Spaces of geodesics of pseudo-Riemannian space forms and normal congruences of hypersurfaces

We describe natural K\"ahler or para-K\"ahler structures of the spaces of geodesics of pseudo-Riemannian space forms and relate the local geometry of hypersurfaces of space forms to that of their normal congruences, or Gauss maps, which are Lagrangian submanifolds. The space of geodesics L(S^{n+1}_{p,1}) of a pseudo-Riemannian space form S^{n+1}_{p,1} of non-vanishing curvature enjoys a K\"ahler or para-K\"ahler structure (J,G) which is in addition Einstein. Moreover, in the three-dimensional case, L(S^{n+1}_{p,1}) enjoys another K\"ahler or para-K\"ahler structure (J',G') which is scalar flat. The normal congruence of a hypersurface s of S^{n+1}_{p,1} is a Lagrangian submanifold \bar{s} of L(S^{n+1}_{p,1}), and we relate the local geometries of s and \bar{s}. In particular \bar{s} is totally geodesic if and only if s has parallel second fundamental form. In the three-dimensional case, we prove that \bar{s} is minimal with respect to the Einstein metric G (resp. with respect to the scalar flat metric G') if and only if it is the normal congruence of a minimal surface s (resp. of a surface s with parallel second fundamental form); moreover \bar{s} is flat if and only if s is Weingarten.


Introduction
After the seminal paper of N. Hitchin ([Hi]) describing the natural complex structure of the space of oriented straight lines of Euclidean 3-space, several invariant structures on the space of geodesics of certain Riemannian manifolds and their submanifolds have recently been explored by different authors (see [AGR], [Ge], [GK1], [GG1], [GG2], [Ho], [Ki], [Sa1], [Sa2]). In [AGK], a unified viewpoint has been given to this question, classifying all invariant Riemannian, symplectic, complex and para-complex structures that may exist on the space of geodesics of a number of spaces: the Euclidean and pseudo-Euclidean spaces, the Riemannian and pseudo-Riemannian space forms and the complex and quaternionic space forms. One of the interesting issues about the spaces of geodesics is that the normal congruence (or Gauss map) of a one-parameter family of parallel hypersurfaces in some space is a Lagrangian submanifold of the corresponding space of geodesics.
The purpose of this paper is twofold: first, to give a more precise picture of the structure of the space of geodesics of pseudo-Riemannian space forms, and second to study in detail the relationships between the pseudo-Riemannian geometry of a one-parameter family of parallel hypersurfaces and that of its normal congruence.
In particular, we describe the natural Kähler or para-Kähler structure of the space of geodesics of pseudo-Riemannian space forms of non-vanishing curvature and prove that the corresponding metric G is Einstein (Theorem 1). The space of geodesics of pseudo-Riemannian three-dimensional space forms, which is fourdimensional, is specific since (i) it is the only dimension for which the space of geodesics of flat pseudo-Euclidean spaces enjoys an invariant metric (see [Sa1], [AGK]), and (ii) in the non-flat case it enjoys another natural complex or paracomplex structure, which in turns defines a neutral metric G ′ . We prove that G ′ is scalar flat and locally conformally flat (Theorem 2).
Next we turn our attention to the relation between one-parameter families of parallel hypersurfaces in pseudo-Riemannian space forms and their normal congruences. We first check that an n-dimensional geodesic congruenceS is Lagrangian if and only it crosses orthogonally a hypersurface S (Theorem 3), and therefore all the hypersurfaces S t parallel to S and to its polar. Given a oneparameter family of parallel hypersurfaces (S t ) and its normal congruenceS, we relate the first and second fundamental forms of (S t ) to those ofS (Theorems 4 and 5). These formulas imply several interesting corollaries:S is totally geodesic (either with respect to G or G ′ ) if and only if the hypersurfaces S t have parallel second fundamental form; in the three-dimensional case,S is minimal with respect to G if and only if one of the parallel surfaces S t is minimal (Corollary 3);S is minimal with respect to G ′ if and only if the parallel surfaces S t are totally geodesic (Corollary 4); the induced metric onS is flat if and only if the surfaces S t are Weingarten (Corollary 5). We also exhibit three families of Lagrangian surfaces which are marginally trapped with respect to G or G ′ . (Corollary 6).
The papers is organised as follows: Section 1 provides some useful preliminaries and Section 2 gives the precise statements of results; Section 3 deals with the geometry of the spaces of geodesics while Section 4 is devoted to normal congruences of hypersurfaces.
The author thanks Nikos Georgiou for interesting observations about the early version of this manuscript.

