Geometry of B¨ acklund Transformations by Yuhao Hu Department of Mathematics Duke University Date: Approved: Robert Bryant, Supervisor Hubert Bray Lenhard L. Ng Leslie Saper Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2018
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Geometry of Backlund Transformations
by
Yuhao Hu
Department of MathematicsDuke University
Date:Approved:
Robert Bryant, Supervisor
Hubert Bray
Lenhard L. Ng
Leslie Saper
Dissertation submitted in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in the Department of Mathematics
in the Graduate School of Duke University2018
Abstract
Geometry of Backlund Transformations
by
Yuhao Hu
Department of MathematicsDuke University
Date:Approved:
Robert Bryant, Supervisor
Hubert Bray
Lenhard L. Ng
Leslie Saper
An abstract of a dissertation submitted in partial fulfillment of the requirements forthe degree of Doctor of Philosophy in the Department of Mathematics
pM, Iq An exterior differential system with manifold M and differentialideal I.
Ik Ă ΛkpT ˚Mq The subbundle corresponding to the degree-k piece of I, whenpM, Iq is an exterior differential system.
xθ1, ..., θ`y The ideal of Ω˚pUq generated by differential forms θ1, ..., θ` (de-fined on U) and their exterior derivatives.
xθ1, ..., θ`yalg The ideal of Ω˚pUq algebraically generated by differential formsθ1, ..., θ` defined on U .
rrθ1, ..., θ`ss The vector subbundle of ΛkpT ˚Uq generated by differential formsθ1, ..., θ` (defined on U) of the same degree k.
CpIq The Cartan system associated to the differential system pM, Iq,i.e., the Frobenius system whose integral curves are precisely theCauchy characteristics of pM, Iq.
X ω The interior product of a smooth vector fieldX with a differentialform ω.
SK Ă TM The distribution spanned by all vector fields X that satisfyX ω “ 0 for any ω P S Ă Ω˚pMq.
viii
Acknowledgements
I’d like to express my deepest gratitude to my advisor, Robert Bryant, for teaching
me how to do mathematics and for his encouragement and guidance during my
research. His influence on me goes beyond mathematics.
In the past years, I’ve benefited a great deal from attending lectures of and talking
with professors in or outside Duke. Special thanks go to
Hubert Bray, Lenny Ng and Leslie Saper, for being on my committee.
Jeanne Clelland, for her inspiring work and her interest in my research.
Clark Bray, for all the training he has given me in mathematical teaching.
David Schaeffer. The memory of your first-year course I’ll always cherish.
I’d like to acknowledge all my friends and fellow students who stood beside me
over the years. Especially, I’d like to thank
Ma Luo and Zhiyong Zhao, for our brotherly 5+ years.
Mendel Nguyen, for mathematics, physics, Chopin and Driade.
Gavin Ball, Ryan Gunderson and Mike Bell, for kindly sharing their research.
Zhennan Zhou, for taking me on hikes and teaching me how to cook.
Sean Lawley, for all our conversations during 2012-14.
Rosa Zhou, for moon, star, and rose.
Thank you, my dear parents. Your thoughts are constantly on me, my well-being.
How many times have you reminded me that I shall never give up. How deeply have
you, by your love, passed on to me the value of a simple and unassuming life.
ix
1
Introduction
In 1882, the Swedish mathematician A.V. Backlund proved the result (see [Bac83],
[BGG03] or [CT80]): Given a surface with a constant Gauss curvature K ă 0 in E3,
one can construct, by solving ODEs, a 1-parameter family of new surfaces in E3 with
the Gauss curvature K. This is the origin of the term “Backlund transformation”.
Classically, a Backlund transformation is a PDE system B that relates solutions
of two other PDE systems E1 and E2. More precisely, such a relation must satisfy
the property: given a solution u of E1 (resp. E2), substituting it in B, one obtains a
PDE system whose solutions can be found by ODE methods and produce solutions
of E2 (resp. E1).
For example, the Cauchy-Riemann system
#
ux ´ vy “ 0,
uy ` vx “ 0(1.1)
is a Backlund transformation; it relates solutions of the Laplace equation ∆z “ 0 for
zpx, yq in the following way: If u satisfies ∆u “ 0, then, substituting it in (1.1), we
obtain a compatible first order system for v, whose solutions can be found by ODE
1
methods and satisfy ∆v “ 0, and vice versa.
As another example, consider the system of nonlinear equations
#
zx ´ zx “ λ sinpz ` zq,
zy ` zy “1λ
sinpz ´ zq,(1.2)
where λ is a nonzero constant. One can show that (1.2) is a Backlund transformation
relating solutions of the sine-Gordon equation
uxy “12
sinp2uq. (1.3)
The system (1.2) is closely connected with the classical Backlund transformation
relating surfaces in E3 with a negative constant Gauss curvature. For details, see
[CT80].
In addition, a Backlund transformation may relate solutions of a parabolic equa-
tion (see [NC82]) or two equations that are nonequivalent (see [CI`09]). Numerous
other examples of Backlund transformations are discussed in [RS02]. Through these
examples, Backlund transformations are found to have rich connection with surface
theory in differential geometry and solitons in mathematical physics.
Among the examples discussed in [RS02], a Backlund transformation relating so-
lutions of the hyperbolic Tzitzeica equation is particularly interesting. The hyperbolic
Tzitzeica equation is the second-order equation for hpx, yq:
plnhqxy “ h´ h´2. (1.4)
This equation was discovered by Tzitzeica in his study of hyperbolic affine spheres
in the affine 3-space A3 (see [Tzi08] and [Tzi09]). He found that the system in α, β,
and h$
’
’
&
’
’
%
αx “ phxα ` λβqh´1 ´ α2,
αy “ βx “ h´ αβ,
βy “ phyβ ` λ´1αqh´1 ´ β2,
(1.5)
2
where λ is an arbitrary nonzero constant, is a Backlund transformation relating
solutions of (1.4). More explicitly, if h solves the hyperbolic Tzitzeica equation
(1.4), then, substituting it in the system (1.5), one obtains a compatible first-order
PDE system for α and β, whose solutions can be found by solving ODEs; for each
solution pα, βq, the function
h “ ´h` 2αβ
also satisfies the hyperbolic Tzitzeica equation (1.4). Furthermore, one can show
that, unlike the systems (1.1) and (1.2), substituting a solution h of (1.4) into (1.5)
yields a system whose solutions depend on 2 parameters instead of 1. Using our
terminology (Definition 2.13 of Chapter 2), one can verify that the system (1.5)
corresponds to a rank -2 Backlund transformation.
An ultimate goal of studying Backlund transformations is solving the Backlund
problem, which was considered by Goursat in [Gou25]. The statement of the Backlund
problem is: Find all pairs of systems of PDEs whose solutions are related by a
Backlund transformation. This problem remains unsolved.
However, recent works of Jeanne Clelland have shed new light on the classification
of Backlund transformations. Her paper [Cle01], in particular, focuses on Backlund
transformations relating solutions of two hyperbolic Monge-Ampere systems, which
are second-order PDEs in the plane arising frequently in differential geometry.
Clelland’s approach in [Cle01] to Backlund transformations involves several key
steps. First, a Backlund transformation relating two hyperbolic Monge-Ampere sys-
tems is formulated as an exterior differential system ([BCG`13]). This allows one to
study a Backlund transformation geometrically as a manifold N with a structure B.
As a result, concepts such as equivalence and symmetry can be easily defined. Sec-
ond, she applies Cartan’s method of equivalence to derive local invariants of such a
Backlund transformation. Third, by assuming all local invariants to be constants, she
3
obtains a complete classification of homogeneous rank-1 Backlund transformations
relating two hyperbolic Monge-Ampere systems, where homogeneous means that the
symmetry group of a Backlund transformation acts transitively on the underlying
manifold.
The aim of the present work is to address the following questions:
i. Up to equivalence, what is the generality of Backlund transformations
relating two hyperbolic Monge-Ampere systems?
ii. Given two hyperbolic Monge-Ampere systems, how to tell whether
they are related by a Backlund transformation?
iii. What can be concluded about the existence of Backlund transforma-
tions in higher ranks?
Of course, similar questions can be asked for Backlund transformations relating two
PDE systems in broader classes (elliptic, hyperbolic, parabolic), but, for clarity, in
this thesis, we focus on Backlund transformations relating two hyperbolic Monge-
Ampere systems.
This thesis is organized as follows.
In Chapter 2, we develop the basic concepts and techniques that are used in this
work. These include the notion of an exterior differential system, in particular, that
of a hyperbolic Monge-Ampere system; the notion of a G-structure, which is at the
heart of an equivalence problem; and the notion of a Backlund transformation.
In the first two sections of Chapter 3, we prove a generality result:
Theorem 3.3. A generic rank-1 Backlund transformation relating two hyperbolic
Monge-Ampere systems can be specified uniquely, up to equivalence, by initial data
consisting of at most 6 functions of 3 variables.
This theorem implies the following
4
Corollary 3.4. There exist hyperbolic Monge-Ampere systems that are not related
to any other hyperbolic Monge-Ampere system by a generic rank-1 Backlund trans-
formation.
Before summarizing the main ideas of proving Theorem 3.3, we briefly explain
what the term “generic” means. Given a rank-1 Backlund transformation pN,Bq
relating two hyperbolic Monge-Ampere systems, there is an intrinsic way to define
a tensor field T on N (Section 3.1). A rank-1 Backlund transformation is said to be
generic if the tensor T takes generic values.
To prove Theorem 3.3, we first show that, given a generic rank-1 Backlund trans-
formation pN,Bq relating two hyperbolic Monge-Ampere systems, where N is a 6-
manifold, there is a canonical way to define a local coframing on N . One can show
that two such coframings are locally equivalent up to diffeomorphism if and only if
the corresponding Backlund transformations are locally equivalent. Then, we apply
a theorem of Cartan ([Bry14]) to show that such local coframings, up to diffeomor-
phism, depend on at most 6 functions of 3 variables.
In the last section of Chapter 3, we focus on the case when two invariants of a
generic rank-1 Backlund transformation are assumed to be specific constants. We
can classify such Backlund transformations. In particular, we find new examples of
Backlund transformations with cohomogeneity 1, 2 and 3.
In Chapter 4, we study how obstructions to the existence of Backlund transfor-
mations may be expressed in terms of the invariants of the underlying hyperbolic
Monge-Ampere systems. Such obstructions can be found by applying techniques
of exterior differential systems ([BCG`13]) to a rank-4 Pfaffian system. We obtain
several results that inform us which pairs of hyperbolic Monge-Ampere systems may
be related by a rank-1 Backlund transformation of a particular type.
