Geometry of B\"acklund Transformations I: Generality

TL;DR

This paper uses Cartan's Method of Equivalence to analyze the generality and classification of rank-1 B"acklund transformations between hyperbolic Monge-Ampère systems, providing new examples and bounds.

Contribution

It establishes an upper bound on the generality of generic rank-1 B"acklund transformations and classifies cases with symmetry groups of codimension 1, 2, or 3, introducing new auto-B"acklund examples.

Findings

Upper bound for the generality of rank-1 B"acklund transformations

Classification results for transformations with symmetry groups of codimension 1, 2, or 3

New examples of auto-B"acklund transformations

Abstract

Using Cartan's Method of Equivalence, we prove an upper bound for the generality of generic rank-1 B\"acklund transformations relating two hyperbolic Monge-Amp\`ere systems. In cases when the B\"acklund transformation admits a symmetry group whose orbits have codimension 1, 2, or 3, we obtain classification results and new examples of auto-B\"acklund transformations.

Equations

$$\left\{\begin{array}[]{l}u_{x}-v_{y}=0,\\[8.00003pt] u_{y}+v_{x}=0\end{array}\right.$$

$$\left\{\begin{array}[]{l}z_{x}-\bar{z}_{x}=\lambda\sin(z+\bar{z}),\\[8.00003pt] z_{y}+\bar{z}_{y}=\lambda^{-1}\sin(z-\bar{z}),\end{array}\right.$$

$$u_{xy}=\frac{1}{2}\sin (2u).$$

$$(A^2-B^2)(z_{xx}-z_{yy})+4AB{z}_{xy}=0,$$

$$[x_2,x_3]=x_1,[x_3,x_1]=x_2,[x_1,x_2]=0.$$

$$(A^2-B^2)(z_{xx}-z_{yy})+4AB{z}_{xy}+(A^2+B^2{)}^2=0,$$

$$A(z_{xx}{z}_{yy}-z_{xy}^2)+B{z}_{xx}+2C{z}_{xy}+D{z}_{yy}+E=0,$$

$$\mathcal{I}=⟨{\theta }^0,{\theta }^1\wedge {\theta }^2,{\theta }^3\wedge {\theta }^4⟩.$$

$$g=\left(\begin{array}[]{ccc}a&{\bf 0}&{\bf 0}\\ {\bf b}_{1}&A&0\\ {\bf b}_{2}&0&B\end{array}\right),\qquad a\neq 0;~{}A,B\in{\rm GL}(2,\mathbb{R});~{}{\bf b}_{1},{\bf b}_{2}\in\mathbb{R}^{2},$$

$$J=\left(\begin{array}[]{ccc}1&{\bf 0}&{\bf 0}\\ {\bf 0}&0&I_{2}\\ {\bf 0}&I_{2}&0\end{array}\right).$$

$$\mathcal{I}=⟨{\eta }^0,{\eta }^1\wedge {\eta }^2,{\eta }^3\wedge {\eta }^4⟩,\bar{\mathcal{I}}=⟨{\bar{\eta }}^0,{\bar{\eta }}^1\wedge {\bar{\eta }}^2,{\bar{\eta }}^3\wedge {\bar{\eta }}^4⟩.$$

$$\displaystyle\mathrm{d}\left(\begin{array}[]{c}\omega^{0}\\ \omega^{1}\\ \omega^{2}\\ \omega^{3}\\ \omega^{4}\end{array}\right)$$

$$\displaystyle\qquad\qquad+\left(\begin{array}[]{c}\omega^{1}\wedge\omega^{2}+\omega^{3}\wedge\omega^{4}\\ (V_{1}+V_{5})\omega^{0}\wedge\omega^{3}+(V_{2}+V_{6})\omega^{0}\wedge\omega^{4}\\ (V_{3}+V_{7})\omega^{0}\wedge\omega^{3}+(V_{4}+V_{8})\omega^{0}\wedge\omega^{4}\\ (V_{8}-V_{4})\omega^{0}\wedge\omega^{1}+(V_{2}-V_{6})\omega^{0}\wedge\omega^{2}\\ (V_{3}-V_{7})\omega^{0}\wedge\omega^{1}+(V_{5}-V_{1})\omega^{0}\wedge\omega^{2}\end{array}\right),$$

$$g=\left(\begin{array}[]{ccc}a&{\bf 0}&{\bf 0}\\ {\bf 0}&A&0\\ {\bf 0}&0&B\end{array}\right),~{}A,B\in{\rm GL}(2,\mathbb{R}),~{}a=\det(A)=\det(B),$$

$$J=\left(\begin{array}[]{ccc}1&{\bf 0}&{\bf 0}\\ {\bf 0}&0&I_{2}\\ {\bf 0}&I_{2}&0\end{array}\right)\in{\rm GL}(5,\mathbb{R}).$$

$$S_{1}:=\left(\begin{array}[]{cc}V_{1}&V_{2}\\ V_{3}&V_{4}\end{array}\right),\quad S_{2}:=\left(\begin{array}[]{cc}V_{5}&V_{6}\\ V_{7}&V_{8}\end{array}\right).$$

$$S_i(u\cdot g)=a{A}^{-1}{S}_i(u)B,(i=1,2)$$

$$S_{1}(u\cdot J)=\left(\begin{array}[]{cc}-V_{4}&V_{2}\\ V_{3}&-V_{1}\end{array}\right),\quad S_{2}(u\cdot J)=\left(\begin{array}[]{cc}V_{8}&-V_{6}\\ -V_{7}&V_{5}\end{array}\right).$$

$${\Sigma }_1:=V_3{\omega }^1{\omega }^3-V_1{\omega }^1{\omega }^4+V_4{\omega }^2{\omega }^3-V_2{\omega }^2{\omega }^4$$

$${\Sigma }_2:=V_7{\omega }^1\wedge {\omega }^3-V_5{\omega }^2\wedge {\omega }^3+V_8{\omega }^1\wedge {\omega }^4-V_6{\omega }^2\wedge {\omega }^4$$

$$\mathrm{d}S_{i}\equiv\left(\begin{array}[]{cc}\phi_{4}&-\phi_{2}\\ -\phi_{3}&\phi_{1}\end{array}\right)S_{i}+S_{i}\left(\begin{array}[]{cc}\phi_{5}&\phi_{6}\\ \phi_{7}&\phi_{8}\end{array}\right)\mod\omega^{0},\omega^{1},\ldots,\omega^{4}.$$

$$\mathrm{d}{\theta }^0$$

$$\mathrm{d}{\bar{\theta }}^0$$

$$T(p):({\theta }^0,{\bar{\theta }}^0,{\theta }^1,{\theta }^2,{\theta }^3,{\theta }^4)\mapsto \left(\frac{1}{A_1}{\theta }^0,\frac{1}{A_4}{\bar{\theta }}^0,{\theta }^3,{\theta }^4,{\theta }^1,{\theta }^2\right),\forall p\in U.$$

$$g$$

$$\text{A}=(a_{ij}),\text{B}=(b_{ij})\in \mathrm{GL}(2,\mathbb{R}).$$

$$\displaystyle\mathrm{d}\left(\begin{array}[]{c}\omega^{1}\\ \omega^{2}\\ \omega^{3}\\ \omega^{4}\\ \omega^{5}\\ \omega^{6}\end{array}\right)=$$

$$\displaystyle+\left(\begin{array}[]{c}A_{1}(\omega^{3}-C_{1}\omega^{1})\wedge(\omega^{4}-C_{2}\omega^{1})+\omega^{5}\wedge\omega^{6}\\ \omega^{3}\wedge\omega^{4}+A_{4}(\omega^{5}-C_{3}\omega^{2})\wedge(\omega^{6}-C_{4}\omega^{2})\\ B_{1}\omega^{1}\wedge\omega^{2}+C_{1}\omega^{5}\wedge\omega^{6}\\ B_{2}\omega^{1}\wedge\omega^{2}+C_{2}\omega^{5}\wedge\omega^{6}\\ B_{3}\omega^{1}\wedge\omega^{2}+C_{3}\omega^{3}\wedge\omega^{4}\\ B_{4}\omega^{1}\wedge\omega^{2}+C_{4}\omega^{3}\wedge\omega^{4}\end{array}\right),$$

$$A_1(u\cdot g)=\frac{\det (\text{A})}{\det (\text{B})}$$

$$\displaystyle\left(\begin{array}[]{c}B_{1}\\ B_{2}\end{array}\right)(u\cdot g)$$

$$\displaystyle\left(\begin{array}[]{c}B_{3}\\ B_{4}\end{array}\right)(u\cdot g)$$

$$\displaystyle\left(\begin{array}[]{c}C_{1}\\ C_{2}\end{array}\right)(u\cdot g)$$

$$\displaystyle\left(\begin{array}[]{c}C_{3}\\ C_{4}\end{array}\right)(u\cdot g)$$

$$W_1:=\mathrm{span}((B_1,B_2),(C_1,C_2)),W_2:=\mathrm{span}((B_3,B_4),(C_3,C_4))$$

Full text

1. 1 Introduction
2. 2 Definitions and Notations
3. 3 Monge-Ampère Systems and Their First Invariants
4. 4 $G$-structure Equations for Bäcklund Transformations
5. 5 An Estimate of Generality
6. 6 Classifications and Examples in Higher Cohomogeneity
7. 7 Concluding Remarks
8. 8 Acknowledgement
9. A Calculations in Theorem 5.2
10. B Invariants of an Euler-Lagrange System

1. Introduction

In 1882, the Swedish mathematician Albert V. Bäcklund proved the result (see [Bäc83], [BGG03] or [CT80]): Given a surface with a constant Gauss curvature $K<0$ in $\mathbb{E}^{3}$, one can construct, by solving ODEs, a 1-parameter family of new surfaces in $\mathbb{E}^{3}$ with the Gauss curvature $K$. This is the origin of the term “Bäcklund transformation”.

Classically, a Bäcklund transformation is a PDE system $\mathcal{B}$ that relates solutions of two other PDE systems $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$. Moreover, such a relation must satisfy the property: given a solution $u$ of $\mathcal{E}_{1}$ (resp. $\mathcal{E}_{2}$), substituting it in $\mathcal{B}$, one would obtain a PDE system whose solutions can be found by ODE methods and produce solutions of $\mathcal{E}_{2}$ (resp. $\mathcal{E}_{1}$). If, in addition, $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ are contact equivalent to each other, then the corresponding Bäcklund transformation is called an auto-Bäcklund transformation.

