The Diffiety Institute Preprint Series

Preprint DIPS–8/98 December 4, 1998

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Coverings and Integrability of the Gauss–Mainardi–Codazzi

Equations

by

Joseph Krasil′shchik, Michal Marvan

Coverings and Integrability of the Gauss–Mainardi–Codazzi Equations

Joseph Krasil′shchik, Michal Marvan

Abstract. Using covering theory approach (zero-curvature representations with the gauge group SL_{2}), we insert the spectral parameter into the Gauss–Mainardi–Codazzi equations in Tchebycheff and geodesic coordinates. For each choice, four integrable systems are obtained.

Introduction

When immersed in the Euclidean space E^{3}, surfaces with metric g = g_{11} dx^{2} + 2g_{12} dx dy + g_{22} dy^{2} and the second fundamental form b = b_{11} dx^{2} + 2b_{12} dx dy + b_{22} dy^{2} satisfy the Gauss–Mainardi–Codazzi equations (GMC) R_{ijl}^{k} = b_{ij}b_{l}^{k} − b_{il}b_{j}^{k} and b_{ij,k} = b_{ik,j}, where R_{ijl}^{k} are components of the curvature tensor. By imposing an algebraic constraint of the form L(g_{ij}, b_{ij}, x, y) = 0 we obtain a special instance of what is called reduced GMC equations. Many reduced GMC systems have been found integrable in the sense of soliton theory, e.g., in works [1, 2, 3, 4, 5, 10, 13], thus leading to integrable classes of surfaces. An example is provided by the so-called linear Weingarten surfaces determined by a linear relation αK + βH = γ (α, β, γ = const) between their Gauss and main curvatures, see the recent book [15] and references therein.

In this paper we apply the methods of [11] to a related problem. Namely, we check GMC equations, written here in Tchebycheff and geodesic coordinates, for existence of a zero-curvature representation (ZCR) with coefficients in the complex Lie algebra sl_{2}. For each choice of the coordinates we have found four cases possessing a non-removable parameter.

2 Joseph Krasil′shchik, Michal Marvan

An sl_{2}-valued ZCR is determined by an sl_{2}-valued form A dx + B dy satisfying A_{,y} − B_{,x} + [A, B] = 0. A gauge-equivalent ZCR is given by A′ dx + B′ dy with A′ = S_{,x}S^{−1} + SAS^{−1}, B′ = S_{,y}S^{−1} + SBS^{−1} for a suitable SL_{2}-matrix S. Any ZCR gauge-equivalent to the zero form is called trivial. A non-removable parameter is a parameter that cannot be removed by a gauge transformation.

1991 Mathematics Subject Classification. 35Q53, 58F07, 53C42.

Key words and phrases. coverings, zero-curvature representations, surface immersions, Gauss–Mainardi–Codazzi equations, spectral parameter.

The first author was partially supported by RFBR grant 97-01-00462 and INTAS grant 96-0793. This work was finished during author’s stay at ESI in October 1998.

The second author was supported from Project VS 96003 (“Global Analysis”) of the Ministry of Education, Youth and Sports, Czech Republic, and grant No. 201/98/0853 from the Czech Grant Agency.

As it happens, the theory of zero-curvature representations is naturally formulated in terms of coverings [9]. To obtain a well-defined construction, one needs to consider a linear covering [16] endowed with an action of a Lie group G. Then a ZCR related to this covering is determined by a closed 𝔤-valued horizontal 1-form. To make exposition self-contained, we expose here a geometrical theory of ZCR based on the covering theory in the category of differential equations.

1. Linear coverings and zero-curvature representations

Let O be a smooth manifold (possibly, infinite-dimensional) with an integrable finite-dimensional distribution C of dimension n. Integrability is understood in the formal sense here: a distribution is said to be integrable, if the module of vector fields lying in this distribution is closed with respect to commutator. Recall (see [9]) that a covering over the pair (O, C) is a locally trivial fiber bundle τ: Õ → O such that

1. The manifold Õ is endowed with an n-dimensional integrable distribution C̃.

2. The mapping dτ|C̃θ̃: C̃θ̃ → T_{θ}O is an isomorphism onto the plane

C_{τ(θ̃)} ⊂ T_{θ}O for any θ̃ ∈ Õ. Our main concern will be with manifolds O of the form E^{∞}, where E^{∞} is the infinite prolongation of a differential equation E ⊂ J^{k}(π), J^{k}(π) being the manifold of k-jets for some locally trivial bundle π: E → M. The corresponding distribution C is the Cartan distribution (see details in [8]).

