Time-Like Isothermic Surfaces

Associated to Grassmannian Systems

M. P. Dussan and M. A. Magid

Received: March 3, 2005 Revised: August 10, 2005

Communicated by Patrick Delorme

Abstract. We establish that, as is the case with space-like isothermic surfaces, time-like isothermic surfaces in pseudo-riemannian space ℝ^{n−j,j} are associated to the O(n − j + 1, j + 1)/O(n − j, j) × O(1, 1)- system.

2000 Mathematics Subject Classification: Primary 53A35; Secondary53C40. Keywords and Phrases: Isothermic surfaces, integrable systems, Grassmannian systems

1 Introduction

There is no doubt that the recent renaissance in interest about isothermic surfaces is principally due to the fact that they constitute an integrable system, as can be seen in several new works where it is shown, for instance, that the theory of isothermic surfaces in ℝ^{3} can be reformulated within the modern theory of soliton theory [4], or can be analyzed as curved flats in the symmetric space O(4, 1)/O(3) × O(1, 1) [3]. Additionally, in a recent work of Burstall [1], we find an account of the theory of isothermic surfaces in ℝ^{n} from both points of view: of classic surfaces geometry as well as from the perspective of the modern theory of integrable systems and loop groups. The key point of this class of surfaces, as well as of the classic pseudospherical surfaces and those with constant mean curvature, is that the Gauss-Codazzi equations are soliton equations, they have a zero-curvature formulation, i.e., the equations should amount to the flatness of a family of connections depending on an auxiliary parameter. It is well known that this special property allows actions of an infinite dimensional group on the space of solutions, called the “dressing action” in the soliton theory. For instance, the geometric transformations found for the surfaces above such as Backlund, Darboux and Ribaucour, arise as the dressing action of some simple elements. More recently, in 1997, Terng in [12] defined a new integrable system, the U/Ksystem (or n-dimensional system associated to U/K), which is very closely related to that of curved flats discovered by Ferus and Pedit [8]. Terng, in [12], showed that the U/K-system admits a Lax connection and initiated the project to study the geometry associated with these systems. In fact, using the existence of this Lax connection, in 2002 Bruck-Du-Park-Terng ([2]) studied the geometry involved in two particular cases of U/K-systems: O(m+n)/O(m)×O(n) and O(m + n, 1)/O(m) × O(n, 1)-systems. For these cases, they found that the isothermic surfaces, submanifolds with constant sectional curvatures and submanifolds admitting principal curvature coordinates are associated to them, and, that the dressing actions of simple elements on the space of solutions corresponded to Backlund, Darboux and Ribaucour transformations for submanifolds. Later, looking for a relation between space-like isothermic surfaces in pseudoriemannian space and the U/K-systems, the first author found in [6] that the class of space-like isothermic surfaces in pseudo-riemannian space ℝ^{n−j,j} for any signature j, were associated to the O(n−j+1, j+1)/O(n−j, j)×O(1, 1)-system. The principal point in this study was the suitable choice of a one maximal abelian subalgebra, which allows one to obtain elliptic Gauss equations, which are appropriate for space-like surfaces. The main goal of this note is to show that time-like isothermic surfaces in the pseudo-riemannian space ℝ^{n−j,j} are also associated to the O(n − j + 1, j + 1)/O(n − j, j) × O(1, 1)-systems, defined by other two maximal abelian subalgebras, that are not conjugate under the Ad(K)-action, where K = O(n − j, j) × O(1, 1). We study the class of time-like surfaces both with diagonal and non-diagonal second fundamental form, in the cases when its principal curvatures are real and distinct and when they are complex conjugates. We show that an isothermic pair i.e, two isothermic time-like surfaces which are dual, in the diagonal or non-diagonal case, are associated to our systems. Additionally, in this paper we present a review of the principal results recently obtained in [7], about the geometric transformations associated to the dressing action of certain elements with two simple poles on the space of solutions of the complex O(n − j + 1, j + 1)/O(n − j, j) × O(1, 1)-system, corresponding to the timelike isothermic surfaces whose second fundamental forms are nondiagonal. The geometric transformations associated to real case of timelike isothermic surfaces with second fundamental forms are diagonal, were already studied in [14]. Finally, we note that all time-like surfaces of constant mean curvature, all timelike rotation surfaces and all time-like members of Bonnet families are examples of time-like isothermic surfaces [11].

2 The U/K-systems

In this section, we introduce the definition of U/K-system given by Terng in [12]. Let U be a semi-simple Lie group, σ an involution on U and K the fixed point set of σ. Then U/K is a symmetric space. The Lie algebra K is the fixed point set of the differential σ_{∗} of σ at the identity, in others words, it is the +1 eigenspace of σ_{∗}. Let now P denote the -1 eigenspace of σ_{∗}. Then we have the Lie algebra of U, U = K ⊕ P and

[K, K] ⊂ K, [K, P] ⊂ P, [P, P] ⊂ K.

Let A be a non-degenerate maximal abelian subalgebra in P, a_{1}, a_{2}, ..., a_{n} a basis for A and A^{⊥} the orthogonal complement of A in the algebra U with respect to the Killing form <, >. Then the U/K-system is the following first order system of non-linear partial differential equations for v : ℝ^{n} → P ∩ A^{⊥}.

[a_{i}, v_{x_{j}}] − [a_{j}, v_{x_{i}}] = [[a_{i}, v], [a_{j}, v]], 1 ≤ i ≠ j ≤ n,  (1)

where v_{x_{j}} = \frac{∂v}{∂x_{j}}.

The first basic result established in [12] is the existence of one-parameter family of connections whose flatness condition is exactly the U/K-system.

Theorem 2.1. ([12]) The following statements are equivalent for a map v : ℝ^{n} → P ∩ A^{⊥}:

i) v is solution of the U/K-system (1). ii)

[\frac{∂}{∂x_{i}} + λa_{i} + [a_{i}, v], \frac{∂}{∂x_{j}} + λa_{j} + [a_{j}, v]] = 0 for all λ ∈ ℂ,  (2)

iii) θ_{λ} is a flat U_{ℂ} = U ⊗ ℂ-connection 1-form on ℝ^{n} for all λ ∈ ℂ, where

θ_{λ} = ∑(a_{i}λ + [a_{i}, v])dx_{i}.  (3)

iv) There exists E so that E^{−1}dE = θ_{λ}.

The one-parameter family of flat connections θ_{λ} given by (3) is called the Lax connection of the U/K-system (1). It is well known that for a flat connection θ = ∑_{i=1}^{n} A_{i}(x)dx_{i}, the trivialization of θ, is a solution E for the following linear system:

E_{x_{i}} = EA_{i}.  (4)

Or equivalently of E^{−1}dE = θ.

3 Main Results

In the next two subsections we establish our results that time-like isothermic surfaces are associated to the Grassmannian system O(n − j + 1, j + 1)/O(n − j, j) × O(1, 1). In fact, using the existence of another two maximal abelian subalgebras in the subspace P, different from that of the space-like case given in [6] in which the first author obtained elliptic Gauss equations, we associate to each of these maximal abelian subalgebras one O(n − j + 1, j + 1)/O(n − j, j) × O(1, 1)-system. As we will see, these systems are not equivalent and for each of these maximal abelian subalgebra we obtain hyperbolic Gauss equations, which are correct for time-like surfaces. Let U/K = O(n − j + 1, j + 1)/O(n − j, j) × O(1, 1), where

O(n − j + 1, j + 1) =

\{X ∈ GL(n + 2)|X^{t} (\begin{matrix}I_{n−j,j} & 0 \\ 0 & J′ \end{matrix}) X = (\begin{matrix}I_{n−j,j} & 0 \\ 0 & J′ \end{matrix})\},

I_{n−j,j} = (\begin{matrix}I_{n−j} & 0 \\ 0 & −I_{j} \end{matrix}) and J′ = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) .

Let U = o(n − j + 1, j + 1) be the Lie algebra of U and σ : U → U be an involution defined by σ(X) = I_{n,2}^{−1}XI_{n,2}. Denote by K, P the +1, -1 eigenspaces of σ respectively, i.e.,

K = \{(\begin{matrix}Y_{1} & 0 \\ 0 & Y_{2} \end{matrix}) |Y_{1} ∈ o(n − j, j), Y_{2} ∈ o(1, 1)\} = o(n − j, j) × o(1, 1),

and

P = \{(\begin{matrix}0 & ξ \\ −J′ξ^{t}I_{n−j,j} & 0 \end{matrix}) |ξ ∈ M_{n×2}\}.