For example, the Cauchy-Riemann system

[TABLE]

is an auto-Bäcklund transformation; it relates solutions of the Laplace equation $\Delta z=0$ for $z(x,y)$ in the following way: If $u$ satisfies $\Delta u=0$, then, substituting it in (1), one obtains a compatible first-order system for $v$, whose solutions can be found by ODE methods and satisfy $\Delta v=0$, and vice versa.

As another example, consider the system of nonlinear equations

[TABLE]

where $\lambda$ is a nonzero constant. One can easily verify that (2) is an auto-Bäcklund transformation relating solutions of the sine-Gordon equation

[TABLE]

The system (2) can be derived from the classical auto-Bäcklund transformation relating surfaces in $\mathbb{E}^{3}$ with a negative constant Gauss curvature. For details, see [CT80].

In general, there may seem to be very few restrictions on the types of PDE systems that admit a Bäcklund transformation. In addition to the elliptic and hyperbolic examples mentioned above, a Bäcklund transformation may exist relating solutions of a parabolic equation (see [NC82]) or an equation of order higher than 2, for example, the KdV equation (see [WE73]). Furthermore, two PDE systems being Bäcklund-related need not be contact equivalent to each other (see [CI09]).

The importance of Bäcklund transformations may, in part, be viewed through their relation to surface geometry and mathematical physics. On the geometry side, Bäcklund transformations allow one to obtain new surfaces with prescribed geometric properties from old. For a variety of such examples, see [RS02]. On the mathematical physics side, a prototypical result is that the Bäcklund transformation (2), when applied to the trivial solution $z(x,y)=0$ of (3), yields a 2-parameter family of 1-soliton solutions of the sine-Gordon equation (see [TU00]). More elaborate techniques have since been developed to find the so-called multi-soliton solutions of nonlinear PDE systems (for example, the KdV equation), using Bäcklund transformations.

An ultimate goal of studying Bäcklund transformations is solving the Bäcklund problem, which was considered by É. Goursat in [Gou25]:

Find all pairs of systems of PDEs whose solutions are related by a Bäcklund transformation.

Although this problem remains largely unsolved, recent works of Clelland and Ivey ([Cle02],[CI05] and [CI09]) and those of Anderson and Fels ([AF12], [AF15]) have pointed out new directions for studying Bäcklund transformations. Instead of aiming at constructing new examples or finding techniques of calculating explicit solutions to PDE systems, they work in a geometric setting that is natural to the study of structural properties (of Bäcklund transformations) that are invariant under contact transformations. Under such settings, a complete classification of Bäcklund transformations, at least in certain cases, is possible by using É. Cartan’s method of equivalence.

The current work is concerned with the geometric aspect of Bäcklund transformations, not so much in the sense of relating to classical surface geometry, as in that of seeing Bäcklund transformations as geometric objects and studying their invariants.

More specifically, we study nontrivial rank-$1$ Bäcklund transformations (see Definition 2.10) relating a pair of hyperbolic Monge-Ampère systems. Since many classical examples belong to this category, it is highly desirable to have a complete classification of Bäcklund transformations of this kind. In [Cle02], by establishing a $G$-structure associated to a Bäcklund 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, within which 11 are analogues of the classical Bäcklund transformation between surfaces in $\mathbb{E}^{3}$ with a negative constant Gauss curvature.

Since homogeneous structures, up to equivalence, depend only on constants, the following question remains to be answered: What kind of initial data do we need to specify in order to determine a rank-1 Bäcklund transformation relating two hyperbolic Monge-Ampère systems?

In Section 5, in the generic case, we use the method of equivalence (see [Gar89], [Bry14]) to prove an upper bound for the magnitude of such initial data:

To determine a generic Bäcklund transformation relating two hyperbolic Monge-Ampère systems, it is sufficient to specify at most $6$ functions of $3$ variables.

It is an immediate consequence of our theorem that most hyperbolic Monge-Ampère systems are not related to any system of the same type by a generic Bäcklund transformation.

A major difficulty in using the method of equivalence to classify Bäcklund transformations lies in verifying the compatibility of large systems of polynomial equations, whose variables are the Bäcklund structure invariants and their covariant derivatives. However, we found that such calculation becomes much more manageable when we set two structure invariants to be specific constants (see Section 6). The corresponding Bäcklund transformations are either homogeneous (corresponding to a case already classified in [Cle02]) or of cohomogeneity 1, 2 or 3. In the cohomogeneity-1 case, we obtain an auto-Bäcklund transformation of a homogeneous Euler-Lagrange system that is contact equivalent to the equation

[TABLE]

where $A=2z_{x}+y$ and $B=2z_{y}-x$. The cohomogeneity-2 case has a subcase that arises when the ‘free derivatives’ associated to the structure are expressed in terms of the primary invariants. In this subcase, the corresponding Lie algebra of symmetry must be of the form $\mathfrak{q}\oplus\mathbb{R}$ where $\mathfrak{q}$ is either $\mathfrak{sl}(3,\mathbb{R})$, $\mathfrak{so}(3,\mathbb{R})$ or the solvable 3-dimensional Lie algebra with basis $\{x_{i}\}_{i=1}^{3}$ satisfying

[TABLE]

In particular, when $\mathfrak{q}$ is solvable, we obtain an auto-Bäcklund transformation; the underlying Monge-Ampère system is Euler-Lagrange, of cohomogeneity 1, and contact equivalent to the equation

[TABLE]

where $A=z_{x}-y$, $B=z_{y}+x$.

2. Definitions and Notations

In this section, we present some definitions and notations to be used later.

2.1. Exterior Differential Systems (c.f. [BCG*+*13])

Definition 2.1**.**

Let $M$ be a smooth manifold, $\mathcal{I}\subset\Omega^{*}(M)$ a graded ideal that is closed under exterior differentiation. The pair $(M,\mathcal{I})$ is said to be an exterior differential system with space $M$ and differential ideal $\mathcal{I}$.

Given an exterior differential system $(M,\mathcal{I})$, we use $\mathcal{I}^{k}$ to denote the degree-$k$ piece of $\mathcal{I}$, namely, $\mathcal{I}^{k}=\mathcal{I}\cap\Omega^{k}(M)$, where $\Omega^{k}(M)$ stands for the $C^{\infty}(M)$-module of differential $k$-forms on $M$. If the rank of $\mathcal{I}^{k}$, restricted to each point, is locally a constant, then the elements of $\mathcal{I}^{k}$ are precisely smooth sections of a vector bundle denoted by $I^{k}$.

Definition 2.2**.**

An integral manifold of an exterior differential system $(M,\mathcal{I})$ is an immersed submanifold $i:N\hookrightarrow M$ satisfying $i^{*}\phi=0$ for any $\phi\in\mathcal{I}$.

Intuitively, an exterior differential system is a coordinate-independent way to express a PDE system; an integral manifold, usually with a certain independence condition satisfied, corresponds to a solution of the PDE system.

Definition 2.3**.**

Two exterior differential systems $(M,\mathcal{I})$ and $(N,\mathcal{J})$ are said to be equivalent up to diffeomorphism, or equivalent, for brevity, if there exists a diffeomorphism $\phi:M\rightarrow N$ such that $\phi^{*}\mathcal{J}=\mathcal{I}.$ Such a $\phi$ is called an equivalence between the two systems. An equivalence between $(M,\mathcal{I})$ and itself is called a symmetry of $(M,\mathcal{I})$.

Definition 2.4**.**

Let $\pi:N\rightarrow M$ be a submersion. A $p$-form $\omega\in\Omega^{p}(N)$ is said to be $\pi$-semi-basic if, for any $x\in N$, $\omega|_{x}\in\pi^{*}(\Lambda^{p}(T^{*}M))$.

Definition 2.5**.**

Let $M$ be a smooth manifold. Let $E\subset\Lambda^{k}(T^{*}M)$ 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 $($smooth$)$ local section $\omega:U\rightarrow E$, where $U\subset M$ is open, the Lie derivative $\mathcal{L}_{X}\omega$ remains a section of $E$ (over $U$).

Notation 1**.**

Let $[\![\theta_{1},\ldots,\theta_{\ell}]\!]$ denote the vector subbundle of $\Lambda^{k}(T^{*}U)$ generated by differential forms $\theta_{1},\ldots,\theta_{\ell}$ (defined on $U$, an open subset of a smooth manifold) of the same degree $k$.

2.2. Hyperbolic Monge-Ampère Systems (c.f. [BGG03])

Among second order PDEs for 1 unknown function of 2 independent variables, Monge-Ampère equations are those of the form

[TABLE]

where $A,B,C,D,E$ are functions of $x,y,z,z_{x},z_{y}$. A Monge-Ampère equation (4) is said to be elliptic (resp., hyperbolic, parabolic) if $AE-BD+C^{2}$ is negative (resp., positive, zero).

A Monge-Ampère equation can be formulated as an exterior differential system on a contact manifold. In the hyperbolic case, we follow [BGH95] to give the following definition.

Definition 2.6**.**

A hyperbolic Monge-Ampère system $(M,\mathcal{I})$ is an exterior differential system, where $M$ is a 5-manifold, $\mathcal{I}$ being locally algebraically generated by $\theta\in\mathcal{I}^{1}$ and $\mathrm{d}\theta,\Omega\in\mathcal{I}^{2}$ satisfying

1. (1)

$\theta\wedge(\mathrm{d}\theta)^{2}\neq 0$; 2. (2)

$[\![\mathrm{d}\theta,\Omega]\!]$, modulo $\theta$, has rank $2$; 3. (3)

$(\lambda\mathrm{d}\theta+\mu\Omega)^{2}\equiv 0\mod\theta$ has two distinct solutions $[\lambda_{i}:\mu_{i}]\in\mathbb{R}\mathbb{P}^{1}$ $(i=1,2)$.

Here, condition $(3)$, in particular, characterizes hyperbolicity: Each integral surface of $(M,\mathcal{I})$ is foliated by two distinct families of characteristics.