Denote by D(N) the Lie algebra of vector fields on a smooth manifold N and by CD(O) ⊂ D(O) the subalgebra of vector fields lying in the distribution C. Then a covering structure in the bundle τ: Õ → O is identified with a flat C-connection in τ, i.e., with a mapping ∇ = ∇^{τ}: CD(O) → D(Õ) such that

1. ∇ is an F(O)-linear mapping, where F(O) is the algebra of smooth functions on O.

2. For any X ∈ CD(O), the field ∇_{X} ∈ D(Õ) projects to X by dτ.

3. For any X, Y ∈ CD(O) one has

∇_{[X,Y]} = [∇_{X}, ∇_{Y}]. (1)

We shall also say that ∇ determines a covering structure in τ.

Two coverings τ_{1} : Õ_{1} → O, τ_{2} : Õ_{2} → O with the connections ∇^{1} and ∇^{2} respectively are said to be equivalent, if there exists an equivalence φ of the bundles τ_{1}, τ_{2}, i.e., a diffeomorphism φ: Õ_{1} → Õ_{2} satisfying τ_{1} = τ_{2} ∘ φ, such that dφ ∘ ∇^{1} = ∇^{2}.

If U ⊂ O is a trivialization of the bundle τ, i.e., a domain such that the bundle τ|_{τ^{−1}(U)} : τ^{−1}(U) → U is trivial, then the covering structure over U is given by the splitting

∇_{X} = X + V_{X}, V_{X} ∈ D(τ^{−1}(U)), (2)

where V_{X} is a τ-vertical vector field, while the flatness condition is expressed in the form

V_{[X,Y]} = [X, V_{Y}] + [V_{X}, Y] + [V_{X}, V_{Y}]. (3)

Let now τ be a vector bundle and LF(Õ) denote the F(O)-module of fiber-wise linear smooth functions on Õ.

Definition 1.1. Let τ: Õ → O be a vector bundle.

1. A covering τ: Õ → O is called linear, if the field ∇_{X} preserves the module LF(Õ) for any X ∈ CD(O) (cf. [16], where the same concept is introduced, though without using the term “covering”).

2. Two linear coverings are said to be equivalent, if there exists their equivalence φ, which is a morphism of vector bundles (i.e., is fiberwise linear).

Let U ⊂ O be a trivialization of the bundle τ and w^{1}, …, w^{r}, … be coordinates along the fiber. Then a covering τ is linear if and only if the field V_{X} represents in the form

V_{X} = ∑_{r} ( ∑_{s} V_{X}^{rs}w^{s})\frac{∂}{∂w^{r}} (4)

for any X ∈ CD(O), where V_{X}^{rs} are smooth functions on O and all internal sums are finite. Thus, any linear covering is locally determined by the system of matrices ∥V_{i}^{rs}∥, where V_{i}^{rs} = V_{X_{i}}^{rs}, X_{i}, i = 1, …, n, being a local basis of the distribution C.

If an automorphism of the vector bundle τ is locally given by a matrix S, then the matrices V_{X} are transformed by the formula

V_{X} ↦ X(S)S^{−1} + SV_{X}S^{−1}, (5)

where X(S) denotes the component-wise action.

Example 1.1. Let τ: Õ → O be a covering. Denote by CΛ^{1}(O) ⊂ Λ^{1}(O) the F(O)-submodule consisting of forms such that i_{X}ω = 0 for any X ∈ CD(O). In a similar way, we obtain the F(Õ)-submodule C̃Λ^{1}(Õ) in Λ^{1}(Õ). Then, since C̃D(Õ) projects to CD(O) by τ_{∗}, one has τ^{∗}(CΛ^{1}(O)) ⊂ C̃Λ^{1}(Õ). Denote by CΛ^{1}(Õ) ⊂ C̃Λ^{1}(Õ) the F(Õ)- submodule generated by τ^{∗}(CΛ^{1}(O)). Then CΛ^{1}(Õ) is stable with respect to the Lie action of vector fields lying in CD(Õ) and we can extend this action to the quotient module C̃Λ^{1}(Õ)/CΛ^{1}(Õ). One can easily see that the corresponding vector bundle τ_{ℓ}: T_{C}^{∗}(Õ) → Õ is endowed in this way with a linear covering structure.