In Chapter 5, motivated by the example (1.5), we study rank-2 Backlund trans-
5
formations relating two hyperbolic Monge-Ampere systems, and partially classify
those that are homogeneous1. The approach to classification is analogous to that in
[Cle01]. The results of classification, as well as the cases that are work in progress,
are summarized in Section 5.4.
Most calculations in Chapters 3,4, and 5 are performed using MapleTM.
1 It turns out that the rank-2 Backlund transformation corresponding to (1.5) is nonhomogeneous.
6
2
Background
2.1 Exterior Differential Systems
Definition 2.1. Let M be a smooth manifold, I Ă Ω˚pMq a graded ideal that
is closed under exterior differentiation. The pair pM, Iq is said to be an exterior
differential system with space M and differential ideal I.
Given an exterior differential system pM, Iq, we use Ik to denote the degree-k
piece of I, namely, Ik “ I X ΩkpMq, where ΩkpMq stands for the C8pMq-module
of differential k-forms on M . If the rank of Ik, restricted to each point, is locally a
constant, then the elements of Ik are precisely smooth sections of a vector bundle
denoted by Ik.
Definition 2.2. An integral manifold of an exterior differential system pM, Iq is a
submanifold i : N ãÑM satisfying i˚φ “ 0 for any φ P I.
By this definition, the requirement of I being closed under exterior differentiation
is natural, because of the identity i˚ ˝ d “ d ˝ i˚.
Intuitively, an exterior differential system is a coordinate-independent way to
express a PDE system; an integral manifold, usually with a certain independence
7
condition satisfied, corresponds to a solution of the PDE system. We illustrate this
point by the following example.
Example 1. A single k-th order PDE in one unknown function u of n independent
variables x “ px1, ..., xnq can be expressed in the form
F px, u, Bα1upxq, ..., Bαkupxqq “ 0, (2.1)
where αi ranges over all multiindices of length i from 1, 2, ..., n. For simplicity, we
regard two multiindices as equivalent if and only if they are reorderings of each
other. Furthermore, if α is a multiindex of length i, then jα, where j P t1, ..., nu, is
a multiindex of length i` 1.
The equation (2.1) corresponds to an exterior differential system pM, Iq induced
from the canonical contact system on the k-jet bundle JkpRn,Rq. To be explicit,
JkpRn,Rq has the standard coordinates px, u, pα1 , ..., pαkq, where αi are multiindices
in the sense above. The canonical contact system C on JkpRn,Rq is algebraically
generated by the 1-forms
θ0 “ du´ pidxi,
θαi “ dpαi ´ pjαidxjpi “ 1, 2, ..., k ´ 1q
and their exterior derivatives.
Let w : Rn Ñ R be a Ck-function. Then w : Rn Ñ JkpRn,Rq, defined by
wpxq “ px,wpxq, Bα1wpxq, ..., Bαkwpxqq
is an n-dimensional integral manifold of the exterior differential system pJkpRn,Rq, Cq.
This w is called the lifting of w to JkpRn,Rq. Conversely, suppose that v : Rn Ñ
JkpRn,Rq is an n-dimensional integral manifold of pJkpRn,Rq, Cq on which x1, x2, ..., xn
are independent functions, in other words, on which dx1 ^ dx2 ^ ¨ ¨ ¨ ^ dxn ‰ 0 ev-
erywhere. We then have
pi ˝ v “B
Bxipu ˝ vq, pij ˝ v “
B2
BxiBxjpu ˝ vq, ...
8
Therefore, locally, a Ck-function w : Rn Ñ R is in one-to-one correspondence with
an integral manifold of pJkpRn,Rq, Cq on which dx1 ^ dx2 ^ ¨ ¨ ¨ ^ dxn ‰ 0.
Now let M Ă JkpRn,Rq be defined by the equation
F px, u, pα1 , ..., pαkq “ 0,
where F is as in (2.1). If ∇F is nonzero at a point, then, by shrinking to a neigh-
borhood of that point, we can assume M to be a smooth manifold. Under this
assmuption, let CM be the restriction of the contact system C to M . Then, by an
argument similar to the above, one can show that an integral manifold of the exte-
rior differential system pM, CMq on which dx1 ^ dx2 ^ ¨ ¨ ¨ ^ dxn ‰ 0 is in one-to-one
correspondence with a solution of (2.1) whose lifting to JkpRn,Rq is contained in M .
Regarding two exterior differential systems, the following notion of equivalence is
natural.
Definition 2.3. Two exterior differential systems pM, Iq and pN,J q are said to be
equivalent up to diffeomorphism, or equivalent, for brevity, if there exists a diffeo-
morphism φ : M Ñ N such that φ˚J “ I. Such a φ is called an equivalence between
both systems. An equivalence between pM, Iq and itself is called a symmetry of
pM, Iq.
Example 2. When J1pRn,Rq with coordinates pxi, u, piq is regarded as a contact
manifold with the contact form
θ “ du´ pidxi,
a symmetry of the exterior differential system pJ1pRn,Rq, Cq is just a contact trans-
formation of the space J1pRn,Rq to itself.
We end this section by presenting the Frobenius Theorem, formulated in terms of
exterior differential systems. This theorem is at the heart of our characterization of
9
a Backlund transformation. For a proof of the Frobenius theorem, see Chapter II of
[BCG`13].
Theorem 2.1. (Frobenius) Let pMn, Iq be an exterior differential system. If, on an
open U Ă M , the differential ideal I is generated by k linearly independent 1-forms
θ1, ..., θk satisfying
dθi ” 0 mod θ1, θ2, ..., θk,
then any p P U has an open neighborhood V Ă U on which there exists a coordinate
system px1, ..., xn´k, y1, ..., ykq such that the ideal I is generated by dy1, dy2, ..., dyk.
Remark 1. One can show ([BCG`13]) that a coordinate system px1, ..., xn´k, y1, ..., ykq
in the conclusion of Theorem 2.1 can be found by solving systems of ODEs. Once
such a coordinate system is obtained, setting the yi pi “ 1, ..., kq to be constants de-
fines an pn´kq-dimensional integral manifold of pM, Iq. Each such integral manifold
is called a leaf associated to the system pM, Iq.
Definition 2.4. Given an exterior differential system pM, Iq. If the assumption in
Theorem 2.1 holds on a neighborhood of every p P M for a constant k, then pM, Iq
is called a rank-k Frobenius system.
2.2 Hyperbolic Monge-Ampere Systems
Among second order PDEs for 1 unknown function of 2 independent variables, the
Monge-Ampere equations are of special interest, as they frequently arise in differential
geometry (see [Bry02]) and the calculus of variations (see [BGG03]).
The general form of a Monge-Ampere equation for zpx, yq is
where A,B,C,D,E are functions of x, y, z, zx, zy. A Monge-Ampere equation (2.2)
is said to be elliptic (resp., hyperbolic, parabolic) if AE´BD`C2 is negative (resp.,
10
positive, zero).
For example, the sine-Gordon equation (1.3) and the Tziteica equation (1.4) are
hyperbolic Monge-Ampere equations; the Cauchy-Riemann equation (1.1) and the
equation1 zxxzyy ´ z2xy “ 1 are elliptic Monge-Ampere equations; in the classical
calculus of variations, the Euler-Lagrange equation for a first-order functional
ż
Ω
Lpx, zpxq,∇zpxqqdx, Ω Ă R2, L : J1pΩ,Rq Ñ R
is always Monge-Ampere (see [BGG03]2).
A Monge-Ampere equation can be formulated as an exterior differential system
on a contact manifold. In the hyperbolic case, we follow [BGH95] to give the
Definition 2.5. A hyperbolic Monge-Ampere system pM, Iq is an exterior differential
system, where M is a 5-manifold, I being locally algebraically generated by θ P I1
and dθ,Ω P I2 satisfying
p1q θ ^ pdθq2 ‰ 0;
p2q rrdθ,Ωss, modulo θ, has rank 2;
p3q pλdθ ` µΩq2 ” 0 mod θ has two distinct solutions rλi : µis P RP1 pi “ 1, 2q.
In these three conditions, the first says that θ is contact; the second says that the
corresponding PDE system is nonempty; the third characterizes hyperbolicity, that
is, each integral surface of pM, Iq is foliated by two distinct families of characteristics.
Example 3. A PDE of the form
zxy “ F px, y, z, zx, zyq
1 cf. Jørgen’s Theorem in [Bry02].
2 In [BGG03], a Monge-Ampere equation is a second order PDE for 1 unknown function of nindependent variables, and it is shown that an Euler-Lagrange equation can be formulated as anEuler-Lagrange exterior differential system, which can be intrinsically characterized. In this thesis,we only consider the case when n “ 2.
11
corresponds to a hyperbolic Monge-Ampere system pM, Iq, where M “ J1pR2,Rq
with the standard coordinates px, y, z, p, qq; I is algebraically generated by
θ “ dz ´ pdx´ qdy,
dθ “ dx^ dp` dy ^ dq,
Ω “ pdp´ F px, y, z, p, qqdyq ^ dx.
In particular, Ω and dθ ` Ω are decomposable 2-forms.
Example 4. The oriented orthonormal frame bundle O over the Euclidean space
E3 consists of elements of the form px; e1, e2, e3q, where x P R3, and pe1, e2, e3q is an
oriented orthonormal frame at x. On O, we have the canonical structure equations
dωi “ ´ωij ^ ωj,
dωij “ ´ωik ^ ω
kj ,
where ωij “ ´ωji . Consider the exterior differential system pO, Iq, where
I “ xω3, dω3, dω12 ´Kω
1^ ω2
yalg, Kconst .
An integral surface of pO, Iq on which ω1 ^ ω2 ‰ 0 corresponds to a surface S in E3
with constant Gauss curvature K and with an orthonormal frame field pe1, e2, e3q
attached to it, e3 being normal to S.
Let X be a nonzero vector field on O that annihilates ω1, ω2, ω3, ω13 and ω2
3. For
each φ in the three algebraic generators of I above, we have
X φ “ 0, LXφ “ 0.
This implies that on the quotient space M5 of O by the flow of X, letting π : O ÑM
be the quotient map, there exists a well-defined differential form α such that π˚α “ φ.