Hypersurfaces in pseudo-Riemannian space forms
Consider the real space R n+2 and endowed with the canonical pseudo-Riemannian metric of signature (p, n + 2 − p), where 0 ≤ p ≤ n + 1: dx 2 i , and the (n + 1)-dimensional quadric where ǫ = ±1. The metric induced on S n+1 p,ǫ by the canonical inclusion S n+1 p,ǫ ֒→ (R n+2 , ., . p ) has signature (p, n + 1 − p) if ǫ = 1 and (p − 1, n + 2 − p) if ǫ = −1, and constant sectional curvature K = ǫ. Conversely, it is known (see [Kr]) that any pseudo-Riemannian manifold with constant sectional curvature is, up to a scaling of the metric, locally isometric to one of these quadrics. The transformation defines an anti-isometry of S n+1 p,ǫ onto S n+1 n+2−p,−ǫ . Is it therefore sufficient to study the case ǫ = 1. The two Riemannian space forms are (i) the sphere S n+1 := S n+1,1 0,1 , which is the only compact quadric, and (ii) the hyperbolic space H n+1 := A(S n+1 n+1,1 ) ∩ {x ∈ R n+2 |x 1 > 0}) (S n+1 1,−1 and S n+1 n+1,1 are the only non-connected quadrics). Analogously, the two Lorentzian space forms are the de Sitter space dS n+1 := S n+1 1,1 and the anti de Sitter space AdS n+1 := S n+1 2,−1 = A(S n+1 n,1 ). Let φ : M n → S n+1 p,1 be a smooth map from an orientable n-dimensional manifold M n . We set g := φ * ., . p for the induced metric on M n . We shall always assume that φ is a pseudo-Riemannian immersion, i.e. g is non-degenerate. This is equivalent to the existence of a unit normal vector field along the immersed hypersurface S := φ(M n ) that we will denote by N. The curvature of S may be equivalently described by two tensors: the second fundamental form h with respect to N , i.e. h(X, Y ) = g(∇ X Y, N ), where ∇ denotes the Levi-Civita connection of ., . p ; the shape operator defined by AX = −dN (X). They are related by the formula: g(AX, Y ) = h(X, Y ). The shape operator A is not necessarily real diagonalizable since it is symmetric with respect to the possibly indefinite metric g. More precisely, A may be of three types: real diagonalizable, complex diagonalizable, or not diagonalizable at all. In the twodimensional case, we shall use the existence of a canonical form for A, i.e. the existence of a frame (e 1 , e 2 ) such that the matrices of g and A take a simple form (see [Ma]): -real diagonalizable case: with non-vanishing λ; -non diagonalizable case:

Parallel hypersurfaces
It will be convenient to introduce some notation: we set (cosǫ, sinǫ) := (cos, sin) if ǫ = 1 and (cosǫ, sinǫ) := (cosh, sinh) if ǫ = −1. Given t ∈ R, the image of when an immersion, is parallel to S. When A is invertible, the map N : M n → S n+1 p,ǫ , where ǫ := |N | 2 p , is an immersion and its image S ′ := N (M n ) is called the polar of S. If ǫ = 1, we have φ π/2 = N, hence the polar of S is parallel to S. If ǫ = −1, S ′ ∈ S n+1 p,−1 = A(S n+1 n+2−p,1 ). In all cases, a unit normal vector field along S t = φ t (M n ) is which, when an immersion, is parallel to S ′ .
We shall denote by arctanǫ the integral of the map 1 1+ǫt 2 , i.e.: The only property of arctanǫ we will need is the following:

Lagrangian submanifolds
We first recall the definition of a Lagrangian submanifold: We refer the reader to [AMT] or [CFG] for material about para-complex geometry (sometimes referred to as split-complex or bi-Lagrangian geometry). By a pseudo-Kähler or a para-Kähler manifold, we mean a manifold equipped with a complex or para-complex structure J and a compatible pseudo-Riemannian metric G, i.e. such that G(J., J.) = ǫG(., .). Here, ǫ = 1 in the complex case and ǫ = −1 in the para-complex case. In other words J is an isometry in the complex case and an anti-isometry in the para-complex case. It is furthermore required that the symplectic form ω := ǫG(J., .) be closed 1 . Observe that the metric G is determined by the pair (J, ω) via the equation G := ω(., J.).
It is well known that the extrinsic curvature of a Lagrangian submanifold in a Kähler manifold (N , J, G) is described by the tri-symmetric tensor h(X, Y, Z) := G(D X Y, JZ), where D denotes the Levi-Civita connection of G (see [An]). It turns out that the same fact holds in the para-Kähler case: Lemma 2 Let L be a non-degenerate, Lagrangian submanifold of a pseudo-Kähler or para-Kähler manifold (N , J, G, ω). Denote by D the Levi-Civita connection of G. Then the map is tensorial and tri-symmetric, i.e.
Proof. The tensoriality of h and its symmetry with respect to the first two slots follow from the tensiorality and the symmetry of the second fundamental form. It remains to prove for example that h(X, Y, Z) = h(X, Z, Y ). From the Lagrangian assumption we have ω(Y, Z) = ǫG(JY, Z) = 0. Differentiating in the X direction gives, using the fact that J is parallel, and the proof is complete.
1 Of course the factor ǫ is unessential here and is put in order to simplify further exposition. In particular, this convention allows to recover, in the case of R 2 , the "natural" objects G := dx 2 + ǫdy 2 , J(∂x, ∂y) := (∂y, −ǫ∂x) and ω := dx ∧ dy.
2 Statement of results 2.1 Structures of the space of geodesics of pseudo-Riemannian space-forms Let x be a point of S n+1 p,1 and v ∈ T x S n+1 p,1 = x ⊥ a unit vector tangent to x. Setting ǫ := |v| 2 p , the unique geodesic γ of S n+1 p,1 passing through x with velocity v is the periodic curve parametrized by γ(t) = cosǫ(t)x + sinǫ(t)v.
Remark 1 It is not difficult to check that G and J are invariant under the natural action of the group SO(n+2−p, p) of isometries of S n+1 p,1 . Such invariant structures have been studied with the Lie algebra formalism in [AGK], where in particular it is proved that such an invariant pseudo-Riemannian metric and complex or para-complex structure are unique on L ± (S n+1 p,1 ), for n ≥ 3. The fact that G is Einstein has been proved in [Le] in the spherical case.
In the three-dimensional case, L ± (S 3 p,1 ) enjoys other natural structures, which may de defined as follows: since the orthogonal two-planex ⊥ admits a canonical orientation (that orientation compatible with the orientations ofx and R 4 ), it enjoys a canonical complex or para-complex structure J ′ (depending of whether the induced metric onx ⊥ is positive or indefinite). Hence we set We therefore get another almost complex or para-complex structure on L ± (S 3 p,1 ). Finally, we introduce one more tensor: we want to define a pseudo-Riemannian structure G ′ on L ± (S 3 p,1 ) with the requirement that the pair (J ′ , G ′ ) induces the same symplectic structure, up to sign, than that of (J, G). In other words, we require that ω(., .) := ǫ ′ G ′ (J ′ ., .) be the same that ω(., .) := ǫG(J., .). Hence, we must have: It turns out that this defines another Kähler or para-Kähler structure: Theorem 2 The two-form G ′ := −ǫG(., J ′ • J.) is symmetric and therefore defines a pseudo-Riemannian metric on L ± (S 3 p,1 ). The Levi-Civita connection of G ′ is the same than that of G and the structures (J, G) and (J ′ , G ′ ) share the same symplectic form ω. Moreover, In all cases, the metric G ′ has neutral signature (2, 2), is scalar flat and locally conformally flat.
Remark 3 The properties of G ′ have been derived in [GG1] in the case of hyperbolic space.
The fact that (J, G) and (J ′ , G ′ ) share both the same Levi-Civita connection and symplectic form implies that they also share some distinguished classes of submanifolds: Corollary 1 Lagrangian surfaces, flat and totally geodesic submanifolds in L ± (S 3 p,1 ) are the same for (J, G) and (J ′ , G ′ ).
Remark 5 Since the two complex or para-complex structures J and J ′ commute, their composition J ′′ := J • J ′ defines one more invariant structure: if J and J ′ are both complex or both para-complex, then J ′′ is complex, and if J and J ′ are of different types, J ′′ is para-complex. The two-form G ′′ := ω(., J ′′ ) is not symmetric, so there is no pseudo-or para-Kähler structure associated to J ′′ .
Observe also that the triple (J, J ′ , J ′′ ) is not a para-quaternionic structure, since J and J ′ commute rather than anti-commute. The case L − (AdS 3 ) excepted, this triple is what is called an almost product bi-complex structure in [Cr].