Definition 2.7**.**

Consider a hyperbolic Monge-Ampère system $(M,\mathcal{I})$. A local coframing $\bm{\theta}=(\theta^{0},\theta^{1},\ldots,\theta^{4})$ defined on an open neighborhood $U\subset M$ is said to be [math]-adapted if, on $U$,

[TABLE]

The condition for [math]-adaptedness as defined above is a pointwise condition on $\bm{\theta}$. In fact, any [math]-adapted coframing associated to $(M,\mathcal{I})$ is a local section of a $G_{0}$-structure $\mathcal{G}_{0}$ on $M$, where $G_{0}\subset{\rm GL}(5,\mathbb{R})$ is the subgroup generated by matrices of the form

[TABLE]

and

[TABLE]

Two hyperbolic Monge-Ampère systems are equivalent if and only if their corresponding $G_{0}$-structures are equivalent.111Two ${G}$-structures $\mathcal{G}$ and $\hat{\mathcal{G}}$ on a manifold $M$ are said to be equivalent if there exists a diffeomorphism $\phi:\mathcal{G}\rightarrow\hat{\mathcal{G}}$ such that $\phi^{*}\hat{\bm{\omega}}=\bm{\omega}$, where $\bm{\omega},\hat{\bm{\omega}}$ are the tautological $1$-forms on $\mathcal{G},\hat{\mathcal{G}}$, respectively.

2.3. Integrable Extensions and Bäcklund Transformations

(c.f. [AF12], [AF15])

Definition 2.8**.**

Let $(M,\mathcal{I})$ be an exterior differential system. A rank-$k$ integrable extension of $(M,\mathcal{I})$ is an exterior differential system $(N,\mathcal{J})$ with a submersion $\pi:N\rightarrow M$ that satisfies the condition: for each $p\in N$, there exists an open neighborhood $U\subset N$ $(p\in U)$ such that

1. (1)

on $U$, the differential ideal $\mathcal{J}$ is algebraically generated by the elements of $\pi^{*}{\mathcal{I}}$ together with 1-forms $\theta_{1},\ldots,\theta_{k}\in\Omega^{1}(U)$, where $k=\dim N-\dim M$; 2. (2)

for any $p\in U$, let $F_{p}$ denote the fiber $\pi^{-1}(\pi(p))$; the $1$-forms $\theta_{1},\ldots,\theta_{k}$ restrict to $T_{p}F_{p}$ to be linearly independent.

Remark 1*.*

In Definition 2.8, one can understand $\mathcal{J}$ as defining a connection on the bundle $\pi:N\rightarrow M$ that is flat over the integral manifolds of $\mathcal{I}$. More specifically, Condition $(1)$ implies that, if $S\subset M$ is an integral manifold of $(M,\mathcal{I})$, then $\mathcal{J}$ restricts to $\pi^{-1}(S)$ to be Frobenius; hence, locally, $\pi^{-1}(S)$ is foliated by integral manifolds of $(N,\mathcal{J})$. Condition $(2)$ implies that, restricting to any integral manifold of $(N,\mathcal{J})$, $\pi$ is an immersion, whose image is an integral manifold of $(M,\mathcal{I})$.

Definition 2.9**.**

A Bäcklund transformation relating two exterior differential systems, $(M_{1},\mathcal{I}_{1})$ and $(M_{2},\mathcal{I}_{2})$, is a quadruple $(N,\mathcal{B};\pi_{1},\pi_{2})$ where, for each $i\in\{1,2\}$, $\pi_{i}:N\rightarrow M_{i}$ makes $(N,\mathcal{B})$ an integrable extension of $(M_{i},\mathcal{I}_{i})$. Such a Bäcklund transformation is represented by the diagram:

Definition 2.10**.**

In Definition 2.9, if $M_{1},M_{2}$ have the same dimension, which is not required in general, then we define the rank of $(N,\mathcal{B};\pi_{1},\pi_{2})$ to be the fiber dimension of either $\pi_{1}$ or $\pi_{2}$. If $(M_{i},\mathcal{I}_{i})$ $(i=1,2)$ are equivalent exterior differential systems, then $(N,\mathcal{B};\pi_{1},\pi_{2})$ is called an auto-Bäcklund transformation of either $(M_{i},\mathcal{I}_{i})$.

Example 1**.**

Let $(N,\mathcal{B};\pi,\bar{\pi})$ be a rank-$1$ Bäcklund transformation relating two hyperbolic Monge-Ampère systems $(M,\mathcal{I})$ and $(\bar{M},\bar{\mathcal{I}})$. On some open subsets $U\subset M$ and $\bar{U}\subset\bar{M}$, we can choose [math]-adapted coframings such that

[TABLE]

Let $V=\pi^{-1}U\cap\bar{\pi}^{-1}\bar{U}$, assumed to be nonempty. It is easy to see that the Cauchy characteristics of the system $\langle\pi^{*}\eta^{0}\rangle$ $($defined on V$$) are precisely the fibers of $\pi|_{V}$; similarly for $\bar{\pi}|_{V}$. Thus, it is natural to regard $(N,\mathcal{B};\pi,\bar{\pi})$ as nontrivial if $\pi^{*}\eta^{0}$ and $\bar{\pi}^{*}\bar{\eta}^{0}$ are linearly independent $1$-forms on $N$. In particular, it follows that, on $V$, the differential ideal $\mathcal{B}$ is algebraically generated by $\pi^{*}\mathcal{I}$ and $\bar{\pi}^{*}\bar{\eta}^{0}$ as well as by $\bar{\pi}^{*}\bar{\mathcal{I}}$ and $\pi^{*}\eta^{0}$.

Definition 2.11**.**

Given a fiber bundle $\pi:E\rightarrow B$, for any $p\in E$, the vertical tangent space of $E$ at $p$ is by definition the kernel of $\pi_{*}:T_{p}E\rightarrow T_{\pi(p)}B$.

Definition 2.12**.**

A Bäcklund transformation $(N,\mathcal{B};\pi_{1},\pi_{2})$ is said to be nontrivial if the two fibrations $\pi_{1},\pi_{2}$ have distinct vertical tangent spaces at each point $p\in N$.

3. Monge-Ampère Systems and Their First Invariants

Let $(M,\mathcal{I})$ be a hyperbolic Monge-Ampère system. Let $\mathcal{G}_{0}$ denote the $G_{0}$-structure on $(M,\mathcal{I})$ (see Definition 2.7). One can reduce (see [BGG03]) $\mathcal{G}_{0}$ to a $G_{1}$-structure $\mathcal{G}_{1}$ on which the tautological $1$-forms $\omega^{0},\omega^{1},\ldots,\omega^{4}$ satisfy the following structure equations:

[TABLE]

where $\phi_{0}=\phi_{1}+\phi_{4}=\phi_{5}+\phi_{8}$, and $G_{1}\subset G_{0}$ is the subgroup generated by

[TABLE]

and

[TABLE]

Definition 3.1**.**

Let $(M,\mathcal{I})$ be a hyperbolic Monge-Ampère system. A $1$-adapted coframing222This is not to be confused with a $1$-adapted coframing in the sense of Definition 5.1. of $(M,\mathcal{I})$ with domain $U\subset M$ is a section $\bm{\eta}:U\rightarrow\mathcal{G}_{1}$.

Following [BGG03], we introduce the notation333These $S_{i}$ are those defined in [BGG03] with the same notation scaled by $1/2$.

[TABLE]

It is shown in [BGG03] that

Proposition 3.1**.**

Along each fiber of $\mathcal{G}_{1}$,

[TABLE]

for any $g={\rm diag}(a;A;B)$ in the identity component of $G_{1}$. Moreover,

[TABLE]

Proposition 31 has a simple interpretation: the matrices $S_{1}$ and $S_{2}$ correspond to two invariant tensors under the $G_{1}$-action. In fact, one can verify that the quadratic form

[TABLE]

and the $2$-form

[TABLE]

are $G_{1}$-invariant, which implies that $\Sigma_{1},\Sigma_{2}$ are locally well-defined on $(M,\mathcal{I})$.

An infinitesimal version of Proposition 31 will be useful: for $i=1,2$,

[TABLE]

An important class of Monge-Ampère systems are the Euler-Lagrange systems. In the classical calculus of variations, an Euler-Lagrange system is a PDE system whose solutions correspond to the stationary points of a given first-order functional. In [BGG03], it is shown:

Proposition 3.2**.**

([BGG03])* A hyperbolic Monge-Ampère system is locally equivalent to an Euler-Lagrange system if and only if $S_{2}$ vanishes.*

Remark 2*.*

Proposition 3.2 says that the property of being Euler-Lagrange is intrinsically defined, that is, it does not depend on the choice of local coordinates.

4. $G$-structure Equations for Bäcklund Transformations

In Definition 2.9, it may appear that $(N,\mathcal{B};\pi_{1},\pi_{2})$ being a Bäcklund transformation imposes conditions on all components in this quadruple. However, when it is a nontrivial rank-$1$ Bäcklund transformation relating two hyperbolic Monge-Ampère systems, one only needs to impose conditions on the exterior differential system444To be more precise, $(N,\mathcal{B})$ is a hyperbolic exterior differential system of type $s=2$ in the sense of [BGH95]. $(N,\mathcal{B})$, as the following proposition shows.