It follows that I descends to M to be an exterior differential system J , in the sense
that I is algebraically generated by the elements of π˚J . Moreover, pM,J q is a
12
hyperbolic Monge-Ampere system if and only if the constant K ă 0. This, in part,
follows from the equality
pλdω3` µpdω1
2 ´Kω1^ ω2
qq2” pλ2
`Kµ2qpdω3
q2 mod ω3.
By Definition 2.5, on each hyperbolic Monge-Ampere system pM, Iq, locally there
exist 1-forms θ, ω1, ω2, ω3, ω4, linearly independent everywhere, such that
I “ xθ, ω1^ ω2, ω3
^ ω4y.
One can show that the pair of Pfaffian systems I10 “ xθ, ω1, ω2y and I01 “ xθ, ω
3, ω4y
are well-defined up to ordering. Restricted to an integral surface of pM, Iq, each
of I10 and I01 becomes a rank-1 Frobenius system whose integral curves are the
characteristics of pM, Iq in the usual sense of hyperbolic PDEs. This motivates the
Definition 2.6. Given a hyperbolic Monge-Ampere system3 pM, Iq, where I “
xθ, ω1 ^ ω2, ω3 ^ ω4y, the systems I10 “ xθ, ω1, ω2y and I01 “ xθ, ω
3, ω4y are called
the characteristic systems associated to pM, Iq.
2.3 Equivalence of G-structures
Definition 2.7. Let M be an n-dimensional manifold. A coframe at p P M is a
linear isomorphism up : TpM Ñ Rn. The set of all coframes on M forms a principal
right GLpn,Rq-bundle with the group action u ¨g :“ g´1u. This is called the coframe
bundle over M , denoted as FpMq. A local section of FpMq defined on U Ă M is
said to be a coframing on U .
Many (local) differential geometric structures on a smooth manifold Mn can be
equivalently expressed by a notion of ‘admissible’ coframings, with the property that
3 The notion of a characteristic system can apply to hyperbolic exterior differential systems ingeneral. See [BGH95].
13
any two such coframings differ pointwise by the action of a Lie group G Ă GLpn,Rq.
For example, a Riemannian metric g on Mn can be viewed as all local coframings
pω1, ω2, ..., ωnq satisfying g “ pω1q2 ` ¨ ¨ ¨ ` pωnq2, any two such coframings (defined
on the same domain) differing by a pointwise action of the orthogonal group Opnq;
a symplectic form Ω on M2n can be viewed as all local coframings pω1, ω2, ..., ω2nq
satisfying Ω “ ω1 ^ ω2 ` ¨ ¨ ¨ ` ω2n´1 ^ ω2n, any two such coframings differing by a
pointwise action of the symplectic group Spp2nq; an almost complex structure J on
M2n is characterized by any basis of p1, 0q-forms on M , whose real and imaginary
parts comprising a local coframing, any two such coframings differing by a pointwise
action of GLpn,Cq Ă GLp2n,Rq; etc. This motivates the following definition.
Definition 2.8. Let G Ă GLpn,Rq be a Lie subgroup. A G-structure on a smooth
manifold Mn is a principal G-subbundle of the coframe bundle FpMq.
When dealing with equivalence between two local geometric structures defined in
terms of coframes, it is necessary to take into account the ambiguity in the choice
of admissible coframings. For example, Let pMi, giq pi “ 1, 2q be two Riemannian
n-manifolds, where gi “ pω1piqq
2 ` ¨ ¨ ¨ ` pωnpiqq
2. These Riemannian structures are
locally equivalent if and only if there exists a (local) diffeomorphism φ : U1 Ñ U2
(Ui Ă Mi open) such that φ˚ωp2q “ γωp1q for some γ : U1 Ñ Opnq, where ωpiq “
pω1piq, ω
2piq, ..., ω
npiqq pi “ 1, 2q. However, the ambiguity represented by γ can be removed
by passing to G-structures, as we will describe below.
Definition 2.9. Let G be aG-structure onM , with π : G ÑM being the submersion.
The tautological 1-form τ on G is the Rn-valued 1-form determined by the equation
τpvq “ upπ˚pvqq, for any u P G and v P TuG.
Definition 2.10. Two G-structures, G1 and G2, with tautological 1-forms τ1, τ2,
respectively, are said to be equivalent if there exists a diffeomorphism φ : G1 Ñ G2,
such that φ˚τ2 “ τ1.
14
The following proposition is proved in [Gar89].
Proposition 2.1. Let Mi pi “ 1, 2q be two smooth n-manifolds, each with a cofram-
ing ωi defined on some open neighborhood Ui Ă Mi. Let G Ă GLpn,Rq be a Lie
subgroup. Let Gi be the G-structure on Ui defined by Gi “ tωippq ¨ h| p P Ui, h P Gu.
There exists a diffeomorphism φ : U1 Ñ U2 and a map g : U1 Ñ G such that
φ˚ω2 “ gω1 if and only if there exists an equivalence of G-structures φ : G1 Ñ G2.
Example 5. Consider a hyperbolic Monge-Ampere system pM, Iq. A local coframing
Θ “ pθ0, θ1, ..., θ4q defined on an open neighborhood U Ă M is said to be 0-adapted
if, on U , the differential ideal I can be expressed as
I “ xθ0, θ1^ θ2, θ3
^ θ4y.
This is a pointwise condition on Θ. Moreover, given any two 0-adapted coframings,
on a common domain, they must relate by a pointwise action of the subgroup G0 Ă
GLp5,Rq generated by matrices of the form
g “
¨
˝
a 0 0b1 A 0b2 0 B
˛
‚, a ‰ 0; A,B P GLp2,Rq; b1,b2 P R2,
and
J “
¨
˝
1 0 00 0 I2
0 I2 0
˛
‚.
Consequently, two hyperbolic Monge-Ampere systems are equivalent if and only if the
corresponding G0-structures are equivalent. This treatment of a differential system
as a geometric structure is what we mean by “geometry of differential systems”.
To close this section, let us mention a useful property of the tautological 1-form
of a G-structure on M : the so-called reproducing property.
15
Proposition 2.2. Let G be a G-structure on M . Let τ be the tautological 1-form on
G. For any local section σ : U Ñ G pU ĂM openq, we have
σ˚τ “ σ,
where, on the left-hand-side of the equality, σ is regarded as a differentiable map,
whereas, on the right-hand-side, it is regarded as a coframing on M .
In particular, if certain differential conditions are satisfied by admissible cofram-
ings, then the reproducing property leads to corresponding restrictions on the equa-
tions satisfied by the tautological 1-form on a G-structure.
2.4 Backlund Transformations
We follow [AF15] to define Backlund transformations, though we will, in later chap-
ters, mostly be concerned with those relating hyperbolic Monge-Ampere systems.
For the latter, a definition can be found in Chapter 4 of [BGG03].
Definition 2.11. Let pM, Iq be an exterior differential system. A rank-k inte-
grable extension of pM, Iq is an exterior differential system pN,J q with a submer-
sion π : N Ñ M that satisfies the condition: for each p P N , there exists an open
neighborhood U Ă N pp P Uq such that
p1q on U , the differential ideal J is algebraically generated by elements of π˚I
together with 1-forms θ1, ..., θk P Ω1pUq, where k “ dimN ´ dimM ;
p2q for any p P U , let Fp denote the fiber π´1pπppqq; the 1-forms θ1, ..., θk restrict
to TpFp to be linearly independent.
In Definition 2.11, roughly speaking, π : N Ñ M is a bundle and J defines a
‘connection’ on this bundle that is flat over the integral manifolds of I. In more
detail, condition p1q implies that, if S Ă M is an integral manifold of pM, Iq, then
J restricts to π´1pSq to be a Frobenius system; hence, π´1pSq is foliated by integral
16
manifolds of pN,J q. Condition p2q implies that, restricting to any integral manifold
of pN,J q, π is an immersion, whose image is an integral manifold of pM, Iq.
Example 6. Let pM, Iq be an exterior differential system. The obvious submersion
π : MˆRÑM induces an integrable extension pMˆR, xπ´1I, dtyq of pM, Iq, where
t is a coordinate on the R-factor.
Proposition 2.3. The composition of two integrable extensions is an integrable ex-
tension.
Proof. Let π1 : pM1, I1q Ñ pM2, I2q and π2 : pM2, I2q Ñ pM3, I3q be integrable
extensions. By definition, I2 is algebraically generated by π˚2I3 and certain 1-forms
α1, ..., αp; I1 is thus algebraically generated by pπ2 ˝ π1q˚I3, π˚1α1, ..., π
˚1αp and q 1-
forms β1, ..., βq. Clearly, for π2 ˝ π1, the first condition in the definition of integrable
extensions is satisfied. To check the second condition, suppose that there exist con-
stants ci, fj, x P M1 such that´
řpi“1 ciπ
˚1αi `
řqj“1 fjβj
¯
pvq “ 0 for any v P TxM1
satisfying π2˚pπ1˚pvqq “ 0. Since π1 is an integrable extension, each TxM1 is a direct
sum of V1 :“ kerxpπ1q and V2 :“ kerxpβ1, ..., βqq. For the previous equality to be
satisfied on V1, all fj must be equal to zero. Since π1˚ restrict to V2 to be an linear
isomorphism, for the equality to hold on V2X kerppπ2 ˝π1q˚q, ci must all be zero.
Definition 2.12. A Backlund transformation relating two exterior differential sys-
tems, pM1, I1q and pM2, I2q, is a quadruple pN,B; π1, π2q where, for each i P t1, 2u,
πi : N Ñ Mi makes pN,Bq an integrable extension of pMi, Iiq. Such a Backlund
transformation is represented by the diagram
pN,Bq
pM1, I1q pM2, I2q
π1 π2
17
Definition 2.13. In Definition 2.12, if M1,M2 have the same dimension, which
is not required in general, then the rank of pN,B; π1, π2q is the fiber dimension of
either π1 or π2. If pMi, Iiq pi “ 1, 2q are equivalent exterior differential systems, then
pN,B; π1, π2q is called an auto-Backlund transformation of either pMi, Iiq.
By Definitions 2.11 and 2.12, it is clear that, given a Backlund transformation
pN,B; π1, π2q relating pMi, Iiq pi “ 1, 2q, we can start with an integral manifold S
of pM1, I1q; restrict B to π´11 S, which becomes a Frobenius system; then solve this
Frobenius system and project, by π2, each leaf associated to it into M2. As a result,
one produces a family of integral manifolds of pM2, I2q. By Remark 1, it is easy to
see that only ODE methods are used in this process. This is what we mean by “a
Backlund transformation allows one to use a known solution of a PDE system and
ODE methods to obtain solutions of a second PDE system.”