Normal congruences of immersed hypersurfaces as Lagrangian submanifolds
Definition 2 Let S be an immersed surface of pseudo-Riemannian space form S n+1 p,1 with unit normal vector N. The normal congruence (or Gauss map)S of S is set of geodesics crossing S orthogonally in the direction N.
Theorem 3 Let φ be a pseudo-Riemannian immersion of an orientable manifold M n in pseudo-Riemannian space form S n+1 p,1 with unit normal vector N.
Whenφ is an immersion, it is Lagrangian with respect to ω. In this case,S is also the normal congruence of the hypersurfaces parallel to S and to its polar S ′ . Conversely, letφ : M n → L ± (S n+1 p,1 ) be an immersion of a simply connected n-manifold. ThenS is the normal congruence of an immersed hypersurface of S n+1 p,1 if and only ifφ is Lagrangian.
In view of this result, it is natural to relate the geometry of a Lagrangian submanifold to that of the corresponding hypersurface of S n+1 p,1 .
Theorem 4 Let φ be a pseudo-Riemannian immersion of an orientable manifold M n in pseudo-Riemannian space form S n+1 p,1 with unit normal vector N. Set |N | 2 p := ǫ, denote by A the shape operator of φ with respect to N and by ∇ g the Levi-Civita connection of g. Then the induced metricḡ :=φ * G, with φ = φ ∧ N , is given by the following formulā g = ǫg + g(A., A.).
In particular,ḡ is non-degenerate if and only if ǫId + A 2 is invertible.
Moreover, the extrinsic curvatures h of S := φ(M n ) and ofh ofS :=φ(M n ) are related by the formulah = ǫ∇ g h.
In particular the normal congruenceS is totally geodesic if and only if S has parallel second fundamental form.
Remark 6 The fact that the tensorh ofS is tri-symmetric is equivalent to the Codazzi equation for the hypersurface S.
Corollary 2 If the shape operator A of S is real diagonalizable (this is always the case if ǫ ′ = 1), the mean curvature vector ofS with respect to G is where κ 1 , ..., κ n are the principal curvatures of S and∇ is the gradient with respect to the induced metricḡ. In particular, if S is isoparametric (i.e. its principal curvatures are constant) or austere (i.e. the set of its principal curvatures is symmetric with respect to 0), then its normal congruenceS is G-minimal.
Corollary 3 If n = 2, the mean curvature vector ofS with respect to G is where H and K denote the mean curvature and the Gaussian curvature of S respectively. In particular,S is G-minimal if and only if it is the normal congruence of a minimal surface.
Remark 7 Corollaries 2 and 3 have been proved in [Pa] in the spherical case. The fact that the mean curvature vector takes the form H = ǫ n J∇β, where β is an S 1 -valued map, is due to the fact that the metric G is Einstein (cf [HR]). The map β is called the Lagrangian angle of the submanifoldS.
In the three-dimensional case, it is natural to study the pseudo-Riemannian geometry of Lagrangian surfaces of L ± (S 3 p,1 ) with respect to the metric G ′ described in Theorem 2.
Theorem 5 Let φ be a pseudo-Riemannian immersion of an orientable surface M 2 in pseudo-Riemannian space form S 3 p,1 with shape operator A and unit normal vector N. Then the induced metricḡ ′ :=φ * G ′ , withφ = φ ∧ N , is given by the following formulaḡ