3.1 Time-like case with diagonal second fundamental form

Here we assume the elements a_{1}, a_{2} ∈ M_{(n+2)×(n+2)}, where

a_{1} = e_{n,n+1} + e_{n,n+2} + e_{n+1,n} + e_{n+2,n}
a_{2} = −e_{1,n+1} + e_{1,n+2} − e_{n+1,1} + e_{n+2,1},

and e_{ij} is the (n + 2) × (n + 2) elementary matrix whose only non-zero entry is 1 in the ij^{th} place. Then it is easy to see that the subalgebra A =< a_{1}, a_{2} > is maximal abelian in P, that T r[a_{1}^{2}]T r[a_{2}^{2}] − T r[a_{1}a_{2}]^{2} = 16 with T r[a_{1}^{2}] = 4, so the induced metric on A is positive definite and finally that

P ∩ A^{⊥} = \{(\begin{matrix}0 & ξ \\ −J′ξ^{t}I_{n−j,j} & 0 \end{matrix}) |ξ ∈ M_{n×2}, ξ_{11} = ξ_{12}, ξ_{n1} = −ξ_{n2}\}.

So using this basis {a_{1}, a_{2}}, the U/K-system (1) for this symmetric space is the following PDE for

ξ = (\begin{matrix}ξ_{1} & ξ_{1} \\ r_{1,1} & r_{1,2} \\ ⋮ & ⋮ \\ r_{n−2,1} & r_{n−2,2} \\ ξ_{2} & −ξ_{2} \end{matrix}) : ℝ^{2} → M_{n×2},

\begin{cases}(r_{i,2})_{x_{1}} − (r_{i,1})_{x_{1}} = −2(r_{i,1} + r_{i,2})ξ_{2}, i = 1, ..., n − 2 \\ (r_{i,2})_{x_{2}} + (r_{i,1})_{x_{2}} = 2(r_{i,2} − r_{i,1})ξ_{1}, i = 1, ..., n − 2 \\ 2((ξ_{1})_{x_{2}} + (ξ_{2})_{x_{1}}) = ∑_{i=1}^{n−2} σ_{i}(r_{i1}^{2} − r_{i2}^{2}) \\ (ξ_{2})_{x_{2}} + (ξ_{1})_{x_{1}} = 0, \end{cases}  (5)

where σ_{i} = 1, i = 1, ..., n − j − 1 and σ_{i} = −1, i = n − j, ..., n − 2. We now denote the entries of ξ by:

(\begin{matrix}ξ_{1} & ξ_{1} \\ ξ_{2} & −ξ_{2} \end{matrix}) = F and (\begin{matrix}r_{1,1} & r_{1,2} \\ ⋮ & ⋮ \\ r_{n−2,1} & r_{n−2,2} \end{matrix}) = G.

For convenience, we call the U/K-system (5) the real O(n − j + 1, j + 1)/O(n − j, j) × O(1, 1)-system, because this system will correspond to time-like surfaces in ℝ^{n−j,j} whose shape operators are diagonalizable. Continuing with the same notation used in [2], the real O(n−j+1, j+1)/O(n− j, j) × O(1, 1)-system II is the PDE for (F, G, B) : ℝ^{2} → gl_{∗}(2) × M_{(n−2)×2} × O(1, 1), where gl_{∗}(2) = {N ∈ M_{2×2}|N_{11} = N_{12}, N_{21} = −N_{22}}

\begin{cases}(r_{i,2})_{x_{1}} − (r_{i,1})_{x_{1}} = −2ξ_{2}(r_{i,1} + r_{i,2}), i = 1, ..., n − 2 \\ (r_{i,1})_{x_{2}} + (r_{i,2})_{x_{2}} = 2ξ_{1}(r_{i,2} − r_{i,1}), i = 1, ..., n − 2 \\ 2((ξ_{1})_{x_{2}} + (ξ_{2})_{x_{1}}) = ∑_{i=1}^{n−2} σ_{i}(r_{i1}^{2} − r_{i2}^{2}) \\ (b_{11})_{x_{1}} − (b_{12})_{x_{1}} = 2ξ_{2}(b_{11} + b_{12}) \\ (b_{21})_{x_{1}} − (b_{22})_{x_{1}} = 2ξ_{2}(b_{22} + b_{21}) \\ (b_{11})_{x_{2}} + (b_{12})_{x_{2}} = −2ξ_{1}(b_{11} − b_{12}) \\ (b_{21})_{x_{2}} + (b_{22})_{x_{2}} = −2ξ_{1}(b_{21} − b_{22}) \end{cases}  (6)

where the matrix B = (b_{ij}) ∈ O(1, 1). Now we recall that if we take

g = (\begin{matrix}A & 0 \\ 0 & B \end{matrix}) solution of g^{−1}dg = θ_{0} and B being the particular case

B = (\begin{matrix}b & 0 \\ 0 & \frac{1}{b} \end{matrix}) = (\begin{matrix}e^{2u} & 0 \\ 0 & e^{−2u} \end{matrix}), we obtain the relation

B^{−1}dB = (\begin{matrix}−2ξ_{1}dx_{2} + 2ξ_{2}dx_{1} & 0 \\ 0 & −2ξ_{2}dx_{1} + 2ξ_{1}dx_{2} \end{matrix}) ,

which implies that ξ_{1} = −u_{x_{2}} and ξ_{2} = u_{x_{1}}, hence the matrix ξ becomes:

ξ = (\begin{matrix}−u_{x_{2}} & −u_{x_{2}} \\ r_{1,1} & r_{1,2} \\ ⋮ & ⋮ \\ r_{n−2,1} & r_{n−2,2} \\ u_{x_{1}} & −u_{x_{1}} \end{matrix}) .

So the real O(n−j +1, j +1)/O(n−j, j)×O(1, 1)-system II is the set of partial differential equations for (u, r_{1,1}, r_{1,2}, …, r_{n−2,1}, r_{n−2,2}):

\begin{cases}(r_{i,2})_{x_{1}} − (r_{i,1})_{x_{1}} = −2(r_{i,1} + r_{i,2})u_{x_{1}}, i = 1, ..., n − 2 \\ (r_{i,1})_{x_{2}} + (r_{i,2})_{x_{2}} = −2(r_{i,2} − r_{i,1})u_{x_{2}}, i = 1, ..., n − 2 \\ 2(u_{x_{1}x_{1}} − u_{x_{2}x_{2}}) = ∑_{i=1}^{n−2} σ_{i}(r_{i1}^{2} − r_{i2}^{2}) \end{cases}  (7)

We observe that the next proposition follows from Proposition 2.5 in [2].

Proposition 3.1. the following statements are equivalent for map (F, G, B) : ℝ^{2} → gl_{∗}(2) × M_{(n−2)×2} × O(1, 1):

(1) (F, G, B) is solution of (6).

(2) θ_{λ}^{II} := g_{2}θ_{λ}g_{2}^{−1} − dg_{2}g_{2}^{−1} is a flat connection on ℝ^{2} for all λ ∈ ℂ, where θ_{λ} is the Lax connection associated to the solution ξ of the system (5) and

g_{2} = (\begin{matrix}I & 0 \\ 0 & B \end{matrix}) is the O(1, 1)-part of the trivialization g = (g_{1}, g_{2}) of θ_{0}.

(3) θ_{λ}^{II} := g_{2}θ_{λ}g_{2}^{−1} − dg_{2}g_{2}^{−1} is flat for λ = 1, where g_{2} is the same as in item (2).

Before showing the relationship between the Grassmannian system and isothermic surfaces we give the definition of a time-like isothermic surface with shape operators diagonalized over ℝ.

Definition 3.1. (Real isothermic surface) Let O be a domain in ℝ^{1,1}. An immersion X : O → ℝ^{n−j,j} is called a real time-like isothermic surface if it has flat normal bundle and the two fundamental forms are:

I = e^{2v}(−dx_{1}^{2} + dx_{2}^{2}), II = e^{v} ∑_{i=2}^{n−1}(g_{i−1,2}dx_{2}^{2} − g_{i−1,1}dx_{1}^{2})e_{i},

with respect to some parallel normal frame {e_{i}}. Or equivalently (x_{1}, x_{2}) ∈ O is conformal and line of curvature coordinate system for X.

We note that each isothermic surface has a dual surface ([11]) and make the following related definition.

Definition 3.2. (Real isothermic time-like dual pair in ℝ^{n−j,j} of type O(1, 1)). Let O be a domain in ℝ^{1,1} and X_{i} : O → ℝ^{n−j,j} an immersion with flat and non-degenerate normal bundle for i = 1, 2. (X_{1}, X_{2}) is called a real isothermic time-like dual pair in ℝ^{n−j,j} of type O(1, 1) if :

(i) The normal plane of X_{1}(x) is parallel to the normal plane of X_{2}(x) and x ∈ O, (ii) there exists a common parallel normal frame {e_{2}, ..., e_{n−1}}, where {e_{i}}_{2}^{n−j} and {e_{i}}_{n−j+1}^{n−1} are space-like and time-like vectors resp. (iii) x ∈ O is a conformal line of curvature coordinate system with respect to {e_{2}, ..., e_{n−1}} for each X_{k} such that the fundamental forms of X_{k} are:

\begin{cases}I_{1} = b^{−2}(−dx_{1}^{2} + dx_{2}^{2}), \\ II_{1} = b^{−1} ∑_{i=2}^{n−1}[−(g_{i−1,1} + g_{i−1,2})dx_{1}^{2} + (g_{i−1,2} − g_{i−1,1})dx_{2}^{2}]e_{i}, \\ I_{2} = b^{2}(−dx_{1}^{2} + dx_{2}^{2}), \\ II_{2} = b ∑_{i=2}^{n−1}[−(g_{i−1,1} + g_{i−1,2})dx_{1}^{2} − (g_{i−1,2} − g_{i−1,1})dx_{2}^{2}]e_{i}, \end{cases}  (8)

where B = (\begin{matrix}b & 0 \\ 0 & b^{−1} \end{matrix}) is in O(1, 1) and a M_{(n−2)×2}-valued map G = (g_{ij}).