Example 7. Let π : pN,J q Ñ pM, Iq be an integrable extension. The quadruple
pN,J ; π, πq is an auto-Backlund transformation of pM, Iq. However, such a Backlund
transformation does not produce new integral manifolds of pM, Iq from a given one.
Example 8. In Example 4, the orthonormal frame bundle O can be viewed as a
subgroup of GLp4,Rq; an element px; e1, e2, e3q P O corresponds to the matrix
ˆ
1 0 0 0x e1 e2 e3
˙
P GLp4,Rq.
Let K be a negative constant. Let θ, r be constants satisfying K “ ´ sin2 θr2. The
element
g “
¨
˚
˚
˝
1 0 0 0r 1 0 00 0 cos θ sin θ0 0 ´ sin θ cos θ
˛
‹
‹
‚
P O
induces a map ψg : O Ñ O defined by
ψgpuq “ ug, @u P O Ă GLp4,Rq.
18
Let I Ă Ω˚pOq be the differential ideal generated by elements of I and ψ˚gI. One can
verify that the quadraple pO, I; π, π ˝ ψgq defines an auto-Backlund transformation
of the system pM,J q. This is the classical Backlund transformation relating surfaces
in E3 with a negative constant Gauss curvature ([Bac83]).
Definition 2.14. Given a fiber bundle π : E Ñ B, for any p P E, the vertical
tangent space of E at p is the kernel of π˚ : TpE Ñ TπppqB.
Definition 2.15. A Backlund transformation pN,B; π1, π2q is called nontrivial if the
two fibrations π1, π2 have distinct vertical tangent spaces at each point p P N .
If pN,B; π1, π2q is a nontrivial Backlund transformation of rank k, then one can
show that, given any p-dimensional integral manifold S of pM1, I1q, π2pπ´11 Sq has
a dimension strictly greater than p; thus, π2pπ´11 Sq must be foliated by a family of
p-dimensional integral manifolds of pM2, I2q.
Example 9. Let pN,B; π, πq be a rank-1 Backlund transformation relating two hy-
perbolic Monge-Ampere systems pM, Iq and pM, Iq. On some open subsets U ĂM
and U Ă M , we have
I “ xη0, η1^ η2, η3
^ η4y, I “ xη0, η1
^ η2, η3^ η4
y.
Let V “ π´1UX π´1U . It is easy to see that the Cauchy characteristics of the system
xπ˚η0y pdefined on V q are precisely the fibers of π|V ; similarly for π|V . Therefore,
if pN,B; π, πq is nontrivial, then π˚η0 and π˚η0 must be linearly independent when
restricted to each tangent space of N . In particular, it follows that, on V , the
differential ideal B is algebraically generated by elements of π˚I and π˚η0, which
needs to be the same as the system algebraically generated by elements of π˚I and
π˚η0.
Definition 2.16. A nontrivial rank-1 Backlund transformation relating two hyper-
bolic Monge-Ampere systems is said to be normal if, in the notations of Example 9,
19
on N , the system rrdη0, dη0ss has rank 2 modulo η0, η0.
Consider two Backlund transformations, one relating pM1, I1q and pM2, I2q, the
other relating pM2, I2q and pM3, I3q, as the following diagram shows.
pN1,B1q pN2,B2q
pM1, I1q pM2, I2q pM3, I3q
π1 π2 π3 π4
The Whitney sum of the fiber bundles π2 : N1 ÑM2 and π3 : N2 ÑM2, denoted as
N1 ‘N2, admits two submersions
p1 : N1 ‘N2 Ñ N1, p2 : N1 ‘N2 Ñ N2
such that π2˝p1 “ π3˝p2. Let B denote the differential ideal on N1‘N2 algebraically
generated by p˚1B1 and p˚2B2.
Proposition 2.4. pN1‘N2,B; π1 ˝ p1, π4 ˝ p2q is a Backlund transformation relating
pM1, I1q and pM3, I3q.
Proof. By taking into account Proposition 2.3, it suffices to show that p1 and
p2 are integrable extensions. Let B1 be algebraically generated by α1, ..., αk and
π˚2I2, B2 by β1, ..., βl and π˚3I2. Hence, B is algebraically generated by p˚1α1, ..., p˚1αk,
p˚2β1, ..., p˚2βl and p˚1π
˚2I2 (the latter being the same as p˚2π
˚3I2). Now suppose that
v is tangent to a fiber of p1. By construction, p2˚v is tangent to a fiber of π3.
If p˚2pβiqpvq “ 0 for all i “ 1, ..., l, it is necessary that p2˚v “ 0, because π3 is
an integrable extension. Since p2, restricting to each fiber of p1, is an immersion,
we have v “ 0. This prove that p1 is an integrable extension. The case for p2 is
similar.
The above discussion suggests that, for exterior differential systems, being Backlund-
related is an equivalence relation. However, it is unknown whether this remains true
if one restricts to the notion of being Backlund-related at a particular rank.
20
We close this section with two more definitions, which will be useful later.
Definition 2.17. Let π : N ÑM be a submersion. A p-form on N that takes value
in π˚pΛppT ˚Mqq is said to be a π-semi-basic p-form.
Definition 2.18. Let M be a smooth manifold. Let E Ă ΛkpT ˚Mq be a vector
subbundle, and X a smooth vector field defined on M . We say that E is invariant
under the flow of X if, for any psmoothq local section ω : U Ñ E, where U Ă M is
open, the Lie derivative LXω remains a section of E defined on U .
Remark 2. It is easy to see that, to verify the condition in Definition 2.18, it suffices
to make a choice of basis sections σ1, ..., σk : U Ñ E passuming that E has rank kq,
and verify that, for each i, LXσi is a linear combination of σ1, ..., σk.
21
3
The Problem of Generality
In this chapter, the objects of study are nontrivial rank-1 Backlund transformations
relating a pair of hyperbolic Monge-Ampere systems. Since many classical exam-
ples belong to this category, it is highly desirable to have a complete classification of
Backlund transformations of this kind. In [Cle01], by establishing a G-structure asso-
ciated to a Backlund transformation, Clelland approached the classification problem
using Cartan’s Method of Equivalence, restricting to the case when all local invariants
of the structure are constants (a.k.a the homogeneous case). Her classification found
15 types, among which 11 are analogues of the classical Backlund transformation
between surfaces in E3 with a negative constant Gauss curvature.
Since homogeneous structures, up to equivalence, only depend on constants, the
following question remains to be answered: What kind of initial data do we need
to specify in order to determine a Backlund transformation? In Section 3.2, in the
generic case, we prove an upper bound for the magnitude of such initial data. In
Section 3.3, we provide several examples of Backlund transformations with higher
cohomogeneity, which we found by specifying only two structure invariants.
22
3.1 G-structure Equations for Backlund Transformations
According to Definition 2.12, it may appear that pN,B; π1, π2q being a Backlund
transformation imposes conditions on all components in this quadruple. However,
when it is a nontrivial rank-1 Backlund transformation relating two hyperbolic
Monge-Ampere systems, one only needs to impose conditions on the exterior dif-
ferential system pN,Bq, as the following proposition shows.
Proposition 3.1. ([Cle01]) An exterior differential system pN6,Bq is a nontrivial
rank-1 Backlund transformation relating two hyperbolic Monge-Ampere systems if
and only if, for each p P N , there exists an open neighborhood V Ă N pp P V q and a
coframing pθ0, θ0, θ1, ..., θ4q, defined on V , satisfying the conditions:
p1q the differential ideal B “ xθ0, θ0, θ1 ^ θ2, θ3 ^ θ4yalg;
p2q the vector bundles E0 “ rrθ0ss, E1 “ rrθ0, θ1, θ2ss and E2 “ rrθ0, θ3, θ4ss are
invariant along the flow of X (see Definition 2.18), where X is a nonvanishing
vector field on V that annihilates θ0, θ1, ..., θ4;
p2q1 the vector bundles E0 “ rrθ0ss, E1 “ rrθ0, θ1, θ2ss and E2 “ rrθ0, θ3, θ4ss are
invariant along the flow of X, where X is a nonvanishing vector field on V that
annihilates θ0, θ1, ..., θ4;
p3q for some nonvanishing functions A1, ..., A4 defined on V ,
dθ0” A1θ
1^ θ2
` A2θ3^ θ4 mod θ0,
dθ0” A3θ
1^ θ2
` A4θ3^ θ4 mod θ0.
Proof. In one direction, assume that pN,B; π, πq is a nontrivial rank-1 Backlund
transformation relating two hyperbolic Monge-Ampere systems pM, Iq and pM, Iq.
Maintaining the notation in Example 9 of Chapter 2, and dropping the pull-back
symbol when there is no confusion, we have that each p P N has an open neighbor-
23
hood V such that, on V ,
B “ xη0, η0, η1^ η2, η3
^ η4yalg “ xη
0, η0, η1^ η2, η3
^ η4yalg.
Thus, there exist nonvanishing functions Ai pi “ 1, ..., 4q defined on V such that
dη0” A1η
1^ η2
` A2η3^ η4 mod η0,
dη0” A3η
1^ η2
` A4η3^ η4 mod η0, η0. (3.1)
By adding appropriate multiples of η0 to η1, ..., η4, we can put (3.1) in the form
dη0” A3η
1^ η2
` A4η3^ η4 mod η0. (3.2)
It is easy to see that the resulting coframing pη0, η0, η1, ..., η4q satisfies the conditions
p1q and p3q. Next, we show that it also satisfies the conditions p2q and p2q1.
Using the congruence (3.2), it is easy to see that the Cartan system1
Cpxη0yq “ xη0, η1, ..., η4
y Ă Ω˚pV q.
As a result, π˚I is included in the intersection
Cpxη0yq X B “ xη0, η1
^ η2, η3^ η4
yalg,
Consequently, by switching pη1, η2q and pη3, η4q if needed, we have the following
relations of rank-3 vector bundles over V :
rrη0, η1, η2ss “ rrη0, η1, η2
ss, (3.3)
rrη0, η3, η4ss “ rrη0, η3, η4
ss. (3.4)
Let X be a vector field on V annihilated by η0, η1, ..., η4. By construction, the right-
hand-sides of (3.3) and (3.4) are invariant under the flow of X; it follows that the
same holds for rrη0, η1, η2ss and rrη0, η3, η4ss. This proves that pη0, η0, η1, ..., η4q satisfies
Condition p2q1. The verification of Condition p2q is similar.