Moreover,
-If A is real diagonalizable, the metricḡ ′ is degenerate at umbilic points of S := φ(M 2 ) and indefinite elsewhere; the null directions ofḡ ′ are the principal directions of S; -If A is complex diagonalizable, the metricḡ ′ is everywhere definite; -If A is not diagonalizable, the metricḡ ′ is everywhere degenerate.
Whenḡ ′ is not degenerate, the extrinsic curvatures h andh of S andS := φ(M 2 ) are related by the formulah = ǫ∇ g h.
In particular the normal congruenceS of S is totally geodesic if and only if S has parallel second fundamental form.
Corollary 4S is G ′ -minimal if and only it is totally geodesic, i.e. S has parallel second fundamental form. In in addition A is real diagonalizable, S is the set of equidistant points to a geodesic of S 3 p,1 .
Corollary 5 The induced metricḡ ′ is flat (and the metricḡ as well by Corollary 1) if and only if the surface S is Weingarten, i.e. there exists a functional relation f (H, K) = 0 satisfied by the mean curvature and the Gaussian curvature of S.
Remark 8 Corollary 4 and 5 have been proved in the case of hyperbolic space in [Ge] and [GG2] respectively. Corollary 5 has been proved in the case of Euclidean space in [GK2].
Corollary 6 If the shape operator A of S is not diagonalizable, then its normal congruenceS is a G-marginally trapped surface, i.e. the mean curvature vector ofS with respect to G is null. If S is a tube (i.e. the set of equidistant points to an arbitrary curve of S 3 p,1 ) or a surface of revolution, then its normal congruencē S is a G ′ -marginally trapped surface. Proof. Letx := x ∧ y ∈ L ± (S n+1 p,1 ) with |x| 2 p = 1 and |y| 2 p = ǫ and let (e 1 , ..., e n ) be an orthonormal basis of the orthogonal complement of x∧y. We set ǫ i := |e i | 2 p and ǫ n+i := ǫǫ i . Then an orthonormal basis (E a ) 1≤a≤2n of TxL ± (S n+1 p,1 ), with G(E a , E a ) = ǫ a , is given by and E n+i := y ∧ e i .
Fix the index i and introduce the curve In particular γ i (0) =x and γ ′ i (0) = E i . Introduce furthermore the following orthonormal frameV = (v 1 , ...,v 2n ) along γ i : Sincev ′ n+j (0) = e i ∧ e j andv ′ i (0) = −ǫ ix are normal to L ± (S n+1 p,1 ), we deduce that the frameV is parallel along γ i . On the other hand, Jv i =v n+i and Jv n+i = −ǫv i .
It follows that D Ei J = JD Ei so J is parallel, and therefore integrable.
We now proceed to compute the second fundamental form of the immersion ι.

Proof. We have
We deduce that, given V, W ∈x ⊥ , The claimed formula follows from the bi-linearity of h ι .

The curvature of G
We use Gauss equation and Proposition 2 in order to compute the curvature tensorR of G: for 1 ≤ a, b, c, d ≤ 2n, we have In particular, we calculate This expression vanishes unless {k, l} = {i, j} and i = j, in which case it becomes A similar computation shows that This expression vanishes unless (i, k) = (j, l), in which case we get Using the symmetry of G (R(., .)., .), we have Morever, This expression vanishes unless {k, l} = {i, j} and i = j, in which case it becomes An easy but tedious calculation shows that G(R(E a , E b )E c , E d ) vanishes if exactly one or three of the indices a, b, c and d belongs to {1, ..., n}.
It is now easy to calculate the Ricci curvature ofR: An analogous calculation shows that Ric(E i , E n+j ) vanishes. Hence the metric G is Einstein, with constant scalar curvatureS = ǫ2n 2 .
Finally, since G is Einstein, the Weyl tensor is given by the formula It is easily seen, for example, that G and therefore G is never conformally flat.
Proof. Elementary using local coordinates and the explicit formula for the Christoffel symbols.
Since G and G ′ have the same Levi-Civita connection, they have the same curvature tensorR. Therefore, It follows that the scalar curvature of G ′ vanishes: It may be interesting to point out that the Ricci curvature of G ′ is non-negative in the case of L(S 3 ), non-positive in the case of L + (AdS 3 ), and indefinite in the other cases.
Finally, since G ′ is scalar flat, its Weyl tensor is given by the formula We may calculate, for example, that It is easily checked in the same manner that the other components of the Weyl tensor vanish. The metric G ′ is therefore locally conformally flat.
4 Normal congruences of hypersurfaces and Lagrangian submanifolds