Our first result, whose proof follows the same lines of Theorem 6.8 or 7.4 in [2], gives us the relationship between the dual pair of real isothermic timelke surfaces in ℝ^{n−j,j} of type O(1, 1) and the solutions of the real O(n − j + 1, j + 1)/O(n − j, j) × O(1, 1)-system II (6):

Theorem 3.1. Suppose (u, r_{1,1}, r_{1,2}, …, r_{n−2,1}, r_{n−2,2}) is solution of (7) and F, B are given by

F = (\begin{matrix}ξ_{1} & ξ_{1} \\ ξ_{2} & −ξ_{2} \end{matrix}) = (\begin{matrix}−u_{x_{2}} & −u_{x_{2}} \\ u_{x_{1}} & −u_{x_{1}} \end{matrix}) , B = (\begin{matrix}e^{2u} & 0 \\ 0 & e^{−2u} \end{matrix}) .

Then: (a) ω =

(\begin{matrix}0 & ϵ_{1}β_{1}dx_{2} & … & ϵ_{n−2}β_{n−2}dx_{2} & 2(−ξ_{1}dx_{1} + ξ_{2}dx_{2}) \\ −β_{1}dx_{2} & 0 & … & 0 & −η_{1}dx_{1} \\ −β_{2}dx_{2} & 0 & … & 0 & −η_{2}dx_{1} \\ ⋮ & ⋮ & … & ⋮ & ⋮ \\ −β_{n−2}dx_{2} & 0 & … & 0 & −η_{n−2}dx_{1} \\ 2(ξ_{2}dx_{2} − ξ_{1}dx_{1}) & −ϵ_{1}η_{1}dx_{1} & … & −ϵ_{n−2}η_{n−2}dx_{1} & 0 \end{matrix})

(9)

where ϵ_{i} = 1 for i < n − j and ϵ_{i} = −1 for i ≥ n − j, and where β_{i} = (r_{i,2} − r_{i,1}), η_{i} = (r_{i,1} + r_{i,2}), i = 1, ..., n − 2, is a flat o(n − j, j)-valued connection 1-form. Hence there exists A : ℝ^{2} → O(n − j, j) such that

A^{−1}dA = ω,  (10)

where ω is given by (9).

(b)

A (\begin{matrix}−dx_{2} & 0 & … & 0 & dx_{1} \\ dx_{2} & 0 & … & 0 & dx_{1} \end{matrix})^{t} B^{−1}

is exact. So there exists a map X : ℝ^{2} → M_{n×2} such that

dX = A (\begin{matrix}−dx_{2} & 0 & … & 0 & dx_{1} \\ dx_{2} & 0 & … & 0 & dx_{1} \end{matrix})^{t} B^{−1}  (11)

(c) Let X_{j} : ℝ^{2} → ℝ^{n−j,j} denote the j-th column of X (solution of 11) and e_{i} denote the i-th column of A. Then (X_{1}, X_{2}) is a dual pair of real isothermic timelike surfaces in ℝ^{n−j,j} of type O(1, 1). I.e. (X_{1}, X_{2}) have the following properties:

(1) e_{1}, e_{n} are resp. space-like and time-like tangent vectors to X_{1} and X_{2}, i.e, the tangent planes of X_{1}, X_{2} are parallel.

(2) {e_{2}, ..., e_{n−1}} is a parallel normal frame for X_{1} and X_{2}, with {e_{2}, ..., e_{n−j}} and {e_{n−j+1}, ..., e_{n−1}} being resp. space-like and time-like vectors.

(3) the two fundamental forms for the immersion X_{k} are:

\begin{cases}I_{1} = e^{−4u}(dx_{2}^{2} − dx_{1}^{2}) \\ II_{1} = e^{−2u} ∑_{i=2}^{n−1}[(r_{i−1,2} − r_{i−1,1})dx_{2}^{2} − (r_{i−1,1} + r_{i−1,2})dx_{1}^{2}]e_{i} \\ I_{2} = e^{4u}(dx_{2}^{2} − dx_{1}^{2}) \\ II_{2} = e^{2u} ∑_{i=2}^{n−1}[−(r_{i−1,2} − r_{i−1,1})dx_{2}^{2} − (r_{i−1,1} + r_{i−1,2})dx_{1}^{2}]e_{i} \end{cases}

Remark 3.1. We observe that we can prove a theorem like Theorem (3.1) for a general solution (F, G, B) of system (6) by taking a generic F = (f_{ij}) and

B = (b_{ij}) = (\begin{matrix}b & 0 \\ 0 & b^{−1} \end{matrix}) ∈ O(1, 1), i.e, we conclude that if (F, G, B) is a solution

of system (6), we obtain a real isothermic timelike dual pair in ℝ^{n−j,j} of type O(1, 1) with I and II fundamental forms like in (8).

Now for the converse, we have the following result.

Theorem 3.2. Let (X_{1}, X_{2}) be a real isothermic time-like dual pair in ℝ^{n−j,j} of type O(1, 1), {e_{2}, ..., e_{n−1}} a common parallel normal frame and (x_{1}, x_{2}) a common isothermal line of curvature coordinates for X_{1} and X_{2}, such that the two fundamental forms I_{k}, II_{k} for X_{k} are given by (8). Set f_{11} = −\frac{b_{x_{2}}}{2b} = f_{12}, f_{22} = −\frac{b_{x_{1}}}{2b} = −f_{21}, and F = (f_{ij})_{2×2}. Then if all entries of G and the g_{i−1,1} + g_{i−1,2}, g_{i−1,2} − g_{i−1,1} are non-zero, then (F, G, B) is a solution of (6).

Proof. From the definition of real isothermic time-like pair in ℝ^{n−j,j}, we have

ω_{1}^{(1)} = −b^{−1}dx_{2}, ω_{n}^{(1)} = b^{−1}dx_{1} ω_{1}^{(2)} = bdx_{2}, ω_{n}^{(2)} = bdx_{1}

is a dual 1-frame for X_{k} and ω_{1α}^{(k)} = l_{α}(g_{α−1,2}−g_{α−1,1})dx_{2}, ω_{nα}^{(k)} = −l_{α}(g_{α−1,1}+ g_{α−1,2})dx_{1} for each X_{k}, where l_{α} = 1 if α = 2, ..., n − j and l_{α} = −1 if α = n − j + 1, ..., n − 1. We observe that ω_{iα}^{(k)}, i = 1, n, α = 2, ..., n − 1 are independent of k. We find that the Levi-civita connection 1-form for the metric I_{k} is:

ω_{1n}^{(k)} = \frac{b_{x_{1}}}{b} dx_{2} + \frac{b_{x_{2}}}{b} dx_{1} = 2(−f_{22}^{(k)}dx_{2} − f_{11}^{(k)}dx_{1}),

which are independent from k. Hence ω_{1n}^{(k)} = ω_{1n}^{(1)} = 2(−f_{22}dx_{2} − f_{11}dx_{1}) = 2(ξ_{2}dx_{2} − ξ_{1}dx_{1}). So the structure equations and the Gauss-Codazzi equations for X_{1}, X_{2} imply that (F, G, B) is a solution of system (6). ■

So, from Theorems (3.1), (3.2) and Remark (3.1), it follows that there exists a correspondence between the solutions (F, G, B) of system (6) and a dual pair of real isothermic timelike surfaces in ℝ^{n−j,j} of type O(1, 1).

Theorem 3.3. The real O(n − j + 1, j + 1)/O(n − j, j) × O(1, 1)-system II (6) is the Gauss-Codazzi equation for a time-like surface in ℝ^{n−j,j} such that:

\begin{cases}I = e^{4u}(dx_{2}^{2} − dx_{1}^{2}) \\ II = e^{2u} ∑_{i=2}^{n−1}[−(r_{i−1,2} − r_{i−1,1})dx_{2}^{2} − (r_{i−1,1} + r_{i−1,2})dx_{1}^{2}]e_{i} \end{cases}  (12)

Proof. We can read from I and II that: ω_{1} = e^{2u}dx_{2}, ω_{n} = e^{2u}dx_{1}, ω_{1,i} = η_{i}(r_{i−1,2} − r_{i−1,1})dx_{2}, and ω_{n,i} = −η_{i}(r_{i−1,2} + r_{i−1,1})dx_{1}, where η_{i} = 1 if i = 2, ..., n − j and η_{i} = −1 if i = n − j + 1, ..., n − 1. Now use the structure equations: dω_{1} = ω_{n} ∧ ω_{1n} and dω_{n} = ω_{1} ∧ ω_{n1}, to obtain:

ω_{1n} = 2(u_{x_{1}}dx_{2} + u_{x_{2}}dx_{1}).