Example 1.2. Consider a construction dual to that of Example 1.1. Let again τ: Õ → O be an arbitrary covering. In the tangent bundle T Õ → Õ, take the subbundle τ^{ℓ}: T^{v}Õ → Õ of τ-vertical tangent vectors. Obviously, the bundle τ^{ℓ} is dual to τ_{ℓ}. If a vector field Y ∈ D(Õ) is such that dτYθ̃ lies in the plane C_{θ}, θ = τ(θ̃), we can define its vertical component Y^{v} by setting

Y v ; θ̃ := Yθ̃ − ∇(dτ(Yθ̃)),

at any point θ̃ ∈ Õ. Take a τ-vertical vector field Z, i.e., a section of the bundle τ^{ℓ}, and a field X ∈ C̃D(Õ). Then the field [X, Z] possesses the above formulated property and the relation

˜∇_{X}(Z) := [X, Z]^{v}

determines a covering structure in the projection τ^{ℓ}: T^{v}Õ → Õ. This covering structure in the bundle τ^{ℓ} is linear and is called the linearization of the covering τ (cf. [7, 12]).

Example 1.3. Let A be a commutative 𝕜-algebra, 𝕜 being a commutative ring, and P be an A-module. Recall that a derivation of the module P is a linear mapping X̄: P → P satisfying X̄(ap) = X(a)p + aX̄(p) for all a ∈ A, p ∈ P, and some derivation X: A → A of the algebra A. Denote the A-module of all such derivations by Der(P). Obviously, Der(P) is a Lie 𝕜-algebra with respect to the commutator.

In particular, let τ: Õ → O be a linear covering. Then, by definition, the covering structure in τ determines the homomorphism of Lie algebras ∇: CD(O) → Der(LF(Õ)), which splits the natural projection Der(LF(Õ)) → D(O), X̄ ↦ X. The kernel of this projection coincides with the module End(LF(Õ)) = Γ(End(τ)), where Γ(·) denotes the module of sections. This kernel is an ideal of the Lie algebra Der(LF(Õ)). Thus, we obtain a linear covering structure in the bundle End(τ).

In local coordinates, an endomorphism φ is represented by a matrix A_{φ} and the action of a field X ∈ CD(O) on A_{φ} is expressed by the formula

X(A_{φ}) + [V_{X}, A_{φ}],

where X(A_{φ}) is understood as the component-wise action.

Assume now that a Lie group G acts in the bundle τ by linear automorphisms of this bundle, i.e., a representation ρ: G → Aut(τ) is given. Let 𝔤 be the corresponding Lie algebra. Then the representation ρ yields the following commutative diagram:

O × 𝔤 ⟶^{ρ̄} T^{v}Õ
pr^{O}↓  ↓_{τ^{ℓ}}
O ⟵_{τ} Õ

Definition 1.2. Let ∇^{0}, ∇^{1} be two linear covering structures in the vector bundle τ: Õ → O.

1. We say that the covering structures ∇^{0} and ∇^{1} differ by a zerocurvature representation (with the gauge group G, or with the gauge algebra 𝔤), if for any X ∈ CD(O) there exists a 𝔤-valued function g_{X} ∈ Γ(pr_{O}) linear in X and such that

∇_{X}^{1} − ∇_{X}^{0} = ρ̄ ∘ g_{X} ∘ τ.

In other words, the vertical vectors ∇_{X}^{1} and ∇_{X}^{0} differ by an element of the Lie algebra 𝔤 at any point θ ∈ O.

2. Let ∇^{0}, ∇^{1} and □^{0}, □^{1} be two pairs satisfying the above definition. They are said to be equivalent, if there exists an element g ∈ G such that ρ(g) is an equivalence of ∇^{0} to □^{0} and of ∇^{1} to □^{1}.

Thus, equivalence in the sense of the previous definition means that the matrices S in (5) belong to the group G.

Remark 1.1. In coordinate computations for particular differential equations E^{∞} = O, the bundle τ is usually trivial, τ = pr_{E^{∞}}: O = E^{∞} × ℝ^{r} → E^{∞}, and the structure ∇^{0} is also trivial (i.e., is such that the matrices V_{X} vanish for all X ∈ CD(E^{∞}), see eq. (4)). This should be the reason why it is never taken into consideration explicitly and is accepted “by default”: only the covering ∇^{1} is included into definition.

From now on we take O = E^{∞}, fix a vector bundle τ: Ẽ^{∞} → E^{∞}, where E ⊂ J^{k}(π) is a differential equation, and assume that a Lie group G acts in τ in the above described way. Consider a covering structure ∇^{0} in the bundle τ. Then ∇^{0} determines a flat connection in the bundle Ẽ^{∞} ⟶^{τ} E^{∞} ⟶^{π_{∞}} M extending the Cartan connection in π_{∞}: E^{∞} → M. This connection is uniquely determined by the corresponding connection form U^{0} and the condition ∇^{0} to be a covering structure is equivalent to the identity [[U^{0}, U^{0}]] = 0, where [[· , ·]] = 0 is the Frölicher–Nijenhuis bracket (see [6] for more details).