Proposition 4.1**.**

[Cle02]* An exterior differential system $(N^{6},\mathcal{B})$ is a nontrivial rank-$1$ Bäcklund transformation relating two hyperbolic Monge-Ampère systems if and only if, for each $p\in N$, there exists an open neighborhood $V\subset N$ $(p\in V)$, a coframing $(\theta^{0},\bar{\theta}^{0},\theta^{1},\ldots,\theta^{4})$ and nonvanishing functions $A_{1},\ldots,A_{4}$ $(A_{1}A_{4}\neq A_{2}A_{3})$ defined on $V$, satisfying the conditions:*

1. $(1)$

the differential ideal $\mathcal{B}=\langle\theta^{0},\bar{\theta}^{0},\theta^{1}\wedge\theta^{2},\theta^{3}\wedge\theta^{4}\rangle_{\rm alg}$; 2. $(2)$

the vector bundles $E_{0}=[\![\theta^{0}]\!]$, $E_{1}=[\![\theta^{0},\theta^{1},\theta^{2}]\!]$ and $E_{2}=[\![\theta^{0},\theta^{3},\theta^{4}]\!]$ are invariant along the flow of $X$ (see Definition 2.5), where $X$ is a nonvanishing vector field on $V$ that annihilates $\theta^{0},\theta^{1},\ldots,\theta^{4}$; 3. $\bar{({2})}$

the vector bundles $\bar{E}_{0}=[\![\bar{\theta}^{0}]\!]$, $\bar{E}_{1}=[\![\bar{\theta}^{0},\theta^{1},\theta^{2}]\!]$ and $\bar{E}_{2}=[\![\bar{\theta}^{0},\theta^{3},\theta^{4}]\!]$ are invariant along the flow of $\bar{X}$, where $\bar{X}$ is a nonvanishing vector field on $V$ that annihilates $\bar{\theta}^{0},\theta^{1},\ldots,\theta^{4}$; 4. $(3)$

the following congruences hold:

[TABLE]

This proposition has the following corollary.

Corollary 4.1**.**

Let $(N^{6},\mathcal{B};\pi_{1},\pi_{2})$ be a nontrivial rank-$1$ Bäcklund transformation relating two hyperbolic Monge-Ampère systems. A coframing defined on an open subset $V\subset N$ that satisfies Conditions $(1)$-$(3)$ in Proposition 4.1 can always be arranged to satisfy the extra condition: $A_{2}=A_{3}=1$.

Proof. This is obtained by scaling $\theta^{0}$ and $\bar{\theta}^{0}$.∎

Definition 4.1**.**

A coframing as concluded in Corollary 4.1 is said to be [math]-adapted to the Bäcklund transformation $(N,\mathcal{B})$.

Given a nontrivial rank-$1$ Bäcklund transformation $(N,\mathcal{B};\pi_{1},\pi_{2})$ relating two hyperbolic Monge-Ampère systems, one can ask whether its [math]-adapted coframings are precisely the local sections of a $G$-structure on $N$. However, this is not true. For example, consider a [math]-adapted coframing $(\theta^{0},\bar{\theta}^{0},\theta^{1},\ldots,\theta^{4})$ defined on an open subset $U\subset N$ with corresponding functions $A_{1},A_{4}$. Let $T:U\rightarrow{\rm GL}(6,\mathbb{R})$ be the transformation:

[TABLE]

It is easy to see that the coframing on the right-hand-side is [math]-adapted. However, the same transformation, when applied to a [math]-adapted coframing with corresponding functions $A_{1}^{\prime},A_{4}^{\prime}$ that are different from $A_{1},A_{4}$, may not result in a [math]-adapted coframing.

One simple strategy, as taken by [Cle02], to avoid this issue is by, in addition to understanding the subbundles $[\![\theta^{0}]\!]$ and $[\![\bar{\theta}^{0}]\!]$ as an ordered pair, fixing an order for the pair of subbundles $[\![\theta^{0},\bar{\theta}^{0},\theta^{1},\theta^{2}]\!]$ and $[\![\theta^{0},\bar{\theta}^{0},\theta^{3},\theta^{4}]\!]$. Once this is considered, all local [math]-adapted coframings respecting such an ordering are precisely the local sections of a $G$-structure, where $G\subset{\rm GL}(6,\mathbb{R})$ is the Lie subgroup consisting of matrices of the form

[TABLE]

Now let $\mathcal{G}$ denote this $G$-structure on $N$. Let $\bm{\omega}=(\omega^{1},\omega^{2},\ldots,\omega^{6})$ be the tautological $1$-form on $\mathcal{G}$. Let $\mathfrak{g}$ be the Lie algebra of $G$. Using the conditions in Proposition 4.1 and the reproducing property of $\bm{\omega}$, one can show that $\bm{\omega}$ satisfies the following structure equations, recorded from [Cle02] with a slight change of notation:

[TABLE]

where the matrix in $\alpha$ and $\beta$ is a $\mathfrak{g}$-valued $1$-form, called a pseudo-connection of $\mathcal{G}$; the second term on the right-hand-side is called the intrinsic torsion of $\mathcal{G}$.

It is easy to see that the intrinsic torsion above, as a map defined on $\mathcal{G}$, takes values in a $10$-dimensional representation of $G$ and is $G$-equivariant. It is proved in [Cle02] that this representation decomposes into 6 irreducible components, as shown by the following equations, where $u\in\mathcal{G}$ is represented as a column, $g$ is as in (40), and $u\cdot g:=g^{-1}u$:

[TABLE]

Definition 4.2**.**

Let $G$ and $\mathcal{G}$ be as above. The Bäcklund transformation555To be precise, this is a Bäcklund transformation with an ordered pair of characteristic systems. corresponding to $\mathcal{G}$ is said to be generic if, at each point $u\in\mathcal{G}$, the intrinsic torsion takes values in a $G$-orbit with the largest possible dimension.

5. An Estimate of Generality

In this section, we address the problem of generality for generic rank-$1$ Bäcklund transformations relating two hyperbolic Monge-Ampère systems. The main ingredients are a $G$-structure reduction procedure described in [Gar89] and a theorem of Cartan described in [Bry14].

Lemma 5.1**.**

Let $(N,\mathcal{B})$ be a nontrivial rank-$1$ Bäcklund transformation relating two hyperbolic Monge-Ampère systems. Let $\mathcal{G}$ be the associated $G$-structure. If $(N,\mathcal{B})$ is generic, then, at each point $u\in\mathcal{G}$, the intrinsic torsion takes values in an $8$-dimensional $G$-orbit.

Proof. Let

[TABLE]

at each point $u\in\mathcal{G}$. By (67), the function $A_{1}A_{4}$ and the dimensions of $W_{1}$ and $W_{2}$ are all invariant under the $G$-action. Let $T$ denote the intrinsic torsion of $\mathcal{G}$. We claim that, for each $u\in\mathcal{G}$, the $G$-orbit of $T(u)$ is at most $8$-dimensional and that this occurs precisely when $W_{1}$ and $W_{2}$ are both $2$-dimensional. To see why this is true, first note that if one of $W_{i}$ $(i=1,2)$ has dimension less than $2$ at $u\in\mathcal{G}$, then the dimension of $u\cdot G$ is at most $7$-dimensional. If both $W_{1},W_{2}$ have dimension $2$ at $u\in\mathcal{G}$, then it is easy to show that there exists a unique $g\in G$ such that, at $u^{\prime}=u\cdot g$,

[TABLE]

where $\epsilon_{i}=\pm 1$ $(i=1,2)$. This completes the proof. ∎

Let $(N,\mathcal{B})$ be a generic rank-$1$ Bäcklund transformation relating two hyperbolic Monge-Ampère systems. By Lemma 5.1, each point $p\in N$ has a connected open neighborhood $U\subset N$ on which a canonical coframing $(\omega^{1},\omega^{2},\ldots,\omega^{6})$ can be determined. Such a coframing satisfies the equation (59), where all differential forms are defined on $U$ instead of $\mathcal{G}$, and the equations (68), where the sign of each $\epsilon_{i}$ is determined. This motivates the following definition.

Definition 5.1**.**

Let $N$ be a $6$-manifold. A coframing $(\omega^{1},\omega^{2},\ldots,\omega^{6})$ defined on an open subset $U\subset N$ is said to be $1$-adapted $($to a generic rank-$1$ Bäcklund transformation relating two hyperbolic Monge-Ampère systems$)$ if there exist $1$-forms $\alpha_{i},\beta_{i}$ $(i=0,\ldots,3)$ and functions $A_{1},A_{4},B_{i},C_{i}$ $(i=1,\ldots,4)$ defined on $U$ such that the equations (59) and (68) are satisfied.

Now we prove the following main theorem that estimates the generality of generic rank-1 Bäcklund transformations relating two hyperbolic Monge-Ampère systems.

Theorem 5.2**.**

Let $N$ be a $6$-manifold. For each $p\in N$, a $1$-adapted coframing (Definition 5.1) defined on a small open neighborhood $U\subset N$ of $p$ can be uniquely determined, up to diffeomorphism, by specifying at most 6 functions of 3 variables.

Proof. Let $U\subset N^{6}$ be a sufficiently small connected open subset. Suppose that $\bm{\omega}=(\omega^{1},\omega^{2},\ldots,\omega^{6})$ is a $1$-adapted coframing on $U$ in the sense of Definition 5.1. It follows that there exist functions $P_{ij}$ $(i=0,\ldots,7;~{}j=1,\ldots,6)$ defined on $U$ such that $\bm{\omega}$ satisfies (59) and (68) with

[TABLE]

There is a standard method to determine the generality of such a coframing $\bm{\omega}$ up to diffeomorphism (see [Bry14]). Our application of such a method involves mainly three steps.

Step 1. By applying $\mathrm{d}^{2}=0$ to (59), we find that $P_{ij}$ are related among themselves and with the coefficients of their exterior derivatives. Repeating this, at a stage, no new relations among the $P_{ij}$ arise.

More explicitly, we can choose $s$ expressions $a^{\alpha}$ $(\alpha=1,\ldots,s)$ from $P_{ij}$, find $r$ expressions $b^{\rho}$ $(\rho=1,\ldots,r)$, real analytic functions $F^{\alpha}_{i}:\mathbb{R}^{r+s}\rightarrow\mathbb{R}$ and $C^{i}_{jk}:\mathbb{R}^{r}\rightarrow\mathbb{R}$ satisfying $C^{i}_{jk}+C^{i}_{kj}=0$, such that

1. (A)

the equation (59), in general, takes the form

[TABLE] 2. (B)

$\mathrm{d}a^{\alpha}$, in general, takes the form:

[TABLE]

moreover, applying $\mathrm{d}^{2}=0$ to (69) yields identities when we take into account both (69) and (70); 3. (C)

there exist functions $G^{\rho}_{j}:\mathbb{R}^{r+s}\rightarrow\mathbb{R}$ such that applying $\mathrm{d}^{2}=0$ to (70) yields identities when we replace $\mathrm{d}b^{\rho}$ by $G^{\rho}_{j}\omega^{j}$ and take into account (69) and (70).