Now from the Gauss equation: dω_{1n} = − ∑_{i=2}^{n−j} ω_{1,i}∧ω_{n,i}+∑_{i=n−j+1}^{n−1} ω_{1,i}∧ω_{n,i}, we have that

u_{x_{1}x_{1}} − u_{x_{2}x_{2}} = \frac{1}{2}[∑_{i=1}^{n−2} σ_{i}(r_{i1}^{2} − r_{i2}^{2})].

The Codazzi equations: dω_{1,i} = −ω_{1n} ∧ ω_{n,i} and dω_{n,i} = −ω_{n1} ∧ ω_{1,i} for i = 2, ..., n − 1, yield, for these values of i,

(r_{i−1,2})_{x_{1}} − (r_{i−1,1})_{x_{1}} = −2(r_{i−1,2} + r_{i−1,1})u_{x_{1}}
(r_{i−1,1})_{x_{2}} + (r_{i−1,2})_{x_{2}} = −2(r_{i−1,2} − r_{i−1,1})u_{x_{2}}.

Collecting our information we see that the Gauss-Codazzi equation is the following system for (u, r_{1,1}, r_{1,2}, …, r_{n−2,1}, r_{n−2,2}):

\begin{cases}(r_{i−1,2})_{x_{1}} − (r_{i−1,1})_{x_{1}} = −2(r_{i−1,2} + r_{i−1,1})u_{x_{1}}, i = 2, ..., n − 1 \\ (r_{i−1,1})_{x_{2}} + (r_{i−1,2})_{x_{2}} = −2(r_{i−1,2} − r_{i−1,1})u_{x_{2}}, i = 2, ..., n − 1 \\ 2(u_{x_{1}x_{1}} − u_{x_{2}x_{2}}) = ∑_{i=1}^{n−2} σ_{i}(r_{i1}^{2} − r_{i2}^{2}) \end{cases}  (13)

Hence if we put

B = (\begin{matrix}e^{2u} & 0 \\ 0 & e^{−2u} \end{matrix}) , F = (\begin{matrix}−u_{x_{2}} & −u_{x_{2}} \\ u_{x_{1}} & −u_{x_{1}} \end{matrix}) , G = (\begin{matrix}r_{1,1} & r_{1,2} \\ ⋮ & ⋮ \\ r_{n−2,1} & r_{n−2,2} \end{matrix}) , (14)

we see that (F, G, B) is solution of the real O(n−j+1, j+1)/O(n−j, j)×O(1, 1)- system II. Conversely, if (F, G, B) is solution of the real O(n−j+1, j+1)/O(n− j, j) × O(1, 1)-system II (6), and we assume B being as in (14), then the fourth and sixth equation of system (6), imply that

ξ_{2} = u_{x_{1}}, ξ_{1} = −u_{x_{2}}

ie, (F, G, B) is the form (14). Finally writing the real O(n − j + 1, j + 1)/O(n − j, j) × O(1, 1)-system II for this (F, G, B) in terms of u and r_{ij} we get equation (13). ■ The next result follows from Theorem (3.1) and Theorem (3.3).

Theorem 3.4. Let O be a domain of ℝ^{1,1}, and X_{2} : O → ℝ^{n−j,j} an immersion with flat normal bundle and (x_{1}, x_{2}) ∈ O an isothermal line of curvature coordinate system with respect to a parallel normal frame {e_{2}, ..., e_{n−1}}, such that I and II fundamental forms are given by (12). Then there exists an immersion X_{1}, unique up to translation, such that (X_{1}, X_{2}) is a real isothermic timelike dual pair in ℝ^{n−j,j} of type O(1, 1). Moreover, the fundamental forms of X_{1}, X_{2} are respectively:

\begin{cases}I_{1} = e^{−4u}(dx_{2}^{2} − dx_{1}^{2}) \\ II_{1} = e^{−2u} ∑_{i=2}^{n−1}[(r_{i−1,2} − r_{i−1,1})dx_{2}^{2} − (r_{i−1,1} + r_{i−1,2})dx_{1}^{2}]e_{i} \\ I_{2} = e^{4u}(dx_{2}^{2} − dx_{1}^{2}) \\ II_{2} = e^{2u} ∑_{i=2}^{n−1}[−(r_{i−1,2} − r_{i−1,1})dx_{2}^{2} − (r_{i−1,1} + r_{i−1,2})dx_{1}^{2}]e_{i} \end{cases}  (15)

It follows from Gauss equation that the Gaussian curvatures of X_{1} and X_{2} of the real isothermic timelike dual pair (15), denoted by K_{G}^{(1)}, K_{G}^{(2)}, and the mean curvatures, denoted by η^{(1)} and η^{(2)}, are given by

K_{G}^{(1)} = −e^{4u} ∑_{i=1}^{n−2} σ_{i}(r_{i,1}^{2} − r_{i,2}^{2}), K_{G}^{(2)} = e^{−4u} ∑_{i=1}^{n−2} σ_{i}(r_{i,1}^{2} − r_{i,2}^{2}),

η^{(1)} = e^{2u} ∑_{i=1}^{n−2} r_{i,2}e_{i+1}, η^{(2)} = e^{−2u} ∑_{i=1}^{n−2} r_{i,1}e_{i+1},

where σ_{i} = 1, i = 1, ..., n − j − 1 and σ_{i} = −1, i = n − j, ..., n − 2.

3.2 Timelike case with non-diagonal second fundamental form

We continue with the same notational convention used in the subsection above. For this new case, we take the elements a_{1}, a_{2} ∈ M_{(n+2)×(n+2)}, to be

a_{1} = e_{1,n+1} + e_{n,n+2} + e_{n+1,n} − e_{n+2,1}

a_{2} = −e_{1,n+2} + e_{n,n+1} + e_{n+1,1} + e_{n+2,n}.

We note that T r[a_{1}^{2}]T r[a_{2}^{2}] − T r[a_{1}a_{2}]^{2} = −16 and T r[a_{1}^{2}] = 0, so that the induced metric on A is time-like. One can see easily that the space A spanned by a_{1} and a_{2} is a maximal abelian subalgebra contain in P, and that

A^{⊥} ∩ P = \{(\begin{matrix}0 & ξ \\ −J′ξ^{t}I_{n−j,j} & 0 \end{matrix}) |ξ ∈ M_{n×2}, ξ_{11} = −ξ_{n2}, ξ_{12} = ξ_{n1}\}.

So the matrix v ∈ A^{⊥} ∩ P if and only if

v = (\begin{matrix}0 & … & … & & 0 & ξ_{1} & ξ_{2} \\ 0 & … & … & & 0 & r_{1,1} & r_{1,2} \\ ⋮ & … & … & & ⋮ & ⋮ & ⋮ \\ 0 & … & … & & 0 & r_{n−2,1} & r_{n−2,2} \\ 0 & … & … & & 0 & ξ_{2} & −ξ_{1} \\ −ξ_{2} & −r_{1,2} & … & r_{n−2,2} & −ξ_{1} & 0 & 0 \\ −ξ_{1} & −r_{1,1} & … & r_{n−2,1} & ξ_{2} & 0 & 0 \end{matrix}) .

Then using this basis {a_{1}, a_{2}}, the U/K-system (1) for this symmetric space is the following PDE for

ξ = (\begin{matrix}ξ_{1} & ξ_{2} \\ r_{1,1} & r_{1,2} \\ ⋮ & ⋮ \\ r_{n−2,1} & r_{n−2,2} \\ ξ_{2} & −ξ_{1} \end{matrix}) : ℝ^{2} → M_{n×2},

\begin{cases}−r_{i,2}_{x_{2}} − r_{i,1}_{x_{1}} & = 2(r_{i,2}ξ_{1} − r_{i,1}ξ_{2}), i = 1, ..., n − 2 \\ −r_{i,1}_{x_{2}} + r_{i,2}_{x_{1}} & = −2(r_{i,1}ξ_{1} + r_{i,2}ξ_{2}), i = 1, ..., n − 2 \\ (−2ξ_{1})_{x_{2}} + (2ξ_{2})_{x_{1}} & = ∑_{i=1}^{n−2} σ_{i}(r_{i,1}^{2} + r_{i,2}^{2}) \\ (2ξ_{2})_{x_{2}} − (2ξ_{1})_{x_{1}} & = 0. \end{cases}  (16)

We now denote the entries of ξ by:

(\begin{matrix}ξ_{1} & ξ_{2} \\ ξ_{2} & −ξ_{1} \end{matrix}) = F and (\begin{matrix}r_{1,1} & r_{1,2} \\ ⋮ & ⋮ \\ r_{n−2,1} & r_{n−2,2} \end{matrix}) = G.