If ∇^{1} is another covering structure in τ, then the connection forms U^{0} and U^{1} differ by a horizontal vector-valued form ω_{01} = U^{1} − U^{0}. From [6] we immediately obtain the following

Proposition 1.1. Two linear covering structures in the vector bundle τ satisfy Definition 1.2 (1) if and only if

1. The form ω_{01} is 𝔤-valued.

2. The identity

∂^{0} = ω_{01} + \frac{1}{2}[[ω_{01}, ω_{01}]] = 0 (6)

holds, where ∂^{0} = [[ω_{01}, ·]] is the differential associated to ∇^{0} by the Frölicher–Nijenhuis bracket.

Remark 1.2. In the situation discussed in Remark 1.1, the action of the differential ∂^{0} on the form ω_{01} coincides with the component-wise action of −d_{h}, where d_{h} is the de Rham horizontal differential on E^{∞}. Thus, eq. (6) can be rewritten as

d_{h}ω_{01} = \frac{1}{2}[[ω_{01}, ω_{01}]]

in this case.

2. Tchebycheff coordinates

This section contains results of computation of sl_{2}-valued ZCR’s for reduced GMC equations in Tchebycheff coordinates. The method used is taken from [11] and shortly explained at the end of this section.

Fix an arbitrary system of Tchebycheff coordinates x, y. Then we have

g = dx^{2} + 2 cos f dx dy + dy^{2},
b = b_{11} dx^{2} + 2b_{12} dx dy + b_{22} dy^{2}.

The Gauss–Mainardi–Codazzi equations [14], eq. (74b,c), are

f_{,xy} = \frac{b_{12}^{2} − b_{11}b_{22}}{sin f} ,
(7)  b_{11,y} = b_{12,x} + \frac{b_{22} − b_{12} cos f}{sin f} f_{,x},
b_{12,y} = b_{22,x} − \frac{b_{11} − b_{12} cos f}{sin f} f_{,y}.

There always exists a nonparametric sl_{2}-valued zero-curvature representation (see [14], eq. (84)), derived from the Gauss–Weingarten equations. In Tchebycheff coordinates we have

A = \frac{i}{2} (\begin{matrix}f_{x} & \frac{e^{if}b_{11} − b_{12}}{sin f} \\ \frac{e^{−if}b_{11} − b_{12}}{sin f} & −f_{x} \end{matrix} ) ,

B = \frac{i}{2} (\begin{matrix}0 & \frac{e^{if}b_{12} − b_{22}}{sin f} \\ \frac{e^{−if}b_{12} − b_{22}}{sin f} & 0 \end{matrix} ) .

(8)

It is well known that this zero-curvature representation does not belong to any 1-parametric family. However, imposing one additional relation between the unknowns x, y, f, b_{11}, b_{12}, b_{22} we obtain four distinct classes of surfaces admitting a 1-parametric ZCR in the above sense. The following proposition may be proved by straightforward computation. We obtained it by methods of [11].

Proposition 2.1. Let X_{1}(x), X_{2}(x), Y_{1}(y), Y_{2}(y) be arbitrary functions and Z be a constant. Denote

L := (b_{11}b_{22} − b_{12}^{2})Z + Y_{1}b_{11} + X_{1}b_{22}
+ [(X_{2} − Y_{2}) sin f − (X_{1} + Y_{1}) cos f]b_{12}
− (X_{1}Y_{1} + X_{2}Y_{2}) sin^{2} f + (X_{1}Y_{2} − X_{2}Y_{1}) sin f cos f.

(9)

If L = 0, then the matrices

A = \frac{1}{2} (\begin{matrix}if_{x} & i\frac{e^{if}b_{11} − b_{12}}{sin f} + X_{1} − iX_{2} \\ i\frac{e^{−if}b_{11} − b_{12}}{(Z + 1) sin f} − \frac{X_{1} + iX_{2}}{Z + 1} & −if_{x} \end{matrix}) ,

B = \frac{1}{2} (\begin{matrix}0 & i\frac{e^{if}b_{12} − b_{22}}{sin f} + (Y_{1} − iY_{2})e^{if} \\ i\frac{e^{−if}b_{12} − b_{22}}{(Z + 1) sin f} − \frac{Y_{1} + iY_{2}}{Z + 1} e^{−if} & 0 \end{matrix})

give a zero-curvature representation for equation (7).

Assume now that the functions X_{1}, X_{2}, Y_{1}, Y_{2}, Z depend on an auxiliary variable t. To be a possible parameter of the ZCR, t must not be a parameter of the equation (7) itself. In other words, the condition L = 0 must not depend on t.

If ∂L/∂t = 0, then one easily derives from (9) that X_{1} = X_{2} = Y_{1} = Y_{2} = Z = 0 and the ZCR is the same as (8), without any parameter.