Step 2. For the tableau of free derivatives associated to $(F^{\alpha}_{i})$, which is a subspace of ${\rm Hom}(\mathbb{R}^{6},\mathbb{R}^{s})$ defined at each point of $\mathbb{R}^{r+s}$, compute its Cartan characters (an array of $6$ integers $(s_{1},s_{2},\ldots,s_{6})$) and the dimension $\delta$ of its first prolongation. For details, see [Bry14]. Moreover, in our case, we verify that $\sum_{i=1}^{6}s_{i}=r$.

Step 3. Restricting to a domain $V\subset\mathbb{R}^{r+s}$ where the Cartan characters are constants, compare $s:=\sum_{j=1}^{6}js_{j}$ with $\delta$. By Cartan’s inequality, there are two possibilities: either $s=\delta$ (called the involutive case) or $s>\delta$.

In the involutive case, one can conclude that (see Theorem $3$ in [Bry14]):

For any $(a_{0},b_{0})\in\mathbb{R}^{s+r}$ there exists a coframing $\bm{\omega}$ and functions $a=(a^{\alpha}),b=(b^{\rho})$ defined on an open neighborhood of $\bm{0}\in\mathbb{R}^{6}$ that satisfy (69), (70) and $(a(\bm{0}),b(\bm{0}))=(a_{0},b_{0})$. Moreover, locally, such a coframing can be uniquely determined up to diffeomorphism by specifying $s_{k}$ functions of $k$ variables, where $s_{k}$ is the last nonzero Cartan character.

In the non-involutive case, which is the case we encounter, a natural step to take is to prolong (see [Bry14]) the system by introducing the derivatives of $b^{\rho}$, carry out similar steps as the above, and obtain new tableaux of free derivatives with Cartan characters $(\sigma_{1},\sigma_{2},\ldots,\sigma_{6})$.

In practice, however, we do not actually prolong, for it is easy to show that, if $s_{k}$ is the last nonzero character in $(s_{1},\ldots,s_{6})$, then $\sigma_{j}=0$ $(j>k)$ and $\sigma_{k}\leq s_{k}$. Using this and the Cartan-Kuranishi Theorem ([BCG*+*13]), one can already conclude that the ‘generality’ of $1$-adapted coframings is bounded from above by $s_{k}$ functions of $k$ variables. (In our case, $k=3$ and $s_{k}=6$.)

For the details of carrying out the steps above, see Appendix A. Most of our calculations are performed using Maple™. ∎

Remark 3*.*

Clearly, two $1$-adapted coframings that are equivalent under a diffeomorphism correspond to equivalent Bäcklund transformations. Because of this, the upper bound for the ‘generality’ of $1$-adapted coframings in Theorem 5.2 applies to the ‘generality’ of generic rank-$1$ Bäcklund transformations relating two hyperbolic Monge-Ampère systems.

Corollary 5.3**.**

There exist hyperbolic Monge-Ampère systems that are not related to any hyperbolic Monge-Ampère system by a generic rank-$1$ Bäcklund transformation.

Proof. A hyperbolic Monge-Ampère system, up to contact equivalence, can be uniquely determined by specifying $3$ functions of $5$ variables.666 One can also apply the same method used in the proof of Theorem 5.2 to verify the stronger statement: Locally, a hyperbolic Euler-Lagrange system, which is Monge-Ampère, can be determined uniquely by specifying $1$ function of $5$ variables. (This is not surprising, as, in our case, a Lagrangian is a function depending on $5$ variables.)

Note that a generic rank-$1$ Bäcklund transformation in consideration completely determines the two underlying hyperbolic Monge-Ampère systems (up to equivalence). The conclusion follows. ∎

6. Classifications and Examples in Higher Cohomogeneity

Following the discussion in the previous section, let $U\subset N^{6}$ be a sufficiently small connected open subset. Let $\bm{\omega}$ be a $1$-adapted coframing defined on $U$ in the sense of Definition 5.1. One can ask, when we specify several structure invariants, can we classify the corresponding Bäcklund transformations, if any?

In the rest of this section, we consider the case when $\epsilon_{1}=\epsilon_{2}=1$ in (68), and when $A_{1}$ and $A_{4}$ (or $P_{81}$ and $P_{84}$ in the new notation) in (59) are specified to be $A_{1}=1$ and $A_{4}=-1$.

The following procedure is similar to that in Appendix A. All calculations below are performed using Maple™.

First, all coefficients in (59) are expressed in terms of the remaining $40$ $P_{ij}$. Defining their derivatives $P_{ijk}$ by

[TABLE]

and applying the identity $\mathrm{d}^{2}=0$ to (59), we obtain a system of $106$ polynomial equations in $P_{ij}$ and $P_{ijk}$, which implies that

[TABLE]

Using these relations and repeating the steps above, we obtain a system of $88$ equations, which implies that

[TABLE]

Using these and repeating, we obtain a system of $86$ equations for the $17$ $P_{ij}$ remaining and $80$ of their $102$ derivatives. This system implies that

[TABLE]

Using these and repeating, we obtain a system of $85$ equations for the $13$ $P_{ij}$ remaining and $64$ of their derivatives. This system implies that

[TABLE]

Using these and repeating, we obtain a system of $61$ equations for $P_{72},P_{44}$, $P_{46},P_{64}$ and $22$ of their derivatives. Solving this system leads to the two cases below.

Case $\bf 1$: $P_{72}\neq 0$

In this case, we have

[TABLE]

Using these and applying $\mathrm{d}^{2}=0$ to the structure equations, we find that

[TABLE]

Using this and repeating, we find that

[TABLE]

It follows that the only primary invariant remaining, $P_{72}$, is a nonzero constant. The structure equations read

[TABLE]

This, after the transformation

[TABLE]

can be readily seen to belong to Case 3D in Clelland’s classification (see [Cle02]). According to [Cle02], if $P_{72}<0$, then $(N,\mathcal{B})$ is a homogeneous Bäcklund transformation relating time-like surfaces of the constant mean curvature

[TABLE]

in $\mathbb{H}^{2,1}$; if $P_{72}>0$, then $(N,\mathcal{B})$ is a homogeneous Bäcklund transformation relating certain surfaces in a $5$-dimensional quotient space of the Lie group ${\rm SO}^{*}(4)$.

Case $\bf 2$: $P_{72}=0$

In this case, all coefficients in (59) are expressed in terms of $P_{44},P_{46}$ and $P_{64}$. Applying $\mathrm{d}^{2}=0$ to the structure equations, no new relations between $P_{44},P_{46}$ and $P_{64}$ arise. Furthermore, $16$ of the $18$ derivatives of $P_{44},P_{46}$ and $P_{64}$ are expressed in terms of these three invariants; two derivatives, $P_{644}$ and $P_{646}$, are free.

It is easy to check that Theorem 3 in [Bry14] applies to $\bm{\omega}$, the expressions $a=(P_{44},P_{46},P_{64})$, $b=(P_{644},P_{646})$, and the functions $C^{i}_{jk}$ and $F^{\alpha}_{i}$ determined during the calculation above. The corresponding tableaux of free derivatives is involutive with Cartan characters $(1,1,0,0,0,0)$. We have thus proved the following theorem.

Theorem 6.1**.**

Locally, a generic rank-$1$ Bäcklund transformation $(N,\mathcal{B})$ relating two hyperbolic Monge-Ampère systems with its $1$-adapted coframing satisfying $\epsilon_{1}=\epsilon_{2}=1$ and $A_{1}=-A_{4}=1$ can be uniquely determined by specifying $1$ function of $2$ variables.

We can study Case 2 in greater detail. For convenience, we introduce the following new notation:

[TABLE]

In this new notation, $\bm{\omega}$ satisfies (59) and (68) where

[TABLE]

and $\epsilon_{1}=\epsilon_{2}=1$, $A_{1}=-A_{4}=1$.

Moreover, the exterior derivatives of $R,S$ and $T$ are

[TABLE]

Using these equations, we can study the symmetry of the corresponding Bäcklund transformation $(N,\mathcal{B})$. Let $U\subset N$ be the domain of a $1$-adapted coframing $\bm{\omega}$. Let the map $\Phi:U\rightarrow\mathbb{R}^{3}$ be defined by

[TABLE]

Lemma 6.2**.**

The map $\Phi$ can never have rank [math]. Moreover, it has

• •

rank $1$ if and only if $2RS=1$ and $T=R^{2}+S^{2}$;

• •

rank $2$ if and only if it does not have rank $1$ and satisfies either

1. (1)

$2RS=1$* or* 2. (2)

$T_{4}=(R+S)(T-1)$* and $T_{6}=(R-S)(T+1)$;*

• •

rank $3$ if and only if it does not have rank $1$ or $2$.

Proof. In $\mathrm{d}R\wedge\mathrm{d}S\wedge\mathrm{d}T$, the coefficients of $\omega^{i}\wedge\omega^{j}\wedge\omega^{k}$ are polynomials in $R,S,T,T_{4}$ and $T_{6}$. These coefficients have the common factor $2RS-1$. Calculating with Maple™, we find that the coefficients of $\omega^{i}\wedge\omega^{j}\wedge\omega^{k}$ in $(2RS-1)^{-1}\mathrm{d}R\wedge\mathrm{d}S\wedge\mathrm{d}T$ all vanish if and only if $T_{4}=(R+S)(T-1)$ and $T_{6}=(R-S)(T+1)$. This justifies the conditions for having rank $2$. The condition for ‘rank-$1$’ can be obtained by setting the coefficients of $\omega^{i}\wedge\omega^{j}$ in $\mathrm{d}R\wedge\mathrm{d}S$, $\mathrm{d}R\wedge\mathrm{d}T$ and $\mathrm{d}S\wedge\mathrm{d}T$ to be all zero. By (73), it is clear that $\mathrm{d}R\neq 0$ everywhere; hence, $\Phi$ cannot have rank [math]. ∎

Definition 6.1**.**

In the current case, if the corresponding Bäcklund transformation $(U,\mathcal{B})$ has a symmetry whose orbits are of dimension $6-k$, then it is said to have cohomogeneity $k$.

