Geometric interpretation of Lagrangian equivalence
Shyuichi IZUMIYA
August 13, 2016
Abstract
As an application of the theory of graph-like Legendrian unfoldings, relations of
the hidden structures of caustics and wave front propagations are revealed.
1 Introduction
Lagrangian equivalence among Lagrangian submanifold germs in the cotangent bundle
was introduced for the study of oscillatory integrals on caustics (cf., [2, 3, 4]). It has been
known that caustic equivalence (i.e., diffeomorphic caustics) does not imply Lagrangian
equivalence. This is one of the main differences from the theory of Legendrian singularities.
In the theory of Legendrian singularities, wave front equivalence (i.e., diffeomorphic wave
fronts) implies Legendrian equivalence generically [11]. Therefore, Legendrian equivalence
is geometric equivalence in this sense. In the real world, the caustics given by reflected rays
are visible. However, the wave front propagations are not visible. Therefore, we can say
that there are hidden structures behind the picture of caustics (cf., [10]). In fact, caustics
are a subject of classical physics (i.e., geometric optics). However, the corresponding
Lagrangian submanifold is deeply related to the semi-classical approximation of quantum
mechanics.
On the other hand, the notion of graph-like Legendrian unfoldings was introduced in
[5]. It belongs to a special class of big Legendrian submanifolds (wave front propagations)
which Zakalyukin introduced in [11]. There have been some developments on this theory
during past two decades[5, 6, 7, 9]. There appeared several equivalence relations among
big Legendrian submanifolds in these articles for different purposes. However, the relation
to Lagrangian equivalence has not been clear so far. One of the main results in this paper
are Theorems 4.3 and 4.5 which reveal the relation between caustics and wave front
propagations. For the proof of Theorem 4.3, we show Propositions 4.1 and 4.2. Actually,
Proposition 4.1 has been proved in [7] by using the notion of generating families which is
a strong tool for the study of Lagrangian equivalence. Proposition 4.2 is the key assertion
in this paper. We avoid to use the notion of generating families to shorten the paper.
2010 Mathematics Subject classification. Primary 58K05,57R45 ; Secondary 58K60
Keywords. wave front propagations, big wave fronts, graph-like Legendrian unfoldings, Caustics
This work was supported by JSPS KAKENHI Grant Number26287009.
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2 Lagrangian singularities
We briefly describe the basic notions of the local theory of Lagrangian singularities
due to [1]. We consider the cotangent bundle π : T ∗Rn → Rn over Rn. Let (x, p) =
(x1, . . . , xn, p1, . . . , pn) be the canonical coordinates on T
∗Rn. Then the canonical sym-
plectic structure on T ∗Rn is given by the canonical two form ω =
∑n
i=1 dpi ∧ dxi. Let
i : L ⊂ T ∗Rn be a submanifold. We say that i is a Lagrangian submanifold if dimL = n
and i∗ω = 0. In this case, the set of critical values of π ◦ i is called the caustic of
i : L ⊂ T ∗Rn, which is denoted by CL.
We now define a natural equivalence relation among Lagrangian submanifold germs.
Let i : (L, p) ⊂ (T ∗Rn, p) and i′ : (L′, p′) ⊂ (T ∗Rn, p′) be Lagrangian submanifold germs.
Then we say that i and i′ are Lagrangian equivalent if there exist a diffeomorphism germ
σ : (L, p) → (L′, p′), a symplectic diffeomorphism germ τ̂ : (T ∗Rn, p) → (T ∗Rn, p′) and a
diffeomorphism germ τ : (Rn, π(p)) → (Rn, π(p′)) such that τ̂ ◦ i = i′ ◦σ and π ◦ τ̂ = τ ◦π,
where π : (T ∗Rn, p) → (Rn, π(p)) is the canonical projection. Here τ̂ is said to be a
symplectic diffeomorphism germ if it is a diffeomorphism germ such that τ̂ ∗ω = ω. We
also say that i and i′ are caustic equivalent if CL and CL′ are diffeomorphic. By definition,
if i and i′ are Lagrangian equivalent then i and i′ are caustic equivalent. In general, the
converse does not hold. This is the reason why we have no geometric interpretation of
Lagrangian equivalence so far. There is the notion of Lagrangian stability of a Lagrangian
submanifold germ (cf., [1]). Here, we do not need the exact definition, so that we omit
the definition.
