Burstall F., Hertrich-Jeromin U., Pedit F., Pinkall U., Curved flats and isothermic surfaces, Math. Z. 225
(1997), 199–209, dg-ga/9411010.
[7] Calapso P., Sulle superficie a linee di curvatura isoterme, Rend. Circ. Mat. Palermo 17 (1903), 273–286.
[8] Cartan É., Sur le problème général de la déformation, C. R. Congrés Strasbourg (1920), 397–406 (or Oeuvres
Complètes, III 1, 539–548).
[9] Cartan É., Sur la déformation projective des surfaces, Ann. Scient. Éc. Norm. Sup. (3) 37 (1920), 259–356
(or Oeuvres Complètes, III 1, 441–538).
[10] Chern S.-S., Wang H.-C., Differential geometry in symplectic space. I, Sci. Rep. Nat. Tsing Hua Univ. 4
(1947), 453–477.
[11] Deconchy V., Hypersurfaces in symplectic affine geometry, Differential Geom. Appl. 17 (2002), 1–13.
[12] Fubini G., Applicabilità proiettiva di due superficie, Rend. Circ. Mat. Palermo 41 (1916), 135–162.
[13] Gálvez J.A., Mart́ınez A., Milán F., Flat surfaces in the hyperbolic 3-space, Math. Ann. 316 (2000), 419–435.
[14] Griffiths P.A., On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence
questions in differential geometry, Duke Math. J. 41 (1974), 775–814.
[15] Guillemin V., Sternberg S., Variations on a theme by Kepler, American Mathematical Society Colloquium
Publications, Vol. 42, American Mathematical Society, Providence, RI, 1990.
[16] Hertrich-Jeromin U., Musso E., Nicolodi L., Möbius geometry of surfaces of constant mean curvature 1 in
hyperbolic space, Ann. Global Anal. Geom. 19 (2001), 185–205, math.DG/9810157.
[17] Jensen G.R., Deformation of submanifolds of homogeneous spaces, J. Differential Geom. 16 (1981), 213–246.
[18] Jensen G.R., Musso E., Rigidity of hypersurfaces in complex projective space, Ann. Sci. École Norm.
Sup. (4) 27 (1994), 227–248.
[19] Kamran N., Olver P., Tenenblat K., Local symplectic invariants for curves, Commun. Contemp. Math., to
appear.
[20] Kokubu M., Umehara M., Yamada K., Flat fronts in hyperbolic 3-space, Pacific J. Math. 216 (2004),
149–175, math.DG/0301224.
[21] McKay B., Lagrangian submanifolds in affine symplectic geometry, Differential Geom. Appl. 24 (2006),
670–689, math.DG/0508118.
[22] Musso E., Deformazione di superfici nello spazio di Möbius, Rend. Istit. Mat. Univ. Trieste 27 (1995),
25–45.
[23] Musso E., Nicolodi L., Deformation and applicability of surfaces in Lie sphere geometry, Tohoku Math. J.
58 (2006), 161–187, math.DG/0408009.
[24] Musso E., Nicolodi L., Conformal deformation of spacelike surfaces in Minkowski space, Houston J. Math.,
to appear, arXiv:0712.0807.
http://arxiv.org/abs/math.SG/0502481
http://arxiv.org/abs/0706.3055
http://arxiv.org/abs/dg-ga/9411010
http://arxiv.org/abs/math.DG/9810157
http://arxiv.org/abs/math.DG/0301224
http://arxiv.org/abs/math.DG/0508118
http://arxiv.org/abs/math.DG/0508118
http://arxiv.org/abs/0712.0807
1 Introduction
2 Preliminaries
2.1 Affine symplectic frames and structure equations
2.2 Oriented Lagrangian planes
2.3 Lagrangian immersions
3 Moving frames on elliptic Lagrangian immersions
3.1 Frame reductions and differential invariants
3.1.1 First order frame fields
3.1.2 Second order frame fields
3.1.3 Third order frame fields
3.2 Adapted frame fields
3.2.1 Invariant functions and integrability equations
4 Symplectic applicability
4.1 Generic Lagrangian immersions
4.2 Applicable elliptic Lagrangian immersions
4.3 The differential equations of applicable Lagrangian immersions
4.4 Totally umbilic Lagrangian immersions
4.5 Complex curves and Lagrangian immersions
5 Examples
References
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