If ∂L/∂t − L/t = 0, then L(t) = tL(1), and the condition L(t) = 0 does not depend on t either. Combining with (9), we arrive at the classification given below. We use the notation

K = \frac{b_{11}b_{22} − b_{12}^{2}}{sin^{2} f}

for the Gauss curvature and

H = \frac{b_{11} − 2b_{12} cos f + b_{22}}{2 sin^{2} f}

for the mean curvature. Functions X_{i}, Y_{i} are as in the proposition above.

Case 1. X_{1} = Y_{1} ≠ 0. This class corresponds to linear Weingarten surfaces determined by a linear relation

K + 2κH + λ = 0

between the two curvatures. Here κ, λ are arbitrary constants.

8 Joseph Krasil′shchik, Michal Marvan

Our t-parametrized zero-curvature representation is

A = \frac{1}{2} (\begin{matrix}if_{x} & i \frac{e^{if}b_{11} − b_{12}}{sin f} + Δ^{−}(κ, λ, t) \\ i \frac{e^{−if}b_{11} − b_{12}}{(t + 1) sin f} − \frac{Δ^{+}(κ, λ, t)}{t + 1} & −if_{x} \end{matrix}) ,

B = \frac{1}{2} (\begin{matrix}0 & i \frac{e^{if}b_{12} − b_{22}}{sin f} + Δ^{−}(κ, λ, t)e^{if} \\ i \frac{e^{−if}b_{12} − b_{22}}{(t + 1) sin f} − \frac{Δ^{+}(κ, λ, t)}{t + 1}e^{−if} & 0 \end{matrix}) ,

where Δ^{±}(κ, λ, t) = κt ± \sqrt{κ^{2}t^{2} + λt}. Case 2. X_{1} = Y_{1} = 0. If

K + 2κ\frac{b_{12}}{sin f} − λ = 0,

where κ, λ are constants, then there is a t-parametrized zero-curvature representation

A = \frac{i}{2} (\begin{matrix}f_{x} & \frac{e^{if}b_{11} − b_{12}}{sin f} − Δ^{+}(κ, λ, t) \\ \frac{e^{−if}b_{11} − b_{12}}{(t + 1) sin f} − \frac{Δ^{+}(κ, λ, t)}{t + 1} & −f_{x} \end{matrix}) ,

B = \frac{i}{2} (\begin{matrix}0 & \frac{e^{if}b_{12} − b_{22}}{sin f} + Δ^{−}(κ, λ, t)e^{if} \\ \frac{e^{−if}b_{12} − b_{22}}{(t + 1) sin f} + \frac{Δ^{−}(κ, λ, t)}{t + 1}e^{−if} & 0 \end{matrix}) .

Case 3a. X_{1} = 0, Y_{1} ≠ 0. If

K sin^{2} f − Y_{1}b_{11} + (Y_{1} cos f + Y_{2} sin f)b_{12} = 0,

then

A = \frac{i}{2} (\begin{matrix}f_{x} & \frac{e^{if}b_{11} − b_{12}}{sin f} \\ \frac{e^{−if}b_{11} − b_{12}}{(t + 1) sin f} & −f_{x} \end{matrix} ) ,

B = \frac{i}{2} (\begin{matrix}0 & \frac{e^{if}b_{12} − b_{22}}{sin f} + (iY_{1} + Y_{2})te^{if} \\ \frac{e^{−if}b_{12} − b_{22}}{(t + 1) sin f} − \frac{iY_{1} − Y_{2}}{t + 1} te^{−if} & 0 \end{matrix}) .

is a t-parametrized zero-curvature representation.

Case 3b. X_{1} ≠ 0, Y_{1} = 0: The relation is

K sin^{2} f + (X_{1} cos f + X_{2} sin f)b_{12} − X_{1}b_{22} = 0

and the t-parametrized zero-curvature representation is

A = \frac{i}{2} (\begin{matrix}f_{x} & \frac{e^{if}b_{11} − b_{12}}{sin f} + (iX_{1} − X_{2})t \\ \frac{e^{−if}b_{11} − b_{12}}{(t + 1) sin f} − \frac{iX_{1} + X_{2}}{t + 1} t & −f_{x} \end{matrix}) ,

B = \frac{i}{2} (\begin{matrix}0 & \frac{e^{if}b_{12} − b_{22}}{sin f} \\ \frac{e^{−if}b_{12} − b_{22}}{(t + 1) sin f} & 0 \end{matrix} ) .

Cases 3a and 3b transform one to another under the x ↔ y symmetry.