Case of cohomogeneity-$1$. This occurs precisely when ${\rm rank}(\Phi)=1$. In this case, locally $\Phi$ is a submersion to either branch of the curve in $\mathbb{R}^{3}$ defined by $2RS=1$ and $T=R^{2}+S^{2}$. Expressing $S$ and $T$ in terms of $R$, we have, on $U\subset N$,

[TABLE]

It is clear that $\mathrm{d}R$ is nowhere vanishing. Since $R$ is the only invariant, each constant value of $R$ determines a 5-dimensional submanifold $N_{R}\subset N$, which has a Lie group structure. The Lie group structure can be determined by setting the right-hand-side of the equation (74) to be zero, obtaining, say,

[TABLE]

then substituting this into the structure equations, yielding equations of $\mathrm{d}\omega^{i}$ $(i=2,\ldots,6)$, expressed in terms of $\omega^{2},\ldots,\omega^{6}$ alone. These are the structure equations on each $N_{R}$. Let $X_{1},X_{2},\ldots,X_{5}$ be the vector fields tangent to $N_{R}$ and dual to $\omega^{2},\ldots,\omega^{6}$, such that $\omega^{i}(X_{j})=\delta^{i}_{j+1}$ $(i-1,j=1,2,\ldots,5)$. We obtain the Lie bracket relations:

[TABLE]

Using these relations, it can be verified that $X_{i}$ $(i=1,\ldots,5)$ generate a 5-dimensional Lie algebra that is solvable but not nilpotent. The derived series has dimensions $(5,3,1,0,\ldots)$. In fact, after introducing the following new basis:

[TABLE]

we obtain the Lie bracket relations:

[TABLE]

with all $[{\bf e}_{i},{\bf e}_{j}]$ that are not on this list being zero. An equivalent way of writing these relations is:

[TABLE]

Now, it is easy to see that the Lie algebra $\bigoplus_{i=1}^{5}\mathbb{R}{\bf e}_{i}$ is isomorphic to the Lie algebra generated by the real and imaginary parts of the vector fields

[TABLE]

on $\mathbb{R}\times\mathbb{C}^{2}$ with coordinates $(\lambda;z,w)$. In fact, an isomorphism is induced by the correspondence

[TABLE]

Next, we describe the hyperbolic Monge-Ampère systems related by the Bäcklund transformation being considered.

Proposition 6.1**.**

A Bäcklund transformation in the current (cohomogeneity-$1$) case is an auto-Bäcklund transformation of a homogeneous Euler-Lagrange system.

Proof. This proof is in two parts. First, we show that the underlying two hyperbolic Monge-Ampère systems are equivalent and are homogeneous. Second, by computing their local invariants, we verify that they are hyperbolic Euler-Lagrange systems in the sense of [BGG03].

Using the structure equations on $U\subset N$, if we let $(\theta^{0},\theta^{1},\ldots,\theta^{4})$ be either

[TABLE]

or

[TABLE]

then we can verify (with Maple™) that $\theta^{i}$ $(i=0,\ldots,4)$, in both cases, satisfy the same structure equations:

[TABLE]

It is easy to verify that (77) are the structure equations on a $5$-manifold $M$ with a hyperbolic Monge-Ampère ideal $\mathcal{I}=\langle\theta^{0},\theta^{1}\wedge\theta^{2},\theta^{3}\wedge\theta^{4}\rangle$. It follows that the expressions (75) and (76) correspond to the pull-back of $\theta^{i}$ under two distinct submersions $\pi_{1},\pi_{2}:U\rightarrow M$. It is easy to see that $(U,\mathcal{B};\pi_{1},\pi_{2})$ is an auto-Bäcklund transformation of the system $(M,\mathcal{I})$. Moreover, $(M,\mathcal{I})$ is homogeneous, since all coefficients in (77) are constants.

Next, we verify that $(M,\mathcal{I})$ is a hyperbolic Euler-Lagrange system. In [BGG03], it is proved that a hyperbolic Monge-Ampère system is Euler-Lagrange if and only if the invariant tensor $S_{2}$ vanishes (see Proposition 3.2). To compute $S_{2}$ in the current case, we choose a new coframing $\bm{\eta}=(\eta^{i})$ and $1$-forms $(\phi_{\alpha})$ below:

[TABLE]

[TABLE]

These $\eta^{i}$ $(i=0,1,\ldots,4)$ and $\phi_{\alpha}$ $(\alpha=1,\ldots,8)$ are chosen such that they satisfy the structure equations (20), and such that $S_{1}$ and $S_{2}$ are as simple as possible. One can verify that, under this choice,

[TABLE]

This completes the proof. ∎

Remark 4*.*

The Monge-Ampère system $(M,\mathcal{I})$ considered in Proposition 6.1 cannot, by a contact transformation, be put in the form

[TABLE]

This is because ${\rm rank}(S_{1})=2$ and $S_{2}=\bm{0}$, which implies that neither of the two characteristic systems of $(M,\mathcal{I})$ contains a rank-$1$ integrable subsystem.

Now, one may wonder whether the homogeneous Monge-Ampère system $(M,\mathcal{I})$ considered in Proposition 6.1 has a symmetry of dimension greater than $5$. Using the method of equivalence, we prove that it is not the case.

Proposition 6.2**.**

The hyperbolic Euler-Lagrange system in Proposition 6.1 has a symmetry of dimension $5$. In addition, any such symmetry is induced from a symmetry of the Bäcklund transformation $(N,\mathcal{B})$.

Proof. Let $(M,\mathcal{I})$ denote the Euler-Lagrange system being considered. To show that $(M,\mathcal{I})$ has a $5$-dimensional symmetry, it suffices to show that there is a canonical way to determine a local coframing on $M$. This can be achieved by applying the method of equivalence. For details, see Appendix B.

By (75) and (76), it is easy to see that the fibers of $\pi_{i}:N\rightarrow M$ $(i=1,2)$ are everywhere transversal to the level sets of the functions $R$. The second half of the statement follows.∎

To end the discussion of the cohomogeneity-$1$ case, we integrate the structure equations (77) to express the corresponding hyperbolic Euler-Lagrange system in local coordinates.

Proposition 6.3**.**

The hyperbolic Monge-Ampère system $(M,\mathcal{I})$ with the differential ideal $\mathcal{I}=\langle\theta^{0},\theta^{1}\wedge\theta^{2},\theta^{3}\wedge\theta^{4}\rangle$, where $\theta^{i}$ satisfy (77), is equivalent to the following hyperbolic Monge-Ampère PDE up to a contact transformation:

[TABLE]

where $A=2z_{x}+y$ and $B=2z_{y}-x$.

Proof. Let $U\subset M$ be a domain on which $\theta^{i}$ are defined. One can verify, using the structure equations (77), that the $1$-forms $\theta^{1}-\theta^{2}-\theta^{3}-\theta^{4}$ and $\theta^{1}+\theta^{3}$ are closed. Hence, by shrinking $U$ if needed, there exist functions $P,Q$ defined on $U$ such that

[TABLE]

Moreover, if we let $\Theta=\theta^{2}+i\theta^{4}$, then, by a straightforward calculation, we obtain

[TABLE]

It follows that there exist functions $X,Y$ on $U$ such that

[TABLE]

Equivalently, we have

[TABLE]

Now we can express $\theta^{1},\ldots,\theta^{4}$ completely in terms of the functions $X,Y,P,Q$. By verifying the equality

[TABLE]

we notice that the right-hand-side of (80) is a symplectic form on a $4$-manifold on which $\theta^{1},\ldots,\theta^{4}$ are well-defined, by (77). Thus, by the theorem of Darboux, locally there exist functions $x,y,p,q$ such that the right-hand-side of (80) is equal to $\mathrm{d}x\wedge\mathrm{d}p+\mathrm{d}y\wedge\mathrm{d}q$.

In fact, in the $XYPQ$-coordinates, the right-hand-side of (80) is equal to

[TABLE]

As a result, we can set

[TABLE]

and write

[TABLE]

for some function $z$, independent of $x,y,p,q$. From these expressions, it is clear that $2p+y$ and $2q-x$ cannot simultaneously vanish.

Now let $A=2p+y$ and $B=2q-x$. We can express $\theta^{1}\wedge\theta^{2}$ in terms of $x,y,z,p,q$:

[TABLE]

Multiplying this expression by $(A^{2}+B^{2})^{2}$ then subtracting the result by $(A^{2}+B^{2})(\mathrm{d}x\wedge\mathrm{d}p+\mathrm{d}y\wedge\mathrm{d}q)$, we obtain

[TABLE]

The vanishing of this $2$-form on integral surfaces (satisfying the independence condition $\mathrm{d}x\wedge\mathrm{d}y\neq 0$) implies that $z$ must satisfy the equation $\eqref{Cohomog1PDE}$. ∎

Case of cohomogeneity-$2$. By Lemma 6.2, this case can only occur when $2RS=1$ and $R^{2}+S^{2}=T$ do not both hold and either

1. (1)

$2RS=1$ or 2. (2)

$T_{4}=(R+S)(T-1)$, $T_{6}=(R-S)(T+1)$

holds. We now focus on the latter case.

Proposition 6.4**.**

When $\Phi$ has rank $2$, and when $T_{4}=(R+S)(T-1)$ and $T_{6}=(R-S)(T+1)$, the map $\Phi:N\rightarrow\mathbb{R}^{3}$ has its image contained in a surface that is defined by either

[TABLE]

or its reciprocal being a constant.