1 Let pM, Iq be an exterior differential system, the Cartan system of pM, Iq is the Frobeniussystem that annihilates the Cauchy characteristic distribution of pM, Iq. See [BCG`13] for details.
24
For the other direction, assume that pθ0, θ0, θ1, ..., θ4q is a coframing defined on
an open set V Ă N and satisfying the conditions p1q-p3q. The quotient of V by the
integral curves of xθ0, θ1, ..., θ4y is a 5-manifold U . Let f : V Ñ U be the projection.
By Condition p2q, there exists a system I “ xη0, η1 ^ η2, η3 ^ η4y defined on U
satisfying
rrf˚η0ss “ rrθ0
ss, rrf˚η0, f˚η1, f˚η2ss “ rrθ0, θ1, θ2
ss, rrf˚η0, f˚η3, f˚η4ss “ rrθ0, θ3, θ4
ss.
It is then easy to show, using Conditions p1q and p3q, that I “ xη0, η1^η2, η3^η4yalg
is a hyperbolic Monge-Ampere ideal. A similar argument applies to the quotient of
V by the integral curves of xθ0, θ1, ..., θ4y. It follows that pN,Bq is a nontrivial rank-1
Backlund transformation relating two hyperbolic Monge-Ampere systems.
Corollary 3.1. Let pN6,B; π1, π2q be a nontrivial rank-1 Backlund transformation
relating two hyperbolic Monge-Ampere systems. A coframing defined on an open
subset V Ă N that satisfies Conditions p1q-p3q in Proposition 3.1 can always be
arranged to satisfy the extra condition: A2 “ A3 “ 1.
Proof. This is obtained by scaling θ0 and θ0.
Definition 3.1. A coframing as concluded in Corollary 3.1 is said to be 0-adapted
to the Backlund transformation pN,Bq.
Given a nontrivial rank-1 Backlund transformation pN,B; π1, π2q, one may wonder
whether its 0-adapted coframings are precisely the local sections of a G-structure
over N . However, this is not true. For example, consider a 0-adapted coframing
pθ0, θ0, θ1, ..., θ4q defined on an open subset U Ă N with corresponding functions
A1, A4. Let T : U Ñ GLp6,Rq be the transformation
T : pθ0, θ0, θ1, θ2, θ3, θ4q ÞÑ pA´1
1 θ0, A´14 θ0, θ3, θ4, θ1, θ2
q. (3.5)
The coframing on the right-hand-side is clearly 0-adapted. However, the same trans-
formation, when applied to a 0-adapted coframing with corresponding functions
25
A11, A14 that are different from A1, A4, may not result in a 0-adapted coframing.
One simple strategy, as taken by [Cle01], to avoid this imperfection is by, in
addition to understanding the subbundles rrθ0ss and rrθ0ss as an ordered pair, fixing
an order for the pair of subbundles rrθ0, θ0, θ1, θ2ss and rrθ0, θ0, θ3, θ4ss. Once this is
considered, all local 0-adapted coframings respecting such an ordering are precisely
the local sections of a G-structure, where G Ă GLp6,Rq is the Lie subgroup consisting
of matrices of the form
g “
¨
˚
˚
˝
detpBq 0 0 00 detpAq 0 00 0 A 00 0 0 B
˛
‹
‹
‚
, A “ paijq, B “ pbijq P GLp2,Rq. (3.6)
Now let G denote this G-structure on N . Let ω “ pω1, ω2, ..., ω6q be the tautolog-
ical 1-form on G. Let g be the Lie algebra of G. Using the conditions in Proposition
3.1 and the reproducing property, one can show that ω satisfies the following struc-
ture equations, recorded from [Cle01] with a slight change of notation:
where rrα, β1, β2ss Ă T ˚G1 has rank 3 and is transversal to the fibers of G1, and
Ω0“ Aω1
^ ω2` ω3
^ pω4` γq ` pP0ω
0`Kγ ` Piω
iq ^ ω0,
Ω0“ ω1
^ pω2` γq ` Aω3
^ ω4` pQ0ω
0` Lγ `Qiω
iq ^ ω0, (5.16)
Γ “ ω0^ ω0
` Ciωi^ ω0
`Diωi^ ω0, pA ą 1q
for functions A,Ci, Di, P0, K, Pi, Q0, L,Qi pi “ 1, ..., 4q defined on G1. In particular, Γ
takes the form above because one can add any linear combination of ω0, ω0, γ, ω1, ..., ω4
to α without changing the form of (5.15) and (5.16).
Let U Ă N be a sufficiently small domain. Let σ : U Ñ G1 be any section. The
pull-back by σ of the tautological 1-form on G1 must then satisfy the Invariance
Property (see (5.13) and (5.14)).
Lemma 5.3. pAq If we express the 2-forms Ω1,Ω2 under the basis tω0^ ω0, ..., ω3^
ω4u, the coefficients of the following terms are zero:
γ ^ ω3, γ ^ ω4, γ ^ ω0, ω0^ ω3, ω0
^ ω4.
Proof. By construction, the vector fields X0, Xγ are annihilated by the 1-forms
ω0, ω1, ω2, ω3, ω4 ` γ and are dual to ω0 and γ. Using this and (5.15), we obtain
LX0ω0” 0 mod ω0,
LX0ωi ” X0 Ωi mod ω0, ω1, ω2
pi “ 1, 2q,
LXγω0” 0 mod ω0,
LXγωi ” Xγ Ωi mod ω0, ω1, ω2pi “ 1, 2q.
The conclusion then follows from the Invariance Property.
By similar arguments, we can prove
Lemma 5.3. pBq If we express the 2-forms Ω1 and Ω2´γ^C2ω0`pC3ω
3`C4ω4q^ω0
under the basis tω0^ ω0, ..., ω3^ω4u, the coefficients of the terms in each pair below
77
are the same:
pγ ^ ω0, ω2^ ω0
q, pγ ^ ω3, ω2^ ω3
q, pγ ^ ω4, ω2^ ω4
q;
the coefficients of the following terms are zero:
ω0^ ω3, ω0
^ ω4.
pCq In the expressions of Ω3 and Ω4, the coefficients of the following terms are zero:
γ ^ ω1, γ ^ ω2, γ ^ ω0, ω0^ ω1, ω0
^ ω2.
pDq In the expressions of Ω3 and Ω4´γ^D4ω0`pD1ω
1`D2ω2q^ ω0, the coefficients
of the terms in each pair below are the same:
pγ ^ ω1, ω4^ ω1
q, pγ ^ ω2, ω4^ ω2
q, pγ ^ ω0, ω4^ ω0
q;
the coefficients of the following terms are zero:
ω0^ ω1, ω0
^ ω2.
By Lemma 5.3, Ω1, ...,Ω4 must take the form
Ω1“ T 1
0ω0^ ω0
` ω0^ pT 1
1ω1` T 1
2 pω2` γqq ` ω0
^ pT 11ω
1` T 1
2ω2q
` γ ^ pR11ω
1`R1
2ω2q `
1
2T 1ijω
i^ ωj, (5.17)
Ω2“ T 2
0ω0^ ω0
` ω0^ pC3ω
3` C4ω
4` T 2
1ω1` T 2
2 pω2` γqq ` ω0
^ pT 21ω
1` T 2
2ω2q
` γ ^ pC2ω0`R2
1ω1`R2
2ω2q `
1
2T 2ijω
i^ ωj, (5.18)
Ω3“ T 3
0 ω0^ ω0
` ω0^ pT 3
3ω3` T 3
4 pω4` γqq ` ω0
^ pT 33ω
3` T 3
4ω4q
` γ ^ pR33ω
3`R3
4ω4q `
1
2T 3ijω
i^ ωj, (5.19)
Ω4“ T 4
0 ω0^ ω0
` ω0^ pD1ω
1`D2ω
2` T 4
3ω3` T 4
4 pω4` γqq ` ω0
^ pT 43ω
3` T 4
4ω4q
` γ ^ pD4ω0`R4
3ω3`R4
4ω4q `
1
2T 4ijω
i^ ωj, (5.20)
78
where T kij “ ´Tkji, and T 1
23, T124, T
223, T
224, T
314, T
324, T
414, T
424 are zero.
The coefficients in Ω1, ...,Ω4 are not all determined. In fact, by adding appropriate
linear combinations of ω0, ω0, γ, ω1, ..., ω4 to β1, we can arrange that
T 212 “ T 2
13 “ T 214 “ T 2
1 “ T 21 “ R2
1 “ 0;
in a similar manner, by adjusting β2, we can arrange that
T 43 “ T 4
3 “ R43 “ T 4
13 “ T 423 “ T 4
34 “ 0.
As a standard step in the method of equivalence, we apply d2 “ 0 to (5.15) and
reduce modulo appropriate differential forms. From this, we obtain relations among
the torsion functions (i.e., the coefficients of Ω0, Ω0,Γ, ...,Ω4).
In particular, expanding the expressions
dpdω0q mod ω0, ω1, dpdω0
q mod ω0, ω2,
dpdω0q mod ω0, ω3, dpdω0
q mod ω0, ω4,
and
dpdγq mod γ, ω0, ω0,
we find the following relations:
T 323 “ P2, T 1
34 “ 0, T 334 “ ´P4 `K ´R3
3 ´R44, D4 “ ´P0 ` T
33 ` T
44 ,
T 313 “ P1, T 2
34 “ 0, T 114 “ ´Q4, T 3
12 “ 0,
T 112 “ ´Q2 ` L´R
11 ´R
22, C2 “ ´Q0 ` T
11 ` T
22 , T 1
13 “ ´Q3, T 412 “ 0,
D2 “ ´1
AC2, C4 “ ´
1
AD4, D1 “ ´
1
AC1, C3 “ ´
1
AD3,
In the equations above, we replace the functions on the left-hand-side by expres-
sions on the right-hand-side. Then we compute
dpdω0q mod ω0, dpdω0
q mod ω0.
79
From this, we find that the expression of dA is determined:
dA
A“ pQ0 ´ T
33 ´ T
44 qω
0` pP0 ´ T
11 ´ T
22 qω
0´ pP1 ´Q1qω
1´ pP2 ´Q2qω
2
` pP3 ´Q3qω3` pP4 ´Q4qω
4` pL´R3
3 ´R44qγ,
with the extra condition: K ´R11´R
22 “ L´R3
3´R44. This shows that the function
A pA ą 1q is an invariant of the G1-structure.