Case 4. 0 ≠ X_{1} ≠ Y_{1} ≠ 0: If

K sin^{2} f − Y_{1}b_{11} + (X_{1}e^{if} + Y_{1}e^{−if})b_{12} − X_{1}b_{22} = 0,

then

A = \frac{i}{2} (\begin{matrix}f_{x} & \frac{e^{if}b_{11} − b_{12}}{sin f} \\ \frac{e^{−if}b_{11} − b_{12}}{(t + 1) sin f} − \frac{2iX_{1}t}{t + 1} & −f_{x} \end{matrix} ) ,

B = \frac{i}{2} (\begin{matrix}0 & \frac{e^{if}b_{12} − b_{22}}{sin f} \\ \frac{e^{−if}b_{12} − b_{22}}{(t + 1) sin f} − \frac{2iY_{1}t}{t + 1}e^{−if} & 0 \end{matrix} ) .

is a t-parametrized zero-curvature representation.

Case 4 is incompatible with the real geometry of E^{3}. A formally real class is obtained after transformation f ↦ if, x ↦ ix, and y ↦ iy.

Two natural questions arise: whether the parameter t cannot be removed by gauge transformation and whether there might be integrable cases outside this classification. To give our answers, we first recall some facts from [11].

In the simplest style, still applicable to the GMC equations, for a system of PDE {F^{l} = 0}_{l=1}^{N} the characteristic N-tuple of a ZCR A dx+ B dy consists of sl_{2} matrices C_{l}, l = 1, …, N, satisfying A_{,y} − B_{,x} + [A, B] = C_{l}F^{l}. Then the gauge-equivalent ZCR with respect to a gauge matrix S has {SC_{l}S^{−1}}_{l=1}^{N} as its characteristic N-tuple. Incidentally, in all four cases the characteristic triple is one and the same:

\frac{i}{2} ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , \frac{i}{2 sin f} ( \begin{matrix}0 & e^{if} \\ \frac{e^{−if}}{t + 1} & 0 \end{matrix} ) , \frac{−i}{2 sin f} ( \begin{matrix}0 & 1 \\ \frac{1}{t + 1} & 0 \end{matrix} ) .

As one obviously cannot remove t by conjugation, t is a true parameter.

To give at least partial answer to our second question, we explain the assumptions we used in derivation of Proposition 2.1. By [11], the characteristic triple C = {C_{1}, C_{2}, C_{3}} also satisfies

R̂^{∗}(C) = 0, (10)

where R^{∗} is the formal adjoint to the operator of universal linearization, and R̂^{∗} is obtained from R^{∗} by replacement of total derivatives D_{x}, D_{y} with “covariant total derivatives” D_{x} −ad_{A}, D_{y} −ad_{B} (or, equivalently, by lifting the operator R^{∗} to the corresponding covering space by means of the connection defining the covering). The procedure then consists in solving the equation R̂^{∗}(C̄) = 0 on unknowns A, B, C̄, where C̄ is a normal form of C with respect to conjugation.

For the above ZCR (8), the characteristic triple C = {C_{1}, C_{2}, C_{3}} is

(11) \frac{i}{2} ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , \frac{i}{2 sin f} ( \begin{matrix}0 & e^{if} \\ e^{−if} & 0 \end{matrix} ) , −\frac{i}{2 sin f} ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

In (11), C_{1} is in Jordan normal form and is diagonal. Then, if the zero-curvature representation (A, B) belongs to a 1-parameter family, then the Jordan form for C_{1} will be diagonal for adjacent members of the family as well. Similarly, since C_{2}, C_{3} are both non-diagonal, they will be so for the adjacent members. By conjugation with an appropriate diagonal matrix leaving C_{1} unchanged, we may set one of the non-diagonal terms of C_{2} or C_{3} to 1. Consequently, during our computation we may assume the characteristic element to be of the form

C_{1} = ( \begin{matrix}r & 0 \\ 0 & −r \end{matrix} ) , C_{2} = ( \begin{matrix}p_{1} & p_{2} \\ p_{3} & −p_{1} \end{matrix} ) , C_{3} = ( \begin{matrix}q_{1} & 1 \\ q_{3} & −q_{1} \end{matrix} ) .

The functions r, p_{1}, p_{2}, p_{3}, q_{1}, q_{3} together with the entries of the sl_{2}- matrices

A = ( \begin{matrix}a_{1} & a_{2} \\ a_{3} & −a_{1} \end{matrix} ) , B = ( \begin{matrix}b_{1} & b_{2} \\ b_{3} & −b_{1} \end{matrix} )

are twelve unknowns that are to be determined along with the yet unknown dependence of b_{22} on x, y, f, b_{11}, b_{12}.