For convenience, we call the U/K-system (16) the complex O(n − j + 1, j + 1)/O(n−j, j)×O(1, 1)-system, because this system will correspond to time-like surfaces in ℝ^{n−j,j} whose shape operators have complex eigenvalues. Now, the complex O(n − j + 1, j + 1)/O(n − j, j) × O(1, 1)-system II is the following PDE for (F, G, B) : ℝ^{2} → gl_{∗}(2) × M_{(n−2)×2} × O(1, 1), where gl_{∗}(2) is the set of matrices 2 × 2 such that −f_{11} = f_{22}, f_{21} = f_{12},

\begin{cases}−r_{i,2}_{x_{2}} − r_{i,1}_{x_{1}} & = 2(r_{i,2}ξ_{1} − r_{i,1}ξ_{2}), \\ −r_{i,1}_{x_{2}} + r_{i,2}_{x_{1}} & = −2(r_{i,1}ξ_{1} + r_{i,2}ξ_{2}), \\ (−2ξ_{1})_{x_{2}} + (2ξ_{2})_{x_{1}} & = ∑_{i=1}^{n−2} σ_{i}(r_{i,1}^{2} + r_{i,2}^{2}) \\ b_{22}_{x_{2}} + b_{21}_{x_{1}} & = −2b_{22}ξ_{1} + 2b_{21}ξ_{2}, \\ b_{12}_{x_{2}} + b_{11}_{x_{1}} & = −2b_{12}ξ_{1} + 2b_{11}ξ_{2}, \\ b_{21}_{x_{2}} − b_{22}_{x_{1}} & = 2b_{21}ξ_{1} + 2b_{22}ξ_{2}, \\ b_{11}_{x_{2}} − b_{12}_{x_{1}} & = 2b_{11}ξ_{1} + 2b_{12}ξ_{2}, \end{cases} (17)

where the matrix B = (b_{ij}) ∈ O(1, 1) and 1 ≤ i ≤ n − 2. Now taking B =

(\begin{matrix}e^{2u} & 0 \\ 0 & e^{−2u} \end{matrix}), and using the fact that

B^{−1}dB = (\begin{matrix}2ξ_{2}dx_{1} + 2ξ_{1}dx_{2} & 0 \\ 0 & −2ξ_{2}dx_{1} − 2ξ_{1}dx_{2} \end{matrix}) ,

we have

ξ = (\begin{matrix}u_{x_{2}} & u_{x_{1}} \\ r_{1,1} & r_{1,2} \\ ⋮ & ⋮ \\ r_{n−2,1} & r_{n−2,2} \\ u_{x_{1}} & −u_{x_{2}} \end{matrix}) .

So the complex O(n − j + 1, j + 1)/O(n − j, j) × O(1, 1)-system II is the PDE for (u, r_{1,1}, r_{1,2}, …, r_{n−2,1}, r_{n−2,2}):

\begin{cases}−r_{i,2}_{x_{2}} − r_{i,1}_{x_{1}} & = 2(r_{i,2}ξ_{1} − r_{i,1}ξ_{2}), \\ −r_{i,1}_{x_{2}} + r_{i,2}_{x_{1}} & = −2(r_{i,1}ξ_{1} + r_{i,2}ξ_{2}), \\ −2u_{x_{2}x_{2}} + 2u_{x_{1}x_{1}} & = ∑_{i=1}^{n−2} σ_{i}(r_{i,1}^{2} + r_{i,2}^{2}). \end{cases}  (18)

Remark 3.2. We recall that the complex O(n−j+1, j+1)/O(n−j, j)×O(1, 1)- system II is the flatness condition for the family:

θ_{λ}^{II} = (\begin{matrix}ω & MB^{−1} \\ BN & 0 \end{matrix})

where B = (b_{ij}) ∈ O(1, 1) and the matrices ω ∈ M_{n×n}, M ∈ M_{n×2}, N ∈ M_{2×n} are given by:

ω = (\begin{matrix}0 & a⃗ & b⃗ & c \\ −a⃗^{t} & 0 & 0 & d⃗^{t} \\ b⃗^{t} & 0 & 0 & e⃗^{t} \\ c & d⃗ & −e⃗ & 0 \end{matrix}) , M = λ (\begin{matrix}dx_{1} & −dx_{2} \\ 0 & 0 \\ ⋮ & ⋮ \\ 0 & 0 \\ dx_{2} & dx_{1} \end{matrix})  (19)

N = λ (\begin{matrix}dx_{2} & 0 & … & 0 & dx_{1} \\ −dx_{1} & 0 & … & 0 & dx_{2} \end{matrix})

where

a⃗ = (a_{1}, …, a_{n−j−1}) and a_{k} = r_{k,1}dx_{2} − r_{k,2}dx_{1}, for 1 ≤ k ≤ n − j − 1,
b⃗ = (b_{n−j}, …, b_{n−2}) and b_{q} = −r_{q,1}dx_{2} + r_{q,2}dx_{1}, for n − j ≤ q ≤ n − 2,
c = −2ξ_{1}dx_{1} − 2ξ_{2}dx_{2}
d⃗ = (d_{1}, …, d_{n−j−1}) and d_{k} = −r_{k,1}dx_{1} − r_{k,2}dx_{2}, for 1 ≤ k ≤ n − j − 1,
e⃗ = (e_{q}, …, e_{n−2}) and e_{q} = −r_{q,1}dx_{1} − r_{q,2}dx_{2}, for n − j ≤ q ≤ n − 2.

We note that a proposition similar to Proposition (3.1), can be proven in this new case. At this point we need the appropriate definition of a complex isothermic surface, i.e., one that has an isothermal coordinate system with respect to which all the shape operators are diagonalized over ℂ.

Definition 3.3. (Complex isothermic surface) Let O be a domain in ℝ^{1,1}. An immersion X : O → ℝ^{n−j,j} is called a complex time-like isothermic surface if it has flat normal bundle and the two fundamental forms are:

I = ±e^{2v}(−dx_{1}^{2} + dx_{2}^{2}), II = ∑_{i=2}^{n−1} e^{v}(g_{i1}(dx_{2}^{2} − dx_{1}^{2}) − 2g_{i2}dx_{1}dx_{2})e_{i},

with respect to some parallel normal frame {e_{i}}.

Remark 3.3. We note that given any complex isothermic surface there is a dual isothermic surface with parallel normal space ([11]). The U/K system generates this pair of dual surfaces, making it clear that they should be considered essentially as a single unit.

Definition 3.4. (Complex isothermic time-like dual pair in ℝ^{n−j,j} of type O(1, 1)). Let O be a domain in ℝ^{1,1} and X_{i} : O → ℝ^{n−j,j} an immersion with flat and non-degenerate normal bundle for i = 1, 2. (X_{1}, X_{2}) is called a complex isothermic timelike dual pair in ℝ^{n−j,j} of type O(1, 1) if :

(i) The normal plane of X_{1}(x) is parallel to the normal plane of X_{2}(x) and x ∈ O, (ii) there exists a common parallel normal frame {e_{2}, ..., e_{n−1}}, where {e_{i}}_{2}^{n−j} and {e_{i}}_{n−j+1}^{n−1} are space-like and time-like vectors resp. (iii) x ∈ O is a isothermal coordinate system with respect to {e_{2}, ..., e_{n−1}}, for each X_{k}, such that the fundamental forms of X_{k} are diagonalizable over ℂ. Namely,

\begin{cases}I_{1} = b^{−2}(dx_{1}^{2} − dx_{2}^{2}), \\ II_{1} = −b^{−1} ∑_{i=1}^{n−2}[g_{i,2}(dx_{2}^{2} − dx_{1}^{2}) + 2g_{i,1}dx_{1}dx_{2}]e_{i+1}, \\ I_{2} = b^{2}(−dx_{1}^{2} + dx_{2}^{2}), \\ II_{2} = b ∑_{i=1}^{n−2}[g_{i,1}(dx_{2}^{2} − dx_{1}^{2}) − 2g_{i,2}dx_{1}dx_{2}]e_{i+1}, \end{cases}  (20)

where B = (\begin{matrix}b & 0 \\ 0 & b^{−1} \end{matrix}) is in O(1, 1) and a M_{(n−2)×2}-valued map G = (g_{ij}).

Theorem 3.5. Suppose (u, r_{1,1}, r_{1,2}, …, r_{n−2,1}, r_{n−2,2}) is solution of (18) and F, B are given by

F = (\begin{matrix}u_{x_{2}} & u_{x_{1}} \\ u_{x_{1}} & −u_{x_{2}} \end{matrix}) , B = (\begin{matrix}e^{2u} & 0 \\ 0 & e^{−2u} \end{matrix}) .

Then: (a) The ω defined by (19) is a flat o(n − j, j)-valued connection 1-form. Hence there exists A : ℝ^{2} → O(n − j, j) such that

A^{−1}dA = ω.  (21)

(b)

A (\begin{matrix}dx_{1} & 0 & … & 0 & dx_{2} \\ −dx_{2} & 0 & … & 0 & dx_{1} \end{matrix})^{t} B^{−1}

is exact. So there exists a map X : ℝ^{2} → M_{n×2} such that

dX = A (\begin{matrix}dx_{1} & 0 & … & 0 & dx_{2} \\ −dx_{2} & 0 & … & 0 & dx_{1} \end{matrix})^{t} B^{−1}.  (22)

(c) Let X_{i} : ℝ^{2} → ℝ^{n−j,j} denote the i-th column of X (solution of (22)) and e_{i} denote the i-th column of A. Then X_{1} and X_{2} are a dual pair of isothermic time-like surfaces in ℝ^{n−j,j} with common isothermal coordinates and second fundamental forms diagonalized over ℂ, so that:

(1) e_{1}, e_{n} are space-like and time-like tangent vectors to X_{1} and X_{2}, i.e, the tangent planes of X_{1}, X_{2} are parallel.