Proof. First note that $R^{2}+S^{2}-T$ and $2RS-1$ cannot be both zero, for this would reduce to the cohomogeneity-1 case; hence, the conclusion has meaning. To see that this statement is true, note that, in the current case, the pull-back of $\mathrm{d}R,\mathrm{d}S$ and $\mathrm{d}T$ via $\Phi$ to $N$ are linearly dependent. To be precise, the $1$-form

[TABLE]

equals to zero when pulled back to $N$. Since the tangent map $\Phi_{*}$ has rank $2$, this can only occur when $\theta\wedge\mathrm{d}\theta=0$. It follows that $\theta$ is integrable. In fact, it is easy to verify that the primitives of $(2RS-1)^{-2}\theta$ and $(R^{2}+S^{2}-T)^{-2}\theta$ are, respectively, the function

[TABLE]

and its reciprocal when $2RS-1$ and $R^{2}+S^{2}-T$ are, respectively, nonzero. This completes the proof.∎

We now study the symmetry of the Bäcklund transformation $(N,\mathcal{B})$ being considered. Let $X_{i}$ $(i=1,2,\ldots,6)$ be the vector fields defined on $U\subset N$ that are dual to $\omega^{i}$ $(i=1,2,\ldots,6)$. Using the expressions of $\mathrm{d}R,\mathrm{d}S$ and $\mathrm{d}T$, it is easy to see that the rank-$4$ distribution on $U$ annihilated by $\mathrm{d}R,\mathrm{d}S$ and $\mathrm{d}T$ is spanned by the vector fields

[TABLE]

These vector fields generate a $4$-dimensional Lie algebra $\mathfrak{l}$ with

[TABLE]

It is easy to verify that $2Y_{1}+Y_{3}$ belongs to the center of $\mathfrak{l}$. The quotient algebra $\mathfrak{q}=\mathfrak{l}/\mathbb{R}(2Y_{1}+Y_{3})$, with the basis ${\bf e}_{1}=[Y_{2}]$, ${\bf e}_{2}=[Y_{3}]$ and ${\bf e}_{3}=[Y_{4}]$, satisfies

[TABLE]

According to the classification of $3$-dimensional Lie algebras, see Lecture 2 in [Bry95] for example, to identify the Lie algebra $\mathfrak{q}$, it suffices to find a normal form of the matrix (note that it is symmetric)

[TABLE]

under the transformation $C\mapsto\det(A^{-1})ACA^{T}$, where $A\in{\rm GL}(3,\mathbb{R})$. Note that $\det(C)=-2(R^{2}+S^{2}-T)$. We have:

Proposition 6.5**.**

If $R^{2}+S^{2}<T$, then $\mathfrak{q}$ is isomorphic to $\mathfrak{so}(3,\mathbb{R})$. If $R^{2}+S^{2}>T$, then $\mathfrak{q}$ is isomorphic to $\mathfrak{sl}(2,\mathbb{R})$. If $R^{2}+S^{2}=T$, then $\mathfrak{q}$ is isomorphic to the solvable Lie algebra with a basis $x_{1},x_{2},x_{3}$ satisfying $[x_{2},x_{3}]=x_{1}$, $[x_{3},x_{1}]=x_{2}$ and $[x_{1},x_{2}]=0$

.

Proof. After a transformation of the form above, $C$ can be put in the form

[TABLE]

By [Bry95], the conclusion follows immediately. ∎

Now consider the case when $\mathfrak{q}$ is solvable, that is, when $R^{2}+S^{2}=T$. By the cohomogeneity-$2$ assumption, we must have $2RS\neq 1$. We proceed to identify the Monge-Ampère systems related by such a Bäcklund transformation.

If we let $(\theta^{0},\theta^{1},\ldots,\theta^{4})$ be

[TABLE]

and let $F$ be defined by

[TABLE]

for which to have meaning we need to restrict to a domain on which $S\neq 0$, then the structure equations on $N$ would imply

[TABLE]

and

[TABLE]

It can be verified that, in the equations (81) and (82), the exterior derivative of the right-hand-sides are zero, by taking into account these equations themselves. By the construction of the $\theta^{i}$ $(i=0,\ldots,4)$, it follows that (81) and (82) are the structure equations of one of the Monge-Ampère systems being related by the Bäcklund transformation $(N,\mathcal{B})$.

On the other hand, it is easy to verify that the transformation

[TABLE]

leaves (81) and (82) unchanged. Thus, by applying such a transformation, if needed, we can assume that $F>0$.

Proposition 6.6**.**

The hyperbolic Monge-Ampère system $(M,\mathcal{I})$ with the differential ideal $\mathcal{I}=\langle\theta^{0},\theta^{1}\wedge\theta^{2},\theta^{3}\wedge\theta^{4}\rangle$, where $\theta^{i}$ satisfy (81) and (82), corresponds to the following hyperbolic Monge-Ampère PDE up to a contact transformation:

[TABLE]

where $A=z_{x}-y$, $B=z_{y}+x$.

Proof. The proof is similar to that of Proposition 6.3. First it is easy to verify that the $1$-forms $F(-2\theta^{0}+\theta^{2}+\theta^{4})-\theta^{1}-\theta^{3}$ and $\theta^{1}-\theta^{2}-\theta^{3}-\theta^{4}$ are closed. Consequently, locally there exist functions $f,g$ such that

[TABLE]

Now the expression of $\mathrm{d}F$ can be written as $\mathrm{d}F=-2F\mathrm{d}f$. This implies that there exists a constant $C>0$ such that $F=Ce^{-2f}$. Using the ambiguity in $f$ (as $f$ is determined up to an additive constant), we can arrange that $C=1$. In addition, if we let $\Theta=e^{-f}(\theta^{2}+i\theta^{4})$, it is easy to verify that $\Theta$ is integrable. To be explicit,

[TABLE]

Thus, there exist functions $X,Y$ such that $\Theta=e^{ig}(\mathrm{d}X+i\mathrm{d}Y)$. From this we obtain

[TABLE]

Using these, differentiating $\theta^{1}+\theta^{3}$ gives

[TABLE]

This implies that there exists a function $Z$, independent of $X,Y,f,g$, such that

[TABLE]

Now, $\theta^{0},\theta^{1},\ldots,\theta^{4}$ can be completely expressed in terms of the functions $X,Y,Z,f,g$. In particular,

[TABLE]

If we make the substitution

[TABLE]

the contact form $\theta^{0}$ is then a nonzero multiple of $\mathrm{d}z-p\mathrm{d}x-q\mathrm{d}y$. The $2$-form $\theta^{3}\wedge\theta^{4}$, when each $\mathrm{d}z$ is replaced by $p\mathrm{d}x+q\mathrm{d}y$, can be expressed as

[TABLE]

where $A=p-y$, $B=q+x$. Note that, by construction, $A,B$ cannot be simultaneously zero. The equation (83) follows. ∎

In the current case, there remain several obvious questions to investigate. What is the Monge-Ampère system corresponding to $\langle\omega^{2},\omega^{3}\wedge\omega^{4},\omega^{5}\wedge\omega^{6}\rangle$? Are the Monge-Ampère systems being Bäcklund-related Euler-Lagrange? Is the Bäcklund transformation an auto-Bäcklund transformation? Answers to these questions can be obtained in a similar way as in the cohomogeneity-$1$ case. We thus have them summarized in the following remark, omitting the details of calculation.

Remark 5*.*

1. A.

Whenever $R\neq 0$,

[TABLE]

form a coframing defined on a $5$-manifold. The system

[TABLE]

descends to correspond to the same equation (83) up to a contact transformation. The system $(N,\mathcal{B})$ is therefore an auto-Bäcklund transformation of the equation (83). 2. B.

One can verify that the hyperbolic Monge-Ampère system in Proposition 6.6 is Euler-Lagrange. In fact, by a transformation of the $\theta^{i}$ $(i=0,\ldots,4)$, the structure equations (81) can be put in the form of (20) with

[TABLE]

Applying the procedure described in Appendix B, we find the corresponding Monge-Ampère invariants

[TABLE]

Using this and the expression of $\mathrm{d}F$, one can show $($in a way that is similar to the proof of Proposition 6.2$)$ that the underlying Euler-Lagrange system has a symmetry of dimension $4$. Such a symmetry is induced from the symmetry of the Bäcklund transformation $(N,\mathcal{B})$. 3. C.

Similar to the reason in Remark 4, (83) is not contact equivalent to any hyperbolic Monge-Ampère PDE of the form

[TABLE]

7. Concluding Remarks

7.1. Regarding Classification in the Rank-$1$ Case

The method outlined in Section 6 can be carried further, for example, by putting weaker restrictions on the invariants. Which new subclasses of Bäcklund transformations relating two hyperbolic Monge-Ampère systems can be classified?

Equation (79) admits the trivial solution $z(x,y)=0$. Which solution do we obtain by applying the Bäcklund transformation found in Proposition 6.1 to this trivial solution? Does this give rise to a $1$-soliton solution to (79)?

7.2. Regarding Monge-Ampère Invariants

It is interesting to ask which pairs of hyperbolic Monge-Ampère systems may be related by a rank-$1$ Bäcklund transformation. This question can be partially answered by studying how obstructions to the existence of Bäcklund transformations may be expressed in terms of the invariants of the underlying hyperbolic Monge-Ampère systems. Some relevant results are presented in [Hu19].

7.3. Bäcklund Transformations of Higher Ranks

Among the examples discussed in [RS02], a Bäcklund transformation relating solutions of the hyperbolic Tzitzeica equation is particularly interesting. The hyperbolic Tzitzeica equation is the second-order equation for $h(x,y)$:

[TABLE]

This equation was discovered by Tzitzeica in his study of hyperbolic affine spheres in the affine $3$-space $\mathbb{A}^{3}$ (see [Tzi08] and [Tzi09]). He found that the system in $\alpha$, $\beta$ and $h$,

[TABLE]

where $\lambda$ is an arbitrary nonzero constant, is a Bäcklund transformation relating solutions of (84). More explicitly, if $h$ solves the hyperbolic Tzitzeica equation (84), then, substituting it in the system (85), one obtains a compatible first-order PDE system for $\alpha$ and $\beta$, whose solutions can be found by solving ODEs; for each solution $(\alpha,\beta)$, the function

[TABLE]

also satisfies the hyperbolic Tzitzeica equation (84).

Unlike the systems (1) and (2), substituting a solution $h$ of (84) into (85) (with fixed $\lambda$) yields a system whose solutions depend on 2 parameters instead of 1. Using our terminology (see Definition 2.10), one can verify that the system (85) corresponds to a rank-$2$ Bäcklund transformation.

Furthermore, in [AF15], it is shown that the following hyperbolic Monge-Ampère equation

[TABLE]

and the wave equation $z_{xy}=0$ admit no rank-$1$ Bäcklund transformation relating their solutions, but a rank-$2$ Bäcklund transformation does exist.