The determining system (10) is rather complex (typeset here, it would fill approximately four pages) and manageable only with the aid of computer algebra. To solve it, we demanded that the expressions a_{1}, b_{1} depend on x, y, f, f_{x}, f_{y}, the expressions a_{2}, a_{3}, b_{2}, b_{3} depend on x, y, f, b_{11}, b_{12} (cf. eq. (8)), and the expressions r, p_{1}, p_{2}, p_{3}, q_{1}, q_{3} depend on x, y, f (cf. (11)). Finally, we demanded that the function L in the constraint L = 0 actually depends on b_{11}, b_{12}, b_{22} and is nonlinear in these variables. Under assumptions stated, the result in Proposition 2.1 is exhaustive.

3. Geodesic coordinates

Assume now that the immersed surface is endowed with the geodesic coordinates, so that the metric is g = dx^{2} + f dy^{2}, f > 0. As before, the second fundamental form is b = b_{11} dx^{2} + 2b_{12} dx dy + b_{22} dy^{2}. Then the Gauss–Mainardi–Codazzi equations are

f_{,xx} = \frac{f_{,x}^{2}}{2f} + 2(b_{12}^{2} − b_{11}b_{22}),
(12)  b_{11,y} = b_{12,x} + \frac{f_{,x}}{2f} b_{12},
b_{12,y} = b_{22,x} + \frac{f_{,y}}{2f} b_{12} − \frac{f_{,x}}{2f} (b_{22} + fb_{11}).

The nonparametric zero-curvature representation is A dx + B dy with

\begin{matrix}A = −\frac{1}{2} ( \begin{matrix}0 & \frac{ib_{12}}{\sqrt{f}} + b_{11} \\ \frac{ib_{12}}{\sqrt{f}} − b_{11} & 0 \end{matrix} ) , \\ B = −\frac{1}{2} ( \begin{matrix}\frac{if_{x}}{2\sqrt{f}} & \frac{ib_{22}}{\sqrt{f}} + b_{12} \\ \frac{ib_{22}}{\sqrt{f}} − b_{12} & −\frac{if_{x}}{2\sqrt{f}} \end{matrix} ) . \end{matrix} (13)

Proposition 3.1. Let Y_{1}(y), Y_{2}(y), Y_{3}(y), Y_{4}(y) be arbitrary functions and Z be a constant. Use the notation

\begin{matrix}L := \frac{1 − Z}{Z√ ; \overline{f}} (b_{11}b_{22} − b_{12}^{2}) − (Y_{1}√ ; \overline{f} + x\frac{∂Y_{2}}{∂y} + Y_{4}) b_{11} \\ + (x\frac{∂Y_{1}}{∂y} + Y_{3}) \frac{1}{√ ; \overline{f}} b_{12} − \frac{Y_{1}}{√ ; \overline{f}} b_{22} \\ −Y_{1}Y_{4} + Y_{2}Y_{3} − xY_{1} \frac{∂Y_{2}}{∂y} + x\frac{∂Y_{1}}{∂y} Y_{2} − (Y_{1}^{2} + Y_{2}^{2})√ ; \overline{f}. \end{matrix} (14)

12 Joseph Krasil′shchik, Michal Marvan

If L = 0, then the matrices

A = −\frac{1}{2} (\begin{matrix}0 & (\frac{ib_{12}}{√ ; \overline{f}} + b_{11} + Δ^{−}) Z \\ \frac{ib_{12}}{√ ; \overline{f}} − b_{11} − Δ^{+} & 0 \end{matrix}) ,

B = −\frac{1}{2}

× (\begin{matrix}\frac{if_{x}}{2√ ; \overline{f}} & (\frac{ib_{22}}{√ ; \overline{f}} + b_{12} + x\frac{∂Δ^{+}}{∂y} + iΔ^{+}√ ; \overline{f} + Y_{3} + iY_{4})Z \\ \frac{ib_{22}}{√ ; \overline{f}} − b_{12} − x\frac{∂Δ^{−}}{∂y} + iΔ^{−}√ ; \overline{f} − Y_{3} + iY_{4} & −\frac{if_{x}}{2√ ; \overline{f}} \end{matrix}) ,

where Δ^{±} = Y_{1} ± iY_{2}, form a zero-curvature representation for equation (12).