3 Theory of the wave front propagations
In this section we describe the basic notions of the theory of wave front propagations (for
details, see [1, 5, 11, 12], etc). We start to consider the general theory of Legendrian
singularities. Let π : PT ∗(Rm) −→ Rm be the projective cotangent bundle over Rm.
This fibration can be considered as a Legendrian fibration with the canonical contact
structure K on PT ∗(Rm). We have the trivialization PT ∗(Rm) ∼= Rm × P (Rm∗) and we
call (x, [ξ]) homogeneous coordinates, where x = (x1, . . . , xm) ∈ Rm and [ξ] = [ξ1 : · · · : ξm]
are homogeneous coordinates of the dual projective space P (Rm∗). Let Φ : (Rm, 0) −→
(Rm, 0) be a diffeomorphism germ. Then we have a unique contact diffeomorphism germ
Φ̂ : PT ∗Rm −→ PT ∗Rm defined by Φ̂(x, [ξ]) = (Φ(x), [ξ ◦ dΦ(x)(Φ−1)]). We say that Φ̂ is a
contact diffeomorphism if dΦ̂((x,[ξ])(K((x,[ξ])) = KΦ̂(x,[ξ]). We call Φ̂ the contact lift of Φ. A
submanifold i : L ⊂ PT ∗(Rm) is said to be a Legendrian submanifold if dimL = m− 1
and dip(TpL ) ⊂ Ki(p) for any p ∈ L. We also call π ◦ i = π|L : L −→ Rm a Legendrian
map and W (L ) = π(L ) a wave front of i : L ⊂ PT ∗(Rm). We say that a point
p ∈ L is a Legendrian singular point if rank d(π|L )p < m − 1. In this case π(p) is the
singular point of W (L ). We say that two Legendrian submanifold germs L and L ′ are
Legendrian equivalent if there exists a diffeomorphism germ Φ : (Rm, 0) −→ (Rm, 0) such
that Φ̂(L ) = L ′ as set germs.
We now consider the case when m = n+1 and distinguish space and time coordinates,
so that we write Rn+1 = Rn ×R and coordinates are denoted by (x, t) = (x1, . . . , xn, t) ∈
Rn × R. For the projective cotangent bundle π : PT ∗(Rn × R) → Rn × R, we have
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homogeneous coordinates ((x1, . . . , xn, t), [ξ1 : · · · : ξn : τ ]) of PT ∗(Rn ×R) ∼= (Rn ×R)×
P ((Rn × R)∗),
For a Legendrian submanifold i : L ⊂ PT ∗(Rn × R), the corresponding wave front
π ◦ i(L ) = W (L ) is called a big wave front. We call
Wt(L ) = π1(π
−1
2 (t) ∩W (L )) (t ∈ R)
a momentary front (or, a small front) for each t ∈ R, where π1 : Rn × R → Rn and
π2 : Rn × R → R are the canonical projections defined by π1(x, t) = x and π2(x, t) = t
respectively. In this sense, we call L a big Legendrian submanifold. We say that a point
p ∈ L is a space-singular point if rank d(π1 ◦ π|L )p < n and a time-singular point if
rank d(π2 ◦π|L )p = 0, respectively. By definition, if p ∈ L is a Legendrian singular point,
then it is a space-singular point of L . The discriminant of the family {Wt(L )}t∈R is
defined as the image of singular points of π1|W (L ). In the general case, the discriminant
consists of three components: the caustic CL = π1(Σ(W (L )), where Σ(W (L )) is the set
of singular points of W (L ) (i.e, the critical value set of the Legendrian mapping π|L ),
the Maxwell stratified set ML , the projection of the closure of the self intersection set of
W (L ); and also of the critical value set ∆L of π1|W (L )\Σ(W (L )).