(2) {e_{2}, …, e_{n−1}} form a parallel normal frame for X_{1} and X_{2} of signature {n − j − 1, j − 1}.

(3) the two fundamental forms for the immersion X_{i} are:

\begin{cases}I_{1} = e^{−4u}(dx_{1}^{2} − dx_{2}^{2}) \\ II_{1} = −e^{−2u} ∑_{i=1}^{n−2}[r_{i,2}(dx_{2}^{2} − dx_{1}^{2}) + 2r_{i,1}dx_{1}dx_{2}]e_{i+1} \\ I_{2} = e^{4u}(dx_{2}^{2} − dx_{1}^{2}) \\ II_{2} = e^{2u} ∑_{i=1}^{n−2}[r_{i,1}(dx_{2}^{2} − dx_{1}^{2}) − 2r_{i,2}dx_{1}dx_{2}]e_{i+1}. \end{cases}  (23)

Remark 3.4. We observe that we can prove a theorem like Theorem (3.5) for a general solution (F, G, B) of system (17) by taking a generic F = (f_{ij}) and

B = (b_{ij}) = (\begin{matrix}b & 0 \\ 0 & b^{−1} \end{matrix}) ∈ O(1, 1), i.e, we conclude that if (F, G, B) is a solution

of system (17), we obtain a complex isothermic timelike dual pair in ℝ^{n−j,j} of type O(1, 1) with I and II fundamental forms like in (20).

Now for the converse, we have the following result.

Theorem 3.6. Let (X_{1}, X_{2}) be a complex isothermic timelike dual pair in ℝ^{n−j,j} of type O(1, 1), {e_{2}, ..., e_{n−1}} a common parallel normal frame and (x_{1}, x_{2}) a common isothermal coordinates for X_{1} and X_{2}, such that the two fundamental forms I_{k}, II_{k} for X_{k} are given by (20). Set f_{11} = \frac{b_{x_{2}}}{2b} = −f_{22}, f_{12} = \frac{b_{x_{1}}}{2b} = f_{21}, and F = (f_{ij})_{2×2}. Then if all entries of G are non-zero, then (F, G, B) is a solution of (17).

Proof. From the definition of complex isothermic timelike dual pair in ℝ^{n−j,j}, we have

ω_{1}^{(1)} = b^{−1}dx_{1}, ω_{n}^{(1)} = b^{−1}dx_{2}, ω_{1}^{(2)} = −bdx_{2}, ω_{n}^{(2)} = bdx_{1}

is a dual 1-frame for X_{k} and ω_{1α}^{(k)} = l_{α}(−g_{α−1,2}dx_{1} + g_{α−1,1}dx_{2}), ω_{nα}^{(k)} = −l_{α}(g_{α−1,1}dx_{1} + g_{α−1,2}dx_{2}) for each X_{k}, where l_{α} = 1 if α = 2, ..., n − j and l_{α} = −1 if α = n−j+1, ..., n−1. We observe that ω_{iα}^{(k)}, i = 1, n, α = 2, ..., n−1 are independent of k. We find that the Levi-civita connection 1-form for the metric I_{k} is:

ω_{1n}^{(k)} = −\frac{b_{x_{2}}}{b} dx_{1} − \frac{b_{x_{1}}}{b} dx_{2},

which are independent from k. Hence ω_{1n}^{(k)} = ω_{1n}^{(1)} = 2(f_{22}dx_{1} − f_{12}dx_{2}) = −2(ξ_{1}dx_{1}+ξ_{2}dx_{2}). So the structure equations and the Gauss-Codazzi equations for X_{1}, X_{2} imply that (F, G, B) is a solution of system (17). ■ So, from Theorems (3.5), (3.6) and Remark (3.4), follows that exists a correspondence between the solutions (F, G, B) of system (17) and a dual pair of complex isothermic timelike surfaces in ℝ^{n−j,j} of type O(1, 1).

Theorem 3.7. The complex O(n − j + 1, j + 1)/O(n − j, j) × O(1, 1)-system II (17) is the Gauss-Codazzi equation for a time-like surface in ℝ^{n−j,j} such that:

\begin{cases}I = e^{4u}(dx_{2}^{2} − dx_{1}^{2}) \\ II = e^{2u} ∑_{i=1}^{n−2}[r_{i,1}(dx_{2}^{2} − dx_{1}^{2}) − 2r_{i,2}dx_{1}dx_{2}]e_{i+1}. \end{cases}  (24)

Proof. For this surface we can read off from the fundamental forms I and II that

\begin{cases}ω_{1} = −e^{2u}dx_{2} \\ ω_{n} = e^{2u}dx_{1} \\ ω_{1 i} = σ_{i}(r_{i−1,1}dx_{2} − r_{i−1,2}dx_{1}) for 2 ≤ i ≤ n − 1 \\ ω_{n i} = −σ_{i}(r_{i−1,1}dx_{1} + r_{i−1,2}dx_{2}) for 2 ≤ i ≤ n − 1. \end{cases}

Using the structure equations, we can see that

ω_{1n} = −2u_{x_{2}}dx_{1} − 2u_{x_{1}}dx_{2},

and that the Gauss and Codazzi equations are the same as (18), since we have

u_{x_{1}} = ξ_{2}, u_{x_{2}} = ξ_{1}.

Hence if we put

B = (\begin{matrix}e^{2u} & 0 \\ 0 & e^{−2u} \end{matrix}) , F = ( \begin{matrix}u_{x_{2}} & u_{x_{1}} \\ u_{x_{1}} & −u_{x_{2}} \end{matrix}) ,  (25)

G = (\begin{matrix}r_{1,1} & r_{1,2} \\ ⋮ & ⋮ \\ r_{n−2,1} & r_{n−2,2} \end{matrix})

we have that (F, G, B) is solution of the complex O(n − j + 1, j + 1)/O(n − j, j)×O(1, 1)-system II (17). Conversely, if (F, G, B) is solution of the complex O(n − j + 1, j + 1)/O(n − j, j) × O(1, 1)-system II (17), and B is as in (25), then the fourth and sixth equation from system (17), imply that

ξ_{2} = u_{x_{1}}, ξ_{1} = u_{x_{2}}

i.e., (F, G, B) has the form (25). Finally writing the O(n − j + 1, j + 1)/O(n − j, j) × O(1, 1)-system II (17) for this (F, G, B), in terms of u and r_{i} we get equation (18). ■ The next result follows from Theorem (3.5) and Theorem (3.7).

Theorem 3.8. Let O be a domain of ℝ^{1,1}, and X_{2} : O → ℝ^{n−j,j} an immersion with flat normal bundle and (x_{1}, x_{2}) ∈ O a isothermal coordinates system with respect to a parallel normal frame {e_{2}, ..., e_{n−1}}, such that I and II fundamental forms are given by (24). Then there exists an immersion X_{1}, unique up to translation, such that (X_{1}, X_{2}) is a complex isothermic timelike dual pair in ℝ^{n−j,j} of type O(1, 1). Moreover, the fundamental forms of X_{1}, X_{2} are respectively:

\begin{cases}I_{1} = e^{−4u}(dx_{1}^{2} − dx_{2}^{2}) \\ II_{1} = −e^{−2u} ∑_{i=1}^{n−2}[r_{i,2}(dx_{2}^{2} − dx_{1}^{2}) + 2r_{i,1}dx_{1}dx_{2}]e_{i+1} \\ I_{2} = e^{4u}(dx_{2}^{2} − dx_{1}^{2}) \\ II_{2} = e^{2u} ∑_{i=1}^{n−2}[r_{i,1}(dx_{2}^{2} − dx_{1}^{2}) − 2r_{i,2}dx_{1}dx_{2}]e_{i+1}. \end{cases}  (26)

Finally, it follows from Gauss equation that the Gaussian curvatures of X_{1} and X_{2} of a complex isothermic timelike dual pair (26), denoted by K_{G}^{(1)}, K_{G}^{(2)}, are given by

K_{G}^{(1)} = e^{4u} ∑_{i=1}^{n−2} σ_{i}(r_{i,1}^{2} + r_{i,2}^{2}), K_{G}^{(2)} = e^{−4u} ∑_{i=1}^{n−2} σ_{i}(r_{i,1}^{2} + r_{i,2}^{2}),

where σ_{i} = 1, i = 1, ..., n − j − 1 and σ_{i} = −1, i = n − j, ..., n − 2.

Example: Next we give an explicit example of a dual pair of complex timelike isothermic surfaces in ℝ^{2,1} and the associated solution to the complex O(3, 2)/O(2, 1) × O(1, 1)-system II.