Denoting

K = \frac{b_{11}b_{22} − b_{12}^{2}}{f} , H = \frac{1}{2} (b_{11} + \frac{b_{22}}{f} )

the Gauss and mean curvature, respectively, we get the following five integrable classes. Case 1. Linear Weingarten surfaces

K + αH + β = 0,

where α, β are arbitrary constants. Then

A = −\frac{1}{2} (\begin{matrix}0 & (t + 1) (\frac{ib_{12}}{\sqrt{f}} + b_{11}) + □^{−} \\ \frac{ib_{12}}{\sqrt{f}} − b_{11} − \frac{□^{+}}{t + 1} & 0 \end{matrix}) ,

B = −\frac{1}{2} (\begin{matrix}\frac{if_{x}}{2\sqrt{f}} & (t + 1) (\frac{ib_{22}}{\sqrt{f}} + b_{12}) + □^{−} i\sqrt{f} \\ \frac{ib_{22}}{\sqrt{f}} − b_{12} + \frac{□^{+}}{t + 1} i\sqrt{f} & −\frac{if_{x}}{2\sqrt{f}} \end{matrix}) ,

where □^{±} = □^{±}(α, β, t) = αt ± \sqrt{α^{2}t^{2} − βt − βt^{2}}. Case 2. The class

K + Y \frac{b_{11}}{\sqrt{f}} + γ = 0,

Coverings and Integrability of the Gauss–Mainardi–Codazzi Equations 13

where Y is an arbitrary function of y and γ is a constant. In this case

A = −\frac{1}{2} (\begin{matrix}0 & (t + 1) (\frac{ib_{12}}{\sqrt{f}} + b_{11} + i\sqrt{\frac{γt}{t + 1}}) \\ \frac{ib_{12}}{\sqrt{f}} − b_{11} + i\sqrt{\frac{γt}{t + 1}} & 0 \end{matrix}) ,

B = −\frac{1}{2} (\begin{matrix}\frac{if_{x}}{2\sqrt{f}} & (t + 1) (\frac{ib_{22}}{\sqrt{f}} + b_{12} + Ȳ − \sqrt{\frac{γft}{t + 1}}) \\ \frac{ib_{22}}{\sqrt{f}} − b_{12} + Ȳ + \sqrt{\frac{γft}{t + 1}} & −\frac{if_{x}}{2\sqrt{f}} \end{matrix}) ,

where Ȳ = i tY /(t + 1).

Case 3. The class

K + Y_{1} \frac{b_{11}}{\sqrt{f}} + Y_{2} \frac{b_{12}}{f} = 0,

where Y_{1}, Y_{2} are arbitrary functions of y. Then

A = −\frac{1}{2} (\begin{matrix}0 & (t + 1) (\frac{ib_{12}}{\sqrt{f}} + b_{11}) \\ \frac{ib_{12}}{\sqrt{f}} − b_{11} & 0 \end{matrix}) ,

B = −\frac{1}{2} (\begin{matrix}\frac{if_{x}}{2\sqrt{f}} & (t + 1) (\frac{ib_{22}}{\sqrt{f}} + b_{12}) + (iY_{1} − Y_{2})t \\ \frac{ib_{22}}{\sqrt{f}} − b_{12} + \frac{(iY_{1} + Y_{2})t}{t + 1} & −\frac{if_{x}}{2\sqrt{f}} \end{matrix}) .

Case 4. The class

K + Y_{1}H − \frac{1}{2}Y (\frac{b_{12}}{f} + i\frac{b_{11}}{\sqrt{f}} ) = 0, Y = x\frac{∂Y_{1}}{∂y} + Y_{2},

where Y_{1}, Y_{2} are arbitrary functions of y. Then

\begin{matrix}A = −\frac{1}{2} (\begin{matrix}0 & (t + 1) (\frac{ib_{12}}{\sqrt{f}} + b_{11}) + Y_{1}t \\ \frac{ib_{12}}{\sqrt{f}} − b_{11} & 0 \end{matrix}) , \\ B = −\frac{1}{2} (\begin{matrix}\frac{if_{x}}{2\sqrt{f}} & (t + 1) (\frac{ib_{22}}{\sqrt{f}} + b_{12}) + tY + iY_{1}t\sqrt{f} \\ \frac{ib_{22}}{\sqrt{f}} − b_{12} & −\frac{if_{x}}{2\sqrt{f}} \end{matrix}) . \end{matrix}

Case 4 is incompatible with the real geometry of E^{3}, but nevertheless we may turn it into a formally real class by allowing f to be negative.

Acknowledgements

The authors are grateful to J. Cieśliński J. and A. Sym for calling their attention to the problem. The send author would also like to thank A. Bobenko and E. Ferapontov for helpful criticism.

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Moscow Institute for Municipal Economy and Independent University of Moscow Correspondence to: 1st Tverskoy-Yamskoy per. 14, Apt. 45, 125047 Moscow, Russia

E-mail address: josephk@glasnet.ru

Dept. of Mathematics, Silesian University at Opava, Bezručovo nám. 13, 746 01 Opava, Czech Republic

E-mail address: Michal.Marvan@fpf.slu.cz