Homogeneity Assumption: Now we assume that the underlying rank-2 Backlund
transformation is homogeneous, that is, its symmetry group acts locally transitively.
Making this assumption will imply that any local structure invariant is a constant.
Following from the homogeneity assumption, dA “ 0. This implies
P3 “ Q3, P4 “ Q4, Q1 “ P1, Q2 “ P2,
K “ R11 `R
22, P0 “ T 1
1 ` T22 , L “ R3
3 `R44, Q0 “ T 3
3 ` T44 .
Further differentiation of the structure equations yields
dpdω1q ” T 1
2ω3^ ω4
^ γ mod ω0, ω0, ω1, ω2,
dpdω1q ” pAT 1
2 ` T12 qω
3^ ω4
^ ω2 mod ω0, ω0, ω1, γ
dpdω2q ”
1
ArAp´T 3
3 ´ T44 ` T
11 q
´ T 33 ´ T
44 ` T
11 ` T
22 sω
3^ γ ^ ω4
` T 20ω
3^ pω4
` γq ^ ω0 mod ω0, ω1, ω2,
and
dpdω3q ” T 3
4ω1^ ω2
^ γ mod ω0, ω0, ω3, ω4,
dpdω3q ” pAT 3
4 ` T34 qω
1^ ω2
^ ω4 mod ω0, ω0, ω3, γ,
dpdω4q ”
1
ArAp´T 1
1 ´ T22 ` T
33 q
´ T 11 ´ T
22 ` T
33 ` T
44 sω
1^ γ ^ ω2
` T 40ω
1^ pω2
` γq ^ ω0 mod ω0, ω3, ω4.
80
These congruences imply
T 12 “ T 1
2 “ T 20 “ T 3
4 “ T 34 “ T 4
0 “ 0,
T 11 “ T 3
3 ` T44 `
1
A2 ´ 1pAT 4
4 ` T22 q, T 3
3 “ T 11 ` T
22 `
1
A2 ´ 1pAT 2
2 ` T44 q.
Using these, we compute
dpdω1q ” T 1
0ω3^ pω4
` γq ^ ω0 mod ω0, ω1, ω2,
dpdω3q ” T 3
0ω1^ pω2
` γq ^ ω0 mod ω0, ω3, ω4,
which implies
T 10 “ T 3
0 “ 0.
In addition, we have
dpdω1q ” pdR1
2 ´ 5R12αq ^ γ ^ ω
2 mod ω0, ω0, ω1, ω3, ω4, (5.21)
dpdω1q ”
R12
A2 ´ 1pAT 4
4 ` T22 qγ ^ ω
0^ ω4
`R1
2D3
Aγ ^ ω0
^ ω3 mod ω1, ω2, (5.22)
dpdω3q ” pdR3
4 ´ 5R34αq ^ γ ^ ω
4 mod ω0, ω0, ω1, ω2, ω3, (5.23)
dpdω3q ”
R34
A2 ´ 1pAT 2
2 ` T44 qγ ^ ω
0^ ω2
`R3
4
AC1γ ^ ω
0^ ω1 mod ω3, ω4. (5.24)
Lemma 5.4. R12 and R3
4 are both zero on an open subset of G1.
Proof. Suppose that locally R12 ‰ 0. The equation (5.22) then implies
D3 “ 0, T 22 “ ´AT
44 .
Following from this, we have
dpdγq ” ´Aω3^ ω4
^ ω0 mod γ, ω1, ω2, ω0,
which is impossible because A ą 1. Therefore, R12 “ 0. An analogous argument
leads to R34 “ 0.
81
Note that there remains freedom to add a multiple of γ to α without changing
the form of the structure equations. Using this, we can arrange
R11 `R
22 “ ´pR
33 `R
44q.
Recall that K “ R11`R
22. We now remind the reader that all torsion coefficients are
expressed in terms of the constant A and the functions
K, P1, P2, Q3, Q4, T 11 , T 2
2 , T 33 , T 4
4 ,
C1, T 22 , T 4
4 , D3; R11, R3
3.
The torsion cannot be absorbed further.
By applying d2 “ 0 to the structure equations, we can find how3 G1 acts on these
remaining torsion functions. For simplicity, we introduce the new notation:
F1 :“A
A2 ´ 1pAT 2
2 ` T44 q F3 :“
A
A2 ´ 1pAT 4
4 ` T22 q.
Infinitesimally, the G1-action on the torsion functions can be expressed as:
dD3 ” ´2αD3 ` F3β2, dF3 ” αF3,
dC1 ” ´2αC1 ` F1β1, dF1 ” αF1,
dP1 ” β1P2 ´ αP1, dP2 ” 2αP2,
dQ3 ” β2Q4 ´ αQ3, dQ4 ” 2αQ4,
dK ” 2αK, dR11 ”2αR1
1, dR33 ” 2αR3
3,
dT 11 ” αT 1
1 , dT 22 ” αT 2
2 , dT 33 ” αT 3
3 , dT 44 ” αT 4
4 ,
where all congruences are modulo the semi-basic 1-forms ω0, ω0, γ, ω1, ..., ω4.
3 What we can obtain by applying d2 “ 0 to the structure equations is essentially an infinitesimalversion of the G1-action on the torsion components. Hence, it only tells us how the identitycomponent of G1 acts. Performing a structure reduction using the action by the identity componentof G1 thus may not distinguish two equivalent coframings that differ by a discrete transformation.However, this would not pose a problem, as we can examine whether there is such an equivalenceat the end of our classification.
82
In the proof of Lemma 5.4, we have seen that, when pD3, F3q or pC1, F1q locally
vanishes, we have a contradiction. In particular, D3 cannot locally vanish, otherwise,
by the equation (5.22), F3 must vanish. For a similar reason, C1 cannot locally vanish.
Hence, for the pair of torsion functions pD3, F3q (and similarly for the pair pC1, F1q),
there are two possibilities:
I. Locally on G1, D3 ‰ 0, F3 “ 0. This implies that G1 scales D3;
II. Locally on G1, F3 ‰ 0. By the G1-action on F3, it is easy to see that
F3 ‰ 0 on the entire group fibers. In this case, it is possible to reduce to
a G12-structure G 12 Ă G1, defined by D3 “ 0. Restricting to G 12, the 1-form
β2 becomes semi-basic.
Lemma 5.5. Case I is empty.
Proof. If F3 “ 0, then D3 scales under the G1-action. We can then reduce to the
subbundle defined by either D3 “ 1 or D3 “ ´1. Now let G2 Ă G1 be defined by
D3 “ 1.
On G2, α is semi-basic. In other words, there exist functions H0, H0, Hγ, H1, ..., H4
on G2 such that
α “ H0ω0` H0ω
0`Hγγ `H1ω
1` ...`H4ω
4.
In addition, the following functions are constant along each fiber of G2
K, P2, Q4, R11, R3
3, T 11 , T 2
2 , T 33 , T 4
4 , F1.
By the homogeneity assumption, they must be constants on G2.
We now compute
dpdγq ”1
Ap2AH2 ` 2AP2 ´ F1H3 ` F1Q3qω
2^ ω3
^ ω0
` p1´ 2Q4 ` 2K ´ 2H4qω3^ ω4
^ ω0
`1
AF1pH4 ´Q4qω
4^ ω2
^ ω0 mod ω0, γ, ω1.
83
If F1 ‰ 0, we must have Q4 “ H4 and 2K “ 4H4 ´ 1. Using this, we compute
dpdγq ” ´1
ApA2
´ 1qω0^ ω3
^ ω4 mod ω0, γ, ω1, ω2,
which is impossible since A ą 1.
It follows that, F1 “ 0 on G2. With this, we compute
dpdγq ” 2pP2ω2`Q4ω
4q ^ ω0
^ ω0` ¨ ¨ ¨ mod γ, ω1.
Therefore, P2 “ Q4 “ 0. Using this, we have
dpdγq ” p1`2K´2H4qω3^ω4
^ω0´
1
ApA2`2K´2H4qω
0^ω3
^ω4 mod γ, ω1, ω0^ω0.
This implies A2 “ 1, which is impossible, by assumption.
By a similar argument, one can show that there is no homogeneous structure in
the case when C1 ‰ 0 and F1 “ 0. This, together with Lemma 5.5, implies that
the only case remaining is when both F3 and F1 are locally nonzero. In this case,
one can reduce to the subbundle on which D3 “ C1 “ 0. There, β1 and β2 both
become semi-basic. Further, we can reduce to an e-structure on which F3 “ 1. On
this e-structure, F1 is a nonzero constant, by homogeneity.
Lemma 5.6. The case when both F3 and F1 are locally nonzero is empty.
Proof. On the e-structure defined by D3 “ C1 “ 0 and F3 “ 1, the 1-forms α, β1
and β2 are semi-basic:
α “ H0ω0` H0ω
0`Hγγ `H1ω
1` ...`H4ω
4,
β1 “M0ω0` M0ω
0`Mγγ `M1ω
1` ...`M4ω
4,
β2 “ N0ω0` N0ω
0`Nγγ `N1ω
1` ...`N4ω
4.
Differentiation gives
dpdγq ” ´1
ApA2
´H3 `N4 `Q3qω3^ ω4
^ ω0` p1´H3 `N4 `Q3qω
3^ ω4
^ ω0
mod γ, ω1, ω2, ω0^ ω0.
84
Clearly, A2 “ 1, which is a contradiction.
Combining Lemmas 5.5 and 5.6, the following theorem is immediate.
Theorem 5.7. There exist no homogeneous rank-2 Backlund transformation satis-
fying all three genericity conditions, ε “ 1, pB1, B2q ‰ 0, and pB3, B4q ‰ 0.
Remark 10. pAq In Theorem 5.7, the condition ε “ 1 can be removed. The case
when ε “ ´1 has a proof that is only a slight modification of the arguments above.
pBq It still remains to be answered whether there exists a (non-homogeneous)
rank-2 Backlund transformation satisfying all three genericity conditions and pB1, B2q ‰
0, pB3, B4q ‰ 0.
5.1.2 Case 2: Bi “ 0 pi “ 1, ..., 4q
In this case, we prove the
Proposition 5.2. Suppose that pN,Bq is a rank-2 Backlund transformation (not
necessarily homogeneous) satisfying all three genericity conditions, ε “ 1 and Bi “ 0
pi “ 1, ..., 4q, then pN,Bq admits a 6-dimensional quotient that is a rank-1 Backlund
transformation relating the same pair of hyperbolic Monge-Ampere systems.