We now consider an equivalence relation among big Legendrian submanifolds which
was independently introduced in [6, 12] for different purposes: Let i : (L , p0) ⊂ (PT ∗(Rn×
R), p0) and i′ : (L ′, p′0) ⊂ (PT ∗(Rn×R), p′0) be big Legendrian submanifold germs. Then
we say that i and i′ are strictly parametrized+ Legendrian equivalent (or, briefly S.P+-
Legendrian equivalent) if there exists a diffeomorphism germs Φ : (Rn × R, π(p0)) →
(Rn×R, π(p′0)) of the form Φ(x, t) = (ϕ1(x), t+α(x)) such that Φ̂(L ) = L ′ as set germs,
where Φ̂ : (PT ∗(Rn×R), p0) → (PT ∗(Rn×R), p′0) is the unique contact lift of Φ. We can
also define the notion of stability of Legendrian submanifold germs with respect to the
above equivalence relation which is analogous to the stability of Legendrian submanifold
germs with respect to Legendrian equivalence (cf. [1, Part III]).
On the other hand, concerning the discriminant, we define the following equivalence
relation among big wave front germs. Let i : (L , p0) ⊂ (PT ∗(Rn × R), p0) and i′ :
(L ′, p′0) ⊂ (PT ∗(Rn ×R), p′0) be big Legendrian submanifold germs. We say that W (L )
and W (L ′) are S.P+-diffeomorphic if there exists a diffeomorphism germ Φ : (Rn ×
R, π(p0)) → (Rn×R, π(p′0)) of the form Φ(x, t) = (ϕ1(x), t+α(x)) such that Φ(W (L )) =
W (L ′) as set germs. We also call Φ a S.P+-diffeomorphism germ. We remark that
S.P+-diffeomorphism among big wave front germs preserves the diffeomorphism types of
CL ∪ML ∪∆L . Since the Legendrian submanifold germ is uniquely determined on the
regular part of the wave front set, we have the following proposition as an easy corollary
of the result in [11].
Proposition 3.1 Let i : (L , p0) ⊂ (PT ∗(Rn × R), p0) and i′ : (L ′, p′0) ⊂ (PT ∗(Rn ×
R), p′0) be big Legendrian submanifold germs such that the sets of critical points of π◦i, π◦i′
are nowhere dense respectively. Then i and i′ are S.P+-Legendrian equivalent if and only
if (W (L ), π(p0)) and (W (L ′), π(p′0)) are S.P
+-diffeomorphic.
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4 Graph-like Legendrian unfoldings
In this section we explain the theory of graph-like Legendrian unfoldings and prove the
main theorems. The notion of graph-like Legendrian unfoldings was introduced in [5]. A
graph-like Legendrian unfolding belongs to a special class of big Legendrian submanifolds.
We remark that PT ∗(Rn × R) is a fiber-wise compactification of the 1-jet space as
follows: We consider an affine open subset Uτ = {((x, t), [ξ : τ ])|τ ̸= 0} of PT ∗(Rn × R).
For any ((x, t), [ξ : τ ]) ∈ Uτ , we have
((x1, . . . , xn, t), [ξ1 : · · · : ξn : τ ]) = ((x1, . . . , xn, t), [−(ξ1/τ) : · · · : −(ξn/τ) : −1]),
so that we may adopt the corresponding affine coordinates ((x1, . . . , xn, t), (p1, . . . , pn)),
where pi = −ξi/τ. On Uτ we have θ−1(0) = K|Uτ , where θ = dt−
∑n
i=1 pidxi. This means
that Uτ can be identified with the 1-jet space which is denoted by J
1
GA(Rn,R) ⊂ PT ∗(Rn×
R). We call the above coordinates a system of graph-like affine coordinates. Throughout
this paper, we use this identification. A big Legendrian submanifold i : L ⊂ PT ∗(Rn×R)
is said to be a graph-like Legendrian unfolding if L ⊂ J1GA(Rn,R). We callW (L ) = π(L )
a graph-like wave front of L , where π : J1GA(Rn,R) −→ Rn×R is the canonical projection.