In a future paper, we will present a partial classification of homogeneous rank-$2$ Bäcklund transformations relating two hyperbolic Monge-Ampère systems. (It will turn out that the rank-$2$ Bäcklund transformation corresponding to (85) is nonhomogeneous.) Based on our classification so far, we expect that those homogeneous Bäcklund transformations (relating two hyperbolic Monge-Ampère systems) that are ‘genuinely’ rank-$2$ are quite few.

8. Acknowledgement

The author would like to thank his PhD thesis advisor, Prof. Robert L. Bryant, for all his guidance and support. He would like to thank Prof. Jeanne N. Clelland for her advice on the current work. Thanks to the referee for reading the manuscript and providing helpful suggestions and comments.

Appendix A Calculations in Theorem 5.2

This Appendix supplements the proof of Theorem 5.2 by providing more details of calculation. Most calculations below are computed using Maple™.

First consider the case when, on $U$, $\epsilon_{1}=\epsilon_{2}=1$. Since $P_{24},P_{33},$ $P_{66},P_{75}$ never appear in the equation (59), we can set them all to zero. Since $P_{14}$ and $P_{23}$ only appear in the term $(P_{14}-P_{23})\omega^{3}\wedge\omega^{4}$, we can set $P_{14}=0$. For similar reasons, we can set $P_{13},P_{55},P_{56}=0$. For convenience, we rename $A_{1}$ as $P_{81}$ and $A_{4}$ as $P_{84}$. Now there are $42$ functions $P_{ij}$ remaining, and they are determined.

For each $P_{ij}$, there exist functions $P_{ijk}$ defined on $U$ satisfying

[TABLE]

We call these $P_{ijk}$ the derivatives of $P_{ij}$.

Now, applying $\mathrm{d}^{2}=0$ to the equation (59), we obtain $106$ polynomial equations expressed in terms of all $42$ $P_{ij}$ and $186$ of all $252$ $P_{ijk}$. These equations imply:

[TABLE]

With these relations, all coefficients in (59) can be expressed in terms of $27$ $P_{ij}$. Repeating the steps above by defining the derivatives $P_{ijk}$ (now $162$ in all) and applying $\mathrm{d}^{2}=0$ to (59), we obtain a system of $91$ polynomial equations in these $27$ $P_{ij}$ and $124$ of all 162 $P_{ijk}$, which imply

[TABLE]

Using these relations and repeating the steps above, we obtain

[TABLE]

All coefficients in (59) are then expressed in terms of $24$ $P_{ij}$.

Now, corresponding to the remaining $24$ $P_{ij}$ are $144$ derivatives $P_{ijk}$. Applying $\mathrm{d}^{2}=0$ to (59) yields a system of $88$ polynomial equations, expressed in terms of the $24$ $P_{ij}$ and $122$ of the $144$ derivatives $P_{ijk}$. This system can be solved for $P_{ijk}$; in the solution, all $P_{ijk}$ are expressed explicitly in terms of the $24$ $P_{ij}$ and $64$ $P_{ijk}$ that are ‘free’.

Let ${a}=(a^{\alpha})$ $(\alpha=1,\ldots,24)$ stand for the $24$ remaining $P_{ij}$; let ${b}=(b^{\rho})$ $(\rho=1,\ldots,64)$ stand for the $64$ ‘free’ $P_{ijk}$. We already have

[TABLE]

for some real analytic functions $F^{\alpha}_{i}$ and $C^{i}_{jk}$ satisfying $C^{i}_{jk}+C^{i}_{kj}=0$.

Now compute the exterior derivatives

[TABLE]

and take into account (86) and (87). From this we obtain $2$-forms $\Omega^{\alpha}$ that are linear combinations of $\mathrm{d}b^{\rho}\wedge\omega^{i}$ and $\omega^{i}\wedge\omega^{j}$. Let $\hat{\Omega}^{\alpha}$ denote the part of $\Omega^{\alpha}$ consisting of linear combinations of $\mathrm{d}b^{\rho}\wedge\omega^{i}$ only. Replacing $\mathrm{d}b^{\rho}$ in $\hat{\Omega}^{\alpha}$ by $G^{\rho}_{i}\omega^{i}$ defines a linear map

[TABLE]

at each point of $U$.

Let $[\Omega]$ denote the equivalence class of $(\Omega^{\alpha})$ in the cokernel of $\phi$. One can show that $[\Omega]$ must vanish and that its vanishing leads to a system of $35$ equations for $a$ and $b$. This system can be solved for $12$ of the $64$ components of $b$. Apply such a solution and update $a^{\alpha}$, $b^{\rho}$ and the functions $F^{\alpha}_{i}$ accordingly.

It is not difficult to verify, using Maple™, that the updated $a^{\alpha}$ $(\alpha=1,\ldots,24)$, $b^{\rho}$ $(\rho=1,\ldots,52)$, $C^{i}_{jk}$ and $F^{\alpha}_{i}$ satisfy the conditions (A)-(C) in Step 1.

For Steps 2 and 3, calculation shows that the tableaux of free derivatives has Cartan characters

[TABLE]

and the dimension of its first prolongation

[TABLE]

The cases when, on $U$, $\epsilon_{1}$ and $\epsilon_{2}$ take other values follow similar steps. In each of these cases, the last nonzero Cartan character, computed at a corresponding stage, is $s_{3}=6$.

Appendix B Invariants of an Euler-Lagrange System

This Appendix supplements the proof of Proposition 6.2.

We start with the $G_{1}$-structure $\pi:\mathcal{G}_{1}\rightarrow M$ of a hyperbolic Monge-Ampère system $(M,\mathcal{I})$ (see [BGG03] or Section 3). Assume that $S_{2}=\bm{0}$ (i.e., the Euler-Lagrange case).

Recall that the $2\times 2$-matrix $S_{1}:\mathcal{G}_{1}\rightarrow\mathfrak{gl}(2,\mathbb{R})$ is equivariant under the $G_{1}$-action. By (30) and (31), it is easy to see that, when $\det(S_{1}(u))>0$ (resp., $\det(S_{1}(u))<0$) at $u\in\mathcal{G}_{1}$, the same is true for $\det(S_{1}(u\cdot g))$ for all $g\in G_{1}$, and the matrix $S_{1}(u)$ lies in the same $G_{1}$-orbit as ${\rm diag}(1,1)$ (resp., ${\rm diag}(1,-1)$).

Now assume that $\det(S_{1})>0$ holds on $\pi^{-1}U\subset\mathcal{G}$ for some domain $U\subset M$. By the discussion above, we can reduce to a subbundle $\mathcal{H}\subset\mathcal{G}_{1}$ defined by $S_{1}={\rm diag}(1,1)$.

It is easy to see that $\mathcal{H}$ is an $H$-structure on $U$ where

[TABLE]

is a (disconnected) $3$-dimensional Lie subgroup. Let the restriction of $\pi:\mathcal{G}_{1}\rightarrow M$ to $\mathcal{H}$ be denoted by the same symbol $\pi$.

One can verify that, restricted to $\mathcal{H}$, the $1$-forms $\phi_{7}-\phi_{3}$, $\phi_{6}-\phi_{2}$, $\phi_{5}-\phi_{1}$ and $\phi_{0}$ in the equation (20) become semi-basic relative to $\pi:\mathcal{H}\rightarrow U$. Hence, there exist functions $Q_{ij}$ defined on $\mathcal{H}$ such that

[TABLE]

where the summations are over $i=0,1,\ldots,4$. There are ambiguities in these $Q_{ij}$ as we can modify them without changing the form of the structure equation (20). Using such ambiguities, we can arrange that

[TABLE]

the remaining $Q_{ij}$ are then determined.

Applying $\mathrm{d}^{2}=0$ to (20) and reducing appropriately, we obtain

[TABLE]

This implies that

[TABLE]

Now all coefficients in the structure equation (20) are expressed in terms of $Q_{0i}$ $(i=0,1,\ldots,4)$ and $Q_{j0}$ $(j=5,6,7)$. By applying $\mathrm{d}^{2}=0$ to (20), it is not difficult to verify that, reduced modulo $\omega^{0},\omega^{1},\ldots,\omega^{4}$, the following congruences hold:

[TABLE]

The congruences (90) tell us how the remaining $Q_{ij}$ transform under the action by the identity component of $H$. Moreover, it is easy to compute directly from (20) to verify that

[TABLE]

hold for any $u\in\mathcal{H}$.

Note that ${H}$ is generated by its identity component and $h_{0}$. Combining (90) and (91), it is easy to see that $Q_{01}Q_{04}-Q_{02}Q_{03}$ and $|Q_{00}|$ are local invariants of the underlying Euler-Lagrange system.

Moreover, using (90) and (91), it is easy to see that the $H$-orbit of

[TABLE]

consists of all $2$-by-$2$ matrices with the same determinant as $q(u)$. Now we are ready to prove the following lemma.

Lemma B.1**.**

If $\det(q)\neq 0$ on $\mathcal{H}$, then there is a canonical way to define a coframing on $U$.

Proof. If the function $L:=\det(q)$ is nonvanishing on $U$, one can reduce to the subbundle $\mathcal{H}_{1}$ of $\mathcal{H}$ defined by $q={\rm diag}(L,1)$. It is easy to see that each fiber of $\mathcal{H}_{1}$ over $U$ contains a single element. ∎

Remark 6*.*

As a result of Lemma B.1, if $\det(q)\neq 0$ 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.

Now we proceed to complete the proof of Proposition 6.2. Recall that the coframing $(\eta^{0},\eta^{1},\ldots,\eta^{4})$ and the $\phi_{\alpha}$ in (78) verify the equation (20), $S_{1}={\rm diag}(1,1)$, and $S_{2}=\bm{0}$. Moreover, we have chosen the $\phi_{\alpha}$ to satisfy (89), where $Q_{ij}$ are computed using (88). By (78), it is immediate that

[TABLE]

Clearly, $\det(q)=Q_{01}Q_{04}-Q_{02}Q_{03}=-1/2\neq 0$. By Lemma B.1 and Remark 6, the hyperbolic Euler-Lagrange system considered in Proposition 6.2 has a symmetry of dimension at most $5$. Because that Euler-Lagrange system is homogeneous, it follows that its symmetry has dimension $5$.