We consider first the Lorentzian helicoid

X(x_{1}, x_{2}) = (x_{2}, sinh(x_{1}) sinh(x_{2}), cosh(x_{2}) sinh(x_{1}))

with normal vector:

N(x_{1}, x_{2}) = \frac{1}{cosh(x_{1})}(− sinh(x_{1}), cosh(x_{2}), sinh(x_{2})).

The dual surface to this surface is:

X̂(x_{1}, x_{2}) = \frac{1}{cosh(x_{1})}(sinh(x_{1}), − cosh(x_{2}), − sinh(x_{2})),

which is a parametrization of part of the standard immersion of the Lorenztian sphere (see[11]). They constitute a dual pair of complex timelike isothermic surfaces in ℝ^{2,1}, with first and second fundamental forms given resp. by

I^{1} = cosh^{2}(x_{1})[−dx_{1}^{2} + dx_{2}^{2}], II^{1} = 2dx_{1}dx_{2},
I^{2} = (1/ cosh^{2}(x_{1}))[dx_{1}^{2} − dx_{2}^{2}], II^{2} = (1/ cosh^{2}(x_{1}))[dx_{2}^{2} − dx_{1}^{2}].

Here

B = ( \begin{matrix}cosh x_{1} & 0 \\ 0 & cosh^{−1} x_{1} \end{matrix} ) , F = ( \begin{matrix}0 & \frac{tanh x_{1}}{2} \\ \frac{tanh x_{1}}{2} & 0 \end{matrix} ) ,

and

G = ( 0, − cosh^{−1} x_{1} ) ,

are a solution of the complex O(3, 2)/O(2, 1) × O(1, 1)-system II. More specifically, taking e^{2u} = cosh x_{1}, we have (u, 0, − cosh^{−1} x_{1}) is a solution of the complex O(3, 2)/O(2, 1) × O(1, 1)-system II.

4 Appendix: Associated Geometric transformations

The first part of this appendix concerns the geometric transformations on surfaces in the pseudo-euclidean space ℝ^{n−j,j} corresponding to the action of an element with two simple poles on the space of local solutions of our complex O(n − j + 1, j + 1)/O(n − j, j) × O(1, 1)-system II (17). In particular, the results which will be established here were proved by the authors in [7], hence we invite the reader to see in [7] the proof’s details. In addition, the reader will find in [7], an explicit example of an isothermic timelike dual pair in ℝ^{2,1} of type O(1, 1) constructed by applying the Darboux transformation to the trivial solution of complex system II (18). We note that the study of the geometric transformations associated to the real case, was already considered in [14].

In the second part of this appendix, we establish the moving frame formulas for timelike surfaces in ℝ^{n−j,j}.

Initially in [7], we made a natural extension of the Ribaucour transformation definition given in [5], and of the definition of Darboux transformation for surfaces in ℝ^{m} for our case of complex timelike surfaces. Later, we found the rational element g_{s,π} whose action corresponds to the Ribaucour and Darboux transformations just as we defined them. We now review the principal results of [7].

We start by defining Ribaucour and Darboux transformations for timelike surfaces in ℝ^{n−j,j} whose shape operators have conjugate eigenvalues as follows: For x ∈ ℝ^{n−j,j} and v ∈ (T ℝ^{n−j,j})_{x}, where let γ_{x,v}(t) = x + tv denote the geodesic starting at x in the direction of v.

Definition 4.1. Let M^{m} and M̃^{m} be Lorentzian submanifolds whose shape operators are all diagonalizable over ℝ or ℂ immersed in the pseudo-riemannian space ℝ^{n−j,j}, 0 < j < n. A sphere congruence is a vector bundle isomorphism P : V(M) → V(M̃) that covers a diffeomorphism ϕ : M → M̃ with the following conditions:

(1) If ξ is a parallel normal vector field of M, then P ∘ ξ ∘ ϕ^{−1} is a parallel normal field of M̃.

(2) For any nonzero vector ξ ∈ V_{x}(M), the geodesics γ_{x,ξ} and γ_{ϕ(x),P (ξ)} intersect at a point that is the same parameter value t away from x and ϕ(x).

For the following definition we assume that each shape operator is diagonalized over the real or complex numbers. We note that there are submanifolds for which this does not hold.

Definition 4.2. A sphere congruence P : V(M) → V(M̃) that covers a diffeomorphism ϕ : M → M̃ is called a Ribaucour transformation if it satisfies the following additional properties:

(1) If e is an eigenvector of the shape operator A_{ξ} of M, corresponding to a real eigenvalue then ϕ_{∗}(e) is an eigenvector of the shape operator A_{P (ξ)} of M̃ corresponding to a real eigenvalue. If e_{1} + ie_{2} is an eigenvector of A_{ξ} on (T M)^{ℂ} corresponding to the complex eigenvalue a + ib (so that e_{1} − ie_{2} corresponds to the eigenvalue a − ib), then ϕ_{∗}(e_{1}) + iϕ_{∗}(e_{2}) is an eigenvector corresponding to a complex eigenvalue for A_{P (ξ)}.

(2) The geodesics γ_{x,e} and γ_{ϕ(x),ϕ_{∗}(e)} intersect at a point that is equidistant to x and ϕ(x) for real eigenvectors e, and γ_{x,e_{j}} and γ_{ϕ(x),ϕ_{∗}(e_{j})} meet for the real and imaginary parts of complex eigenvectors e_{1} + ie_{2}, i.e., for j = 1, 2.

Definition 4.3. Let M, M̃ be two timelike surfaces in ℝ^{n−j,j} with flat and non-degenerate normal bundle, shape operators that are diagonalizable over ℂ and P : V(M) → V(M̃) a Ribaucour transformation that covers the map ϕ : M → M̃. If, in addition, ϕ is a sign-reversing conformal diffeomorphism then P is called a Darboux transformation.

In definition (4.3), by a sign-reversing conformal diffeomorphism we mean that the time-like and space like vectors are interchanged and the conformal coordinates remain conformal. Next we define the rational element

g_{s,π}(λ) = (π + \frac{λ − is}{λ + is}(I − π))(\overline{π} + \frac{λ + is}{λ − is}(I − \overline{π}))  (27)

where 0 ≠ s ∈ ℝ, π is the orthogonal projection of ℂ^{n+2} onto the span of (\begin{matrix}W \\ iZ \end{matrix})

with respect to the bi-linear form ⟨, ⟩_{2} given by

⟨U, V ⟩_{2} = \overline{u}_{1}v_{1} + ... + \overline{u}_{n−j}v_{n−j} − \overline{u}_{n−j+1}v_{n−j+1} − ... − \overline{u}_{n}v_{n} + \overline{u}_{n+1}v_{n+2} + \overline{u}_{n+2}v_{n+1},

for W ∈ ℝ^{n−j,j}, Z ∈ ℝ^{1,1} unit vectors. It is easy to see that g_{s,π} belongs to the group:

G_{−} = {g : S^{2} → U_{ℂ} | g is meromorphic, g(∞) = I and satisfies

the reality conditions},

where U_{ℂ} = O(n − j + 1, j + 1; ℂ) and the reality conditions are the following, for a map g : ℂ → U_{ℂ}:

\{ \begin{matrix}\overline{g(\overline{λ})} = g(λ) \\ I_{n,2} g(−λ) I_{n,2} = g(λ) \\ g(λ)^{t} (\begin{matrix}I_{n−j,j} & 0 \\ 0 & J′ \end{matrix}) g(λ) = (\begin{matrix}I_{n−j,j} & 0 \\ 0 & J′ \end{matrix}) . \end{matrix}  (28)

With this, we have:

Theorem 4.1. Let (X_{1}, X_{2}) be a complex isothermic timelike dual pair in ℝ^{n−j,j} of type O(1, 1) corresponding to the solution (u, G) of the system (18),

and let ξ = (\begin{matrix}F \\ G \end{matrix}) the corresponding solution of the system (16), where

F = (\begin{matrix}u_{x_{2}} & u_{x_{1}} \\ u_{x_{1}} & −u_{x_{2}} \end{matrix}) , B = (\begin{matrix}e^{2u} & 0 \\ 0 & e^{−2u} \end{matrix}) .