We define a mapping Π : J1GA(Rn,R) −→ T ∗Rn by Π(x, t, p) = (x, p), where (x, t, p) =
(x1, . . . , xn, t, p1, . . . , pn). Then we have the following proposition. In [7] we have shown
that Π|L : L −→ T ∗Rn is immersive, so that Π(L ) is a Lagrangian submanifold in T ∗Rn.
Moreover, the discriminant of the family of momentary fronts is CL ∪ML for a graph-
like Legendrian unfolding L ⊂ J1GA(Rn,R). By using the notion of generating families,
we can show the following assertion: For any Lagrangian submanifold germ (L, z) ⊂
T ∗Rn, there exists a graph-like Legendrian unfolding germ (L , p) ⊂ J1GA(Rn,R) such
that (Π(L ),Π(p)) = (L, z).We now compare the equivalence relations between graph-like
Legendrian unfoldings and induced Lagrangian submanifold germs. By using the notion of
graph-like generating families of graph-like Legendrian unfoldings and generating families
of Lagrangian submanifold germs respectively, we showed the following proposition in [7].
Proposition 4.1 ([7]) Let (L1, p1), (L2, p2) be graph-like Legendrian unfoldings. If
(Π(L1),Π(p1)) and (Π(L2).Π(p2)) are Lagrangian equivalent, then (L1, p1) and (L2, p2)
are S.P+-Legendrian equivalent.
The above proposition asserts that Lagrangian equivalence is a stronger equivalence rela-
tion than S.P+-Legendrian equivalence. The S.P+-Legendrian equivalence relation among
graph-like Legendrian unfoldings preserves both the diffeomorphism types of caustics and
Maxwell stratified sets. On the other hand, if we observe the real caustics of rays, we
cannot observe the structure of wave front propagations and the Maxwell stratified sets.
In this sense, there are hidden structures behind the picture of real caustics. By the
above proposition, Lagrangian equivalence preserves not only the diffeomorphism type of
caustics, but also the hidden geometric structure of wave front propagations. It has been
believed that the converse assertion of Proposition 4.1 does not hold since a long time
ago at least for me. Actually, [8] showed the converse assertion only for the case that
Π(L1) and Π(L2) are Lagrange stable. Therefore, we have not tried to show the converse
assertion so far. However, we have the following result.
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Proposition 4.2 Let (L1, p1), (L2, p2) be graph-like Legendrian unfoldings. If (L1, p1)
and (L2, p2) are S.P+-Legendrian equivalent, then (Π(L1),Π(p1)) and (Π(L2),Π(p2)) are
Lagrangian equivalent.
Proof. In order to simplify the arguments, we use x = (x1, . . . , xn), ξ = (ξ1, . . . , ξn) and
p = (p1, . . . , pn). With this notation, we write that ξ · x =
∑n
i=1 ξixi and θ = dt− p · dx =
dt−
∑n
i=1 pidxi etc.
Without loss of generality, we assume that Π(p1) = Π(p2) = 0 ∈ Rn. By the as-
sumption, there exists a diffeomorphism germ Φ : (Rn × R, 0) −→ (Rn × R, 0) of the
form Φ(x, t) = (ϕ1(x), t + α(x)) such that Φ̂(L1) = L2. Then we have Φ−1(x, t) =
(ϕ−11 (x), t− α(ϕ−11 (x))), so that the Jacobi matrix is
JΦ(x)Φ
−1 =
 ∂ϕ
−1
1
∂x
(ϕ1(x)) 0
−∂α ◦ ϕ
−1
1
∂x
(ϕ1(x)) 1
 .
It follows that
Φ̂((x, t), [ξ : τ ]) =
(
Φ(x, t),
[
ξ · ∂ϕ
−1
1
∂x
(ϕ1(x))− τ
∂α ◦ ϕ−11
∂x
(ϕ1(x)) : τ
])
.
Since τ ̸= 0,[
ξ · ∂ϕ
−1
1
∂x
(ϕ1(x))−
∂α ◦ ϕ−11
∂x
(ϕ1(x)) : τ
]
=
[
− ξ
τ
· ∂ϕ
−1
1
∂x
(ϕ1(x)) +
∂α ◦ ϕ−11
∂x
(ϕ1(x)) : −1
]
.