Proof. Using previous notation, the vector fields Xγ, Yγ coincide when Bi “ 0
pi “ 1, ..., 4q. Shrink N if needed, and let N 1 be the quotient of N by the flow
of Xγ. The Invariance Property ((5.13) and (5.14)) then implies that the vector
bundles rrω0, ω1, ω2ss, rrω0, ω3, ω4ss, rrω0, ω1, ω2ss and rrω0, ω3, ω4ss annihilate Xγ and
are invariant under the flow of Xγ. Hence, their intersections are the pull-backs of
vector subbundles of T ˚N 1. In particular, there locally exist 1-forms η0, η0, η1, ..., η4
on N 1 such that the following relations hold (dropping the pull-back symbol): rrω0ss “
rrη0ss, rrω0ss “ rrη0ss, rrω1, ω2ss “ rrη1, η2ss, rrω3, ω4ss “ rrη3, η4ss. This implies that pull-
back of the corresponding Monge-Ampere systems are respectively xη0, η1^η2, η3^η4y
85
and xη0, η1^η2, η3^η4y. It follows that pN 1,B1q, with B1 “ xη0, η0, η1^η2, η3^η4y and
the obvious submersions to M, M , defines a rank-1 Backlund transformation.
Remark 11. One can prove that the conclusion in Proposition 5.2 remains true when
ε “ ´1.
5.2 Assuming Genericity Conditions 1, 2
Without assuming the third genericity condition, we need a new definition of 0-
adapted coframings.
Definition 5.3. Let pN,B; π, πq be a rank-2 Backlund transformation (relating two
hyperbolic Monge-Ampere systems) satisfying only the first two genericity conditions
and ε “ 1. A coframing pω0, ω0, γ, ω1, ..., ω4q defined on an open subset U Ă N is
said to be 0-adapted if it satisfies
rrω0ss “ rrπ˚θss, rrω0
ss “ rrπ˚θss, rrω0, ω0, γss “ B1
and
dω0” Aω1
^ ω2` ω3
^ ω4` pB3ω
3`B4ω
4q ^ γ mod ω0, (5.25)
dω0” ω1
^ ω2` Aω3
^ ω4` pB1ω
1`B2ω
2q ^ γ mod ω0, (5.26)
dγ ” Ciωi^ ω0
`Diωi^ ω0 mod γ, (5.27)
with A ą 1.
Lemma 5.8. Given a rank-2 Backlund transformation pN,Bq (relating two hyper-
bolic Monge-Ampere systems) satisfying only the first two genericity conditions and
ε “ 1, its 0-adapted coframings are local sections of a G-structure G over N , where
G Ă GLp7,Rq is generated by
g “
¨
˚
˚
˚
˚
˝
detpbq 0 0 0 00 detpaq 0 0 00 0 c 0 00 0 0 a 00 0 0 0 b
˛
‹
‹
‹
‹
‚
,detpaq “ detpbq ‰ 0, c ‰ 0
a “ paijq, b “ pbijq P GLp2,Rq.(5.28)
86
Proof. We omit this proof since the arguments are similar to Lemma 5.1.
It is easy to show that, on G,
ˆ
B1
B2
˙
pu ¨ gq “c
detpaqaT
ˆ
B1
B2
˙
puq,
ˆ
B3
B4
˙
pu ¨ gq “c
detpbqbT
ˆ
B3
B4
˙
puq.
It follows that one can normalize Bi pi “ 1, ..., 4q to be one of the following:
Case 1: B1 “ B3 “ 1, B2 “ B4 “ 0;
Case 2: Bi “ 0 pi “ 1, ..., 4q;
Case 3: B2, B3, B4 “ 0, B1 “ 1;
Case 31: B1, B2, B4 “ 0, B3 “ 1.
Lemma 5.9. A rank-2 Backlund transformation in the current case arises as a
1-parameter family of rank-1 Backlund transformations relating the same pair of
hyperbolic Monge-Ampere systems if and only if γ is integrable.
Proof. Suppose that γ is integrable. It is easy to see that each leaf of γ is a
rank-1 Backlund transformation. Conversely, suppose that pN,Bq is constructed in
the natural way from a 1-parameter family of rank-1 Backlund transformations. Let
t be the parameter. Then we have that B1 is generated by the pull-back of θ, θ and
dt. It follows that, given a local 0-adapted coframing on N , there exists a linear
combination ψ “ γ ` λω0 ` µω0 that is integrable. Computing dψ and reducing
modulo ω0, ω0 and γ, it is easy to see, by (5.25)-(5.27), that λ and µ are zero. It
follows that γ must be integrable.
We will only be interested in the case when γ is not integrable. In the current
case, this is given by the condition: locally the functions Ci, Di pi “ 1, ..., 4q are not
all zero.
87
5.2.1 Case: pB1, B2q “ pB3, B4q “ p1, 0q
In this case, we reduce to a G1-structure G1 Ă G, where the subgroup G1 Ă G is
of G2 with nonzero weights in φ only. If they all vanish, then it can be verified that
4 Let G Ă GLpn,Rq be a Lie subgroup, and let G be a G-structure over M . A G-equivariantfunction f : G Ñ R, where G acts on R linearly, is called a relative invariant of G.
93
the structure equations are incompatible with the identity d2 “ 0. On the other
hand, instead of asking which of these relative invariants are nonzero and going into
various cases, we can always choose a nonzero function U on G2, expressed in terms
of these relative invariants, satisfying
dU ” Uφ mod ω0, ω0, γ, ω1, ..., ω4.
Then there exist constants hγ, h2, h3, h4, n0, n0, nγ, n2, n4 such that
Hγ “ hγU,
H2 “ h2U,
H3 “ h3U´1,
H4 “ h4U,
N0 “ n0U´2,
N0 “ n0U´2,
Nγ “ nγU´1,
N2 “ n2U´1,
N4 “ n4U´1,
and constants `, c1, t34, r
34, r
44 such that
L “ `U, C1 “ c1U´2, T 3
4 “ t34U2, R3
4 “ r34U
3, R44 “ r4
4U.
Since U is expressed in terms of the relative invariants and has weight 1 in φ, we can
perform a structure reduction that leads to a subbundle G3 defined by U “ 1.
On G3, φ is semi-basic, namely,
φ “ Z0ω0` Z0ω
0` Zγγ `
4ÿ
i“1
Ziωi,
for Z0, Z0, ..., Z4 defined on G3. We find that
dZ0 ” 0, dZ0 ” 0, dZγ ” 0,
dZ1 ” Z2β1, dZ2 ” 0, dZ3 ” 0, dZ4 ” 0,
all congruences being modulo the semi-basic 1-forms. By homogeneity, Z0, Z0, Zγ,
Z2, Z3, Z4 are constants. Taking this into account, it can be verified that the structure
equations are incompatible with d2 “ 0.
94
Consequently, D4 must be zero on G1. Moreover, note that the equations (5.25)-
(5.27) allows us to switch pω0, ω1, ω2q with pω0, ω3, ω4q; applying this, pC1, C2, C3, C4q
exchanges with pD3, D4, D1, D2q; and pB1, B2q exchanges with pB3, B4q. Since we are
in a case when pB1, B2q “ pB3, B4q, we can conclude from Lemma 5.11 that
Lemma 5.12. Assuming homogeneity, C2 must be zero on G1.
Therefore, the functions D3 and C1 are relative invariants of G1, both having
weight 2 in α.
Case of D3 ‰ 0.
Without loss of generality, we can assume that D3 ‰ 0 on G1. By Lemma 5.10,
R12, T
12 must be zero.
Depending on the sign of D3, we can reduce to the subbundle G2 defined by either
D3 “ 1 or D3 “ ´1. On G2, there exist functions H0, H0, Hγ, H1, ..., H4 such that
α “ H0ω0` H0ω
0`Hγγ `
4ÿ
i“1
Hiωi, (5.42)
Moreover, the torsion functions on G1 restrict to G2 to satisfy
dL ” φL, dT 33 ” T 3
3 φ, dT 33 ” T 3
3 φ`1
2T 3
4 β2,
dC1 ” 0, C2 “ 0, D3 “ ˘1, D4 “ 0,
T 12 “ 0, dT 3
4 ” 2T 34 φ, R1
2 “ 0, dR34 ” 2R3
4φ, dR44 ” R4
4φ´1
2R3
4β2,
(5.43)
where all congruences are modulo the semi-basic 1-forms. By homogeneity, C1 must
be a constant; L, T 33 , T
34 , R
34 are now relative invariants. We have two cases:
I. L, T 33 , T
34 , R
34 are all zero;
II. not all of L, T 33 , T
34 , R
34 are zero.
95
Now consider the case when G2 is defined by D3 “ 1. A “`” sign will be used to
indicate that we are in this case.
`I. If L, T 33 , T
34 , R
34 are identically zero on G2, then, by (5.43), T 3
3 , R44 are relative
invariants. It is easy to verify that
dH0 ” H0φ,
dH0 ” H0φ,
dHγ ” Hγφ,
dH1 ” H2β1,
dH2 ” H2φ,
dH3 ” H4β2,
dH4 ” H4φ,
(5.44)
modulo the semi-basic 1-forms. We can always choose U to be a function defined on
G2, satisfying
dU ” Uφ, mod ω0, ω0, γ, ω1, ..., ω4,
in the following manner: noting that T 33 , R
44, H0, Hγ, H0, H2, H4 are relative invariants
with nonzero weights in φ only, if they are all zero, we simply choose U “ 0; otherwise,
we choose U be an appropriate combination of these relative invariants satisfying the
equation above and the condition U ‰ 0. There are thus two subcases to consider:
`I1. T 33 , R
44, H0, Hγ, H0, H2, H4 are all zero;
`I2. not all of T 33 , R
44, H0, Hγ, H0, H2, H4 are zero.
`I1. In this case, by (5.44) and the homogeneity assumption, H1 and H3 are con-
stants on G2. We compute
dpdγq “ ´dφ^ γ ` p1´ C1qω1^ ω3
^ γ
´2
ApAC1H3 `H1qω
3^ ω1
^ ω0`
2
ApAH1 ` C1H3qω
3^ ω1
^ ω0.
Since A ą 1, we must have
H1 “ C1H3 “ 0; (5.45)
in particular, either C1 or H3 is zero.
96
Lemma 5.13. C1 is nonzero.