Let g_{s,π} defined in (27), and Ŵ, Ẑ as in Main Lemma 4.1 (see below), for the

solution ξ of the system (16). Let (Ẽ^{♯}^{II}, Ã^{♯}, B̃^{♯}) = g_{s,π}.(E^{II}, A, B) the action of g_{s,π} over (E^{II}, A, B) where A, B, Ã^{♯}, B̃^{♯} are the entries of

E(x, 0) = (\begin{matrix}A(x) & 0 \\ 0 & B(x) \end{matrix}) , Ẽ^{♯}(x, 0) = (\begin{matrix}Ã^{♯}(x) & 0 \\ 0 & B̃^{♯}(x) \end{matrix})

and E^{II} is the frame corresponding to the solution (F, G, B) of the complex system II (18). Write A = (e_{1}, ..., e_{n}) and Ã^{♯} = (ẽ_{1}, ..., ẽ_{n}). Set

\begin{cases}X̃_{1} = X_{1} + \frac{2}{s} ẑ_{2}e^{−2u} ∑_{i=1}^{n} ŵ_{i}e_{i}, \\ X̃_{2} = X_{2} + \frac{2}{s} ẑ_{1}e^{2u} ∑_{i=1}^{n} ŵ_{i}e_{i}, \end{cases}  (29)

Then

(i) (ũ, G̃) is the solution of system (18), corresponding to X̃ = (X̃_{1}, X̃_{2}), where e^{4ũ} = \frac{4z_{2}^{4}}{e^{4u}} and G̃ = (r̃_{ij}) is defined by Main Lemma 4.1, for the new solution ξ̃ of the system (16). (ii) The fundamental forms of pair (X̃_{1}, X̃_{2}) are respectively

\begin{cases}Ĩ_{1} = e^{4ũ}(−dx_{1}^{2} + dx_{2}^{2}) \\ ĨI_{1} = e^{2ũ} ∑_{i=1}^{n−2}[r̃_{i,1}(dx_{2}^{2} − dx_{1}^{2}) − 2r̃_{i,2}dx_{1}dx_{2}]ẽ_{i+1}. \\ Ĩ_{2} = e^{−4ũ}(dx_{1}^{2} − dx_{2}^{2}) \\ ĨI_{2} = −e^{−2ũ} ∑_{i=1}^{n−2} r̃_{i,2}(dx_{2}^{2} − dx_{1}^{2}) + 2r̃_{i,1}dx_{1}dx_{2}]ẽ_{i+1}. \end{cases}

(iii) The bundle morphism P (e_{k}(x)) = ẽ_{k}(x), k = 2, ..., n − 1 covering the map X_{i} → X̃_{i} is a Darboux transformation for each i = 1, 2. Proof. For (i) and (ii) we just observe that

dX̃ = Ã^{♯} (\begin{matrix}dx_{1} & 0 & … & 0 & dx_{2} \\ −dx_{2} & 0 & … & 0 & dx_{1} \end{matrix})^{t} B̃^{♯}^{−1},

and calculate. For (iii) we observe that the map ϕ : X_{i} → X̃_{i} is sign-reversing conformal because the coordinates (x_{1}, x_{2}) are isothermic for X_{i} and X̃_{i} but timelike and spacelike vectors are interchanged. The rest of the properties of Darboux transformation follows from Lemma 4.2 below. ■

Lemma 4.1. (Main Lemma) Let ξ = (\begin{matrix}F \\ G \end{matrix}) be a solution of the system (16),

and E(x, λ) a frame of ξ such that E(x, λ) is holomorphic for λ ∈ ℂ. Let g_{s,π}

the map defined by (27) and π̃(x) the orthogonal projection onto ℂ (W̃ ; iZ̃) (x)

with respect to ⟨, ⟩_{2}, where

(W̃ ; iZ̃) (x) = E(x, −is)^{−1} (\begin{matrix}W \\ iZ \end{matrix}) .  (30)

Let Ŵ = \frac{W̃}{∥W̃∥_{n−j,j}} and Ẑ = \frac{Z̃}{∥Z̃∥_{1,1}} , Ẽ(x, λ) = g_{s,π}(λ)E(x, λ)g_{s,π̃(x)}(λ)^{−1},

ξ̃ = ξ − 2s(ŴẐ^{t}J′)_{∗},  (31)

where (ϑ_{∗}) is the projection onto the span of {a_{1}, a_{2}}^{⊥}. Then ξ̃ is a solution of system (16), Ẽ is a frame for ξ̃ and Ẽ(x, λ) is holomorphic in λ ∈ ℂ.

For the Proof of the Main Lemma see ([7]).

Writing the new solution given by Main Lemma 4.1 as ξ̃ = (F̃ ; G̃), one sees the

components of ξ̃ are:

\begin{cases}f̃_{11} = −f̃_{22} = f_{11} − s(ŵ_{1}ẑ_{2} − ŵ_{n}ẑ_{1}), \\ f̃_{12} = f̃_{21} = f_{12} − s(ŵ_{1}ẑ_{1} + ŵ_{n}ẑ_{2}), \\ r̃_{i1} = r_{i1} − 2sŵ_{1+i}ẑ_{2} \\ r̃_{i2} = r_{i2} − 2sŵ_{1+i}ẑ_{1}, \end{cases}  (32)

for F = (f_{ij})_{2×2}, G = (r_{ij})_{(n−2)×2}, F̃ = (f̃_{ij})_{2×2}, G̃ = (r̃_{ij})_{(n−2)×2}.

Lemma 4.2. Let ξ = (\begin{matrix}F \\ G \end{matrix}) solution of (16), E frame of ξ, E(x, 0) =

(\begin{matrix}A(x) & 0 \\ 0 & B(x) \end{matrix}), (F, G, B) a solution corresponding to complex O(n − j + 1, j +

1)/O(n − j, j) × O(1, 1)-system II, and

(F̃, G̃, B̃^{♯}, Ẽ^{♯}^{II}) = g_{s,π}.(F, G, B, E^{II}), Ã^{♯} = g_{s,π}.A

the action of g_{s,π} over the solution (F, G, B) and the matrix A, resp.. Let e_{i}, ẽ_{i} denote the i-th column of A and Ã^{♯} resp. Then we have

(i) X = (X_{1}, X_{2}) and X̃ = (X̃_{1}, X̃_{2}) are complex isothermic timelike dual pairs in ℝ^{n−j,j} of type O(1, 1) such that {e_{2}, ...e_{n−1}} and {ẽ_{2}, ..., ẽ_{n−1}} are parallel normal frames for X_{j} and X̃_{j} respectively for j = 1, 2, where {e_{α}}_{α=2}^{n−j} and {e_{α}}_{α=n−j+1}^{n−1} are spacelike and timelike vectors resp. (ii) The solutions of the complex O(n−j +1, j +1)/O(n−j, j)×O(1, 1)-system II corresponding to X and X̃ are (F, G, B) and (F̃, G̃, B̃^{♯}) resp. (iii) The bundle morphism P (e_{k}(x)) = ẽ_{k}(x) k = 2, ..., n − 1, is a Ribaucour Transformation covering the map X_{j}(x) ↦ X̃_{j}(x) for each j = 1, 2. (iv) There exist smooth functions ψ_{ik} such that X_{i} + ψ_{ik}e_{k} = X̃_{i} + ψ_{ik}ẽ_{k} for 1 ≤ i ≤ 2 and 1 ≤ k ≤ n.

For the proof of Lemma 4.2, see ([7]).

Now we begin the second part of this appendix, where we review the method of moving frames for time-like surfaces in the Lorentz space ℝ^{n−j,j}. Set

e_{A} · e_{B} = σ_{AB} = I_{n−j,j} = (\begin{matrix}I_{n−j} & 0 \\ 0 & −I_{j} \end{matrix}) .

We also let σ_{i} := σ_{ii}.

For the time-like immersion X set dX = ω_{1}e_{1} + ω_{n}e_{n}, so that a space-like unit tangent vector to the surface is e_{1}, a time-like unit vector to the surface is e_{n} and the normal space is spanned by e_{α}, for 2 ≤ α ≤ n − 1. Define

de_{B} = ∑_{A} ω_{AB}e_{A}.  (33)

This gives ω_{AB} = σ_{A}e_{A} · de_{B} and

ω_{AB}σ_{A} + ω_{BA}σ_{B} = 0.  (34)

From d(dX) = 0 we get:

dω_{1} = ω_{n} ∧ ω_{1n}  (35)
dω_{n} = ω_{1} ∧ ω_{n1}  (36)
ω_{1} ∧ ω_{α1} + ω_{n} ∧ ω_{αn} = 0,  (37)

for α as above. In addition, by Cartan’s Lemma we have:

ω_{1α} = h_{11}^{α}ω_{1} + h_{1n}^{α}ω_{n}, ω_{nα} = h_{n1}^{α}ω_{1} + h_{nn}^{α}ω_{n}.

This makes the first fundamental form:

I : ω_{1}^{2} − ω_{n}^{2}  (38)

and the second fundamental form is:

II :  − ∑_{k=1,n} ∑_{α} ω_{kα}σ_{k} ω_{k} σ_{α} e_{α} =  (39)
− ∑_{α} (h_{11}^{α}ω_{1} + h_{1n}^{α}ω_{n}) ω_{1} σ_{α} e_{α} + ∑_{α} (h_{n1}^{α}ω_{1} + h_{nn}^{α}ω_{n}) ω_{n} σ_{α} e_{α}.

We also have: dω_{CA} = − ∑_{B} ω_{CB} ∧ ω_{BA}, which yield the Gauss and Codazzi equations. The Gauss equation comes from examining dω_{1n}, while the Codazzi equations are from dω_{1α} and dω_{nα}.

References

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M. P. Dussan

Department of Mathematics IME - USP São Paulo, Brazil dussan@ime.usp.br M. A. Magid Department of Mathematics Wellesley College Wellesley, MA mmagid@wellesley.edu 550

Documenta Mathematica 10 (2005)