We consider the graph-like affine coordinates ((x, t), p) ∈ J1GA(Rn,R), where p = −
ξ
τ
.
Then we have Φ̂(J1GA(Rn,R)) = J1GA(Rn,R) and
Φ̂((x, t), p) =
(
ϕ1(x), t+ α(x), p ·
∂ϕ−11
∂x
(ϕ1(x)) +
∂α ◦ ϕ−11
∂x
(ϕ1(x))
)
.
We now define a map ϕ̃1 : T
∗Rn −→ T ∗Rn by
ϕ̃1(x, p) =
(
ϕ1(x), p ·
∂ϕ−11
∂x
(ϕ1(x)) +
∂α ◦ ϕ−11
∂x
(ϕ1(x))
)
.
Since Φ̂ is a contact diffeomorphism germ, there exists a function germ µ : J1GA(Rn,R) −→
R with µ(x, t, p) ̸= 0 such that Φ̂∗θ = µθ. Therefore, we have
dt+ dα− ϕ̃1
∗
(p · dx) = µ(dt− p · dx) = µdt− µ(p · dx),
so that µ ≡ 1. It follows that −p · dx = dα− ϕ̃1
∗
(p · dx). Thus we have
ϕ̃1
∗
(ω) = ϕ̃1
∗
(d(p · dx)) = dϕ̃1
∗
(p · dx) = d(p · dx) = ω.
This means that ϕ̃1 is a symplectic diffeomorphism germ (i.e., Lagrangian diffeomorphism
germ). Since Π ◦ Φ̂|J1GA(Rn,R) = ϕ̃1 ◦ Π|J1GA(Rn,R), we have Π(L2) = Π ◦ Φ̂(L1) =
ϕ̃1(Π(L1)). 2
By Propositions 4.1 and 4.2, we conclude the following theorem.
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Theorem 4.3 The graph-like Legendrian unfoldings (L1, p1) and (L2, p2) are S.P+-
Legendrian equivalent if and only if the Lagrangian submanifold germs (Π(L1),Π(p1))
and (Π(L2),Π(p2)) are Lagrangian equivalent.
As a corollary of the above theorem, we have the following result:
Corollary 4.4 ([9]) The graph-like Legendrian unfolding (L , p) is S.P+-Legendrian sta-
ble if and only if the Lagrangian submanifold germ (Π(L ),Π(p)) is Lagrangian stable.
In [9] we apply the infinitesimal characterization of S.P+-Legendrian stability. How-
ever, the assertion of Corollary 4.4 follows directly from Theorem 4.3.
On the other hand, by Corollary 4.4 and the fact π ◦ Π = π1 ◦ π, the set of Leg-
endrian singular points of a graph-like Legendrian unfolding L coincides with the set
of Lagrangian singular points of π|Π(L ). Moreover, for generic graph-like Legendrian
unfolding germ (L , p), the set of singular points of π|L is nowhere dense. Hence we can
apply Proposition 3.1 to our situation and obtain the following geometric interpretation
of Lagrangian equivalence as a corollary of Theorem 4.3.
Theorem 4.5 Let (L1, p1) and (L2, p2) be graph-like Legendrian unfolding germs such
that the sets of singular points of π|L1, π|L2 are nowhere dense respectively. Then the
following conditions are equivalent:
(1) (Π(L1),Π(p1)) and (Π(L2),Π(p2) are Lagrangian equivalent.
(2) Graph-like wave fronts (W (L1), π(p1)) and (W (L2), π(p2)) are S.P+-diffeomorphic.
We remark that the condition (1) implies the contain (2) in the above theorem without
any assumptions.
Let (L , p) be a graph-like Legendrian unfolding germ. We consider a representative
L̃ of (L , p) on π−1(W ), where W ⊂ Rn ×R is an open neighborhood of π(p) ∈ Rn ×R.
Then we have a representative W (L̃ ) = W (L̃ ) ∩W of the set germ (W (L ), π(p)).
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Shyuichi Izumiya
Department of Mathematics
Hokkaido University
Sapporo 060-0810, Japan
e-mail: izumiya@math.sci.hokudai.ac.jp
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