Proof. If C1 “ 0 on G2, then
dpdω0q “ ´dφ^ ω0
` ω3^ ω1
^ ω0, (5.46)
dpdω0q “ ´dφ^ ω0
` ω3^ ω1
^ ω0, (5.47)
dpdγq “ ´dφ^ γ ` ω1^ ω3
^ γ. (5.48)
The equation (5.46) and (5.47) imply that, on G2, there exists a function K such
that
dφ “ ω3^ ω1
`Kω0^ ω0,
which, however, is incompatible with (5.48). The conclusion follows.
By Lemma 5.13 and (5.45), we must have H3 “ 0 on G2. Applying d2 “ 0 to the
structure equations yields
dpdω0q “ ´dφ^ ω0
` p1´ C1qω3^ ω1
^ ω0, (5.49)
dpdω0q “ ´dφ^ ω0
` p1´ C1qω3^ ω1
^ ω0, (5.50)
dpdγq “ ´dφ^ γ ` p1´ C1qω1^ ω3
^ γ. (5.51)
Among these three equations, (5.49) and (5.50) imply that there exists a function K
on G2 satisfying
dφ “ p1´ C1qω3^ ω1
`Kω0^ ω0;
in order for (5.51) to hold, we must have
C1 “ 1, K “ 0.
As a result, φ is (determined and) integrable. Restricting to a leaf of xφy, we have
φ “ 0. Such a leaf then can be regarded as a bundle (over an open subset U Ă N)
97
on which the following structure equations are satisfied (A ą 1 being a constant)
dω0 “ Aω1 ^ ω2 ` ω3 ^ pω4 ` γq,
dω0 “ ω1 ^ pω2 ` γq ` Aω3 ^ ω4,
dγ “ ω1 ^
ˆ
ω0 ´1
Aω0
˙
` ω3 ^
ˆ
ω0 ´1
Aω0
˙
,
dω1 “ 0,
dω2 “ ´β1 ^ ω1 ´
1
Aω0 ^ ω3,
dω3 “ 0,
dω4 “ ´β2 ^ ω3 ´
1
Aω0 ^ ω1.
(5.52)
Remark 12. In terms of the method of equivalence, one can check that the structure
equation (5.52) has constant torsion and involutive tableau, the Cartan characters of
the tableau being p2, 0, ..., 0q. By a theorem of Cartan, any two Backlund transfor-
mations in this case are equivalent; the symmetry of such a Backlund transformation
depends on 2 functions of 1 variable.
We have thus proven:
Proposition 5.3. Up to equivalence, there is a unique local model for a rank-2
homogeneous Backlund transformation in Case p`I1q.
Remark 13. We will see later that the Backlund transformation corresponding to
(5.52) is an auto-Backlund transformation of the linear equation zxy “ z.
`I2. In this case, by the construction of U , there exist constants t33, r44, h0, h0, hγ, h2, h4,
such that
T 33 “ t33U, R4
4 “ r44U,
H0 “ h0U, H0 “ h0U, Hγ “ hγU, H2 “ h2U, H4 “ h4U.
Moreover, it is easy to see that one can reduce to the subbundle G3 defined by U “ 1.
98
On G3, there exist functions Z0, Z0, Zγ, Z1, ..., Z4 such that
φ “ Z0ω0` Z0ω
0` Zγγ `
4ÿ
i“1
Ziωi,
By applying d2 “ 0 to the structure equations, one can verify that
dZ0 ” 0,
dZ0 ” 0,
dZγ ” 0,
dZ1 ” Z2β1,
dZ2 ” 0,
dZ3 ” Z4β2,
dZ4 ” 0,
(5.53)
modulo semi-basic 1-forms. Therefore, by homogeneity, Z0, Z0, Zγ, Z2, Z4 are con-
stants. Using the structure equations, we find
dpdγq ” ´2h2ω2^ ω3
^ ω0´ 2h4ω
4^ ω3
^ ω0 mod ω0, γ, ω1.
Evidently, h2 and h4 must be zero. By (5.44) and the homogeneity assumption, H1
and H3 are constants.
By expanding
dpdω0q, dpdω0
q, dpdγq, dpdω2q, dpdω4
q all mod ω1, ω3,
dpdω1q mod ω3, dpdω3
q mod ω1
we find that either Z2, Z4 are both zero, or h0, h0, hγ, r44, t
33 are all zero.
However, noting that h2, h4 are already zero, we cannot have h0, h0, hγ, r44, t
33 to
be all zero in the current case (by the assumption for Case (`I2)).
Now assume that Z2, Z4 are both zero. In this case, Z1, Z3 become invari-
ants, by (5.53), and are thus constants by the homogeneity assumption. Expanding
where, after adding a linear combination of the semi-basic 1-forms to φ, one can
arrange
Ω0“ ω1
^ ω2` ω3
^ ω4` ω3
^ γ ` pP0ω0`Kγ ` Piω
iq ^ ω0,
Ω0“ ω1
^ ω2` ω3
^ ω4` ω1
^ γ ` pQ0ω0` Lγ `Qiω
iq ^ ω0,
Γ “ ω1^ ω2
´ ω3^ ω4
` Ciωi^ ω0
`Diωi^ ω0.
5 One can verify that the classical Backlund transformation (1.5) relating solutions of the hyper-bolic Tzitzeica equation (1.4) belongs Case 1. By computing the corresponding structure invariants,one can show that it is non-homogeneous.
112
Lemmas 5.2 and 5.3 apply to the current case without change. Following from
this, Ω1, ...,Ω4 have expressions (5.17)-(5.20) where T kij “ ´Tkji for all k, i, j “ 1, ..., 4,
and T 123, T
124, T
223, T
224, T
314, T
324, T
414, T
424 are zero.
Furthermore, one can add a linear combination of the semi-basic 1-forms to α to
arrange
T 21 “ T 2
1 “ R21 “ T 2
12 “ T 213 “ T 2
14 “ 0.
By adjusting β, we can arrange
T 43 “ T 4
3 “ R43 “ T 4
13 “ T 423 “ T 4
34 “ 0
Finally, by adding a suitable multiple of γ to φ, we can arrange
K “ ´L.
The torsion cannot be absorbed further.
Applying d2 “ 0 to the structure equations, we obtain the congruences
Applying d2 “ 0 to (4.1) and reducing appropriately, we obtain
d2ω1” pQ63 ´Q04qω
0^ ω3
^ ω4 mod ω1, ω2,
d2ω2” pQ03 ´Q74qω
0^ ω3
^ ω4 mod ω1, ω2,
d2ω3” pQ02 `Q61qω
0^ ω1
^ ω2 mod ω3, ω4,
d2ω4” p´Q01 ´Q72qω
0^ ω1
^ ω2 mod ω3, ω4.
This implies that
Q61 “ ´Q02, Q63 “ Q04, Q72 “ ´Q01, Q74 “ Q03.
Now all coefficients in the structure equation (4.1) are expressed in terms of Q0i
pi “ 0, 1, ..., 4q and Qj0 pj “ 5, 6, 7q. By applying d2 “ 0 to (4.1), it is not hard to
verify that, reduced modulo ω0, ω1, ..., ω4, the following congruences hold:
d
ˆ
Q01 Q03
Q02 Q04
˙
”
ˆ
φ1 φ3
φ2 ´φ1
˙ˆ
Q01 Q03
Q02 Q04
˙
, dpQ00q ” 0,
d
¨
˝
Q50
Q60
Q70
˛
‚”
¨
˝
0 φ3 ´φ2
2φ2 ´2φ1 0´2φ3 0 2φ1
˛
‚
¨
˝
Q50
Q60
Q70
˛
‚. (B.3)
135
The equations (B.3) tell us how the remaining Qij transform under the action by
the identity component of H. Moreover, it is easy to compute directly from (4.1) to
verify that
ˆ
Q01 Q03
Q02 Q04
˙
pu ¨ h0q “
ˆ
´Q01 Q03
Q02 ´Q04
˙
puq, Q00pu ¨ h0q “ ´Q00puq, (B.4)
¨
˝
Q50
Q60
Q70
˛
‚pu ¨ h0q “
¨
˝
´Q50
Q60
Q70
˛
‚puq, h0 “ diagp´1,´1, 1, 1,´1q P H
hold for any u P H.
Note that H is generated by its identity component and h0. Combining (B.3)
and (B.4), it is easy to see that Q01Q04´Q02Q03 and |Q00| are local invariants of the
underlying Euler-Lagrange system.
Moreover, using (B.3) and (B.4), it is easy to see that the H-orbit of
qpuq :“
ˆ
Q01 Q03
Q02 Q04
˙
puq, u P H
consists of all 2-by-2 matrices with the same determinant as qpuq. Now we are ready
to prove the
Lemma B.1. If detpqq ‰ 0 on H, then there is a canonical way to define a coframing
on U .
Proof. If the function L :“ detpqq is nonvanishing on U , one can reduce to the
subbundle H1 of H defined by q “ diagpL, 1q. It is easy to see that each fiber of H1
over U contains a single element.
Remark 15. As a result of Lemma B.1, if detpqq ‰ 0 on U , then the corresponding
hyperbolic Euler-Lagrange system has a symmetry of dimension at most 5. This is
a consequence of applying the Frobenius Theorem.
136
Now we proceed to complete proving Proposition 3.4. Recall that the coframing
pη0, η1, ..., η4q and the φα in (3.19) verify the equation (4.1), S1 “ diagp1, 1q, and
S2 “ 0. Moreover, we have chosen the φα to satisfy (B.2), where Qij are computed
using (B.1). By (3.19), it is immediate that
Q00 “ Q02 “ 0, Q01 “ ´Q04 “1?
2, Q03 “ 1,
Q70 “ 1, Q60 “ ´1, Q50 “?
2.
Clearly, detpqq “ Q01Q04 ´ Q02Q03 “ ´1
2‰ 0. By Lemma B.1 and Remark 15,
the hyperbolic Euler-Lagrange system considered in Proposition 3.4 has a symmetry
of dimension at most 5. Since such an Euler-Lagrange system is homogeneous, it
follows that its symmetry has dimension 5.
137
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Biography
Yuhao Hu was born on Jan 8, 1991 in Jingdezhen, China. During 2008-2012, he
was a student at the Shing-Tung Yau class of mathematics at Zhejiang University,
where he obtained a Bachelor of Science degree in June, 2012. After graduation, he
went to Duke University for PhD studies in mathematics. He conducted research in
differential geometry and the geometry of differential equations under the supervision
of Professor Robert Bryant. After graduating from Duke, he will work as a postdoc
with Professor Jeanne Clelland at the University of Colorado at Boulder. As a hobby,