Lagrangian submanifolds are normal congruences (proof of Theorem 3)
Let φ : M n → S n+1 p,1 an immersed, orientable hypersurface with non-degenerate metric and unit normal vector N and introduce the map In the following, we shall often allow the abuse of notation of identifying a tangent vector X to M n with its image dφ(X), a vector tangent to S n+1 p,1 , therefore an element of R n+2 . We furthermore setX := dφ(X), so that It follows that hence parallel hypersurfaces have the same normal congruence. Conversely, letS an n-dimensional geodesic congruence, i.e. the image of an immersionφ : M n → L ± (S n+1 p,1 ). We shall investigate under which condition there exists an hypersurface S of S n+1 p,1 which intersects orthogonally the geodesicsφ(x), ∀x ∈ M n . For this purpose setφ(x) := e 1 (x)∧e 2 (x) with |e 1 | 2 p = 1 and |e 2 | 2 p = ǫ. Let φ : M n → S n+1 p,1 such that φ(x) ∈φ(x), ∀x ∈ M n . Therefore there exits t : M n → S 1 , such that φ(x) = e 1 (x)cosǫ(t(x)) + e 2 (x)sinǫ(t(x)).
Hence,S is the normal congruence of S if and only there exists t : M n → S 1 such that de 1 , e 2 p = −ǫdt. Since M n is simply connected, it is sufficient to have d de 1 , e 2 p = 0. Observe that On the other hand, dφ = de 1 ∧ e 2 + e 1 ∧ de 2 , and We conclude that t, and thus φ as well, exists if and only ifφ is Lagrangian. Of course, the choice of different constants of integration when solving t corresponds to different, parallel hypersurfaces. Using the description of the metric G given in Section 3.1, we have:

Geometry of Lagrangian submanifolds with respect to the Einstein metric
We now discuss the degeneracy ofḡ : suppose there exist X such that g(X, Y ) = ǫg(X, Y ) + g(AX, AY ) = g(ǫX + A 2 X, Y ) vanishes ∀ Y ∈ T M. Since the metric g is non-degenerate, it follows that ǫX + A 2 X vanishes. Hence ǫId + A 2 is not invertible. If A is diagonalizable, the eigenvalues of A 2 are non-negative, so we must have ǫ = −1. Next, denoting by ∇ (resp. D) the flat connection of R n+2 (resp. Λ 2 (R n+2 )),

The mean curvature vector in the diagonalizable case (proof of Corollary 2)
Assume that A is real diagonalizable and let (e 1 , ..., e n ) be an orthonormal frame (e 1 , ..., e n ) on (T M, g), with ǫ i := g(e i , e i ) and such that Ae i = κ i e i , where κ 1 , ..., κ n are the principal curvatures of S. We introduce the notation ω i jk := g(∇ ei e j , e k ). In particular ω i jk is antisymmetric in its lower indices. It follows that g(e i , e j ) = 0 if i = j, andḡ(e i , e i ) = ǫǫ i + ǫ i κ 2 i = ǫ i (ǫ + κ 2 i ).
For further use, observe that the tri-symmetry ofh, or equivalently the Codazzi equation of the immersion φ implies Since the basis (e 1 , ..., e n ) is orthogonal with respect to the metricḡ, we have where β := − n j=1 arctanǫ(κ j ), which implies that H = ǫ n J∇β. Clearly the immersionφ is G-minimal if and only the map β is constant. This happens of course if the principal curvatures of S are constant, i.e. it is isoparametric. Moreover, if S is austere, i.e. the set of the principal curvatures is symmetric with respect to 0, the Lagrangian angle β vanishes because the function arctanǫ is odd. This completes the proof of Corollary 2.

The mean curvature vector in the two-dimensional case (proof of Corollary 3)
Here and in the next section, we shall make use of canonical form of A (see Section 1.1 and [Ma]).

The real diagonalizable case
We use the computation of the previous section: which is the required expression of the Lagrangian angle β. We now prove that if β is constant, the assumptions of Lemma 1 are satisfied. Assume by contradiction that (ǫ, 2H K−ǫ ) = (−1, ±1). It follows that κ1+κ2 κ1κ2+1 = ±1, which in turn implies that |κ 1 | or |κ 2 | = 1. Therefore, −Id + A 2 is not invertible, and the metricḡ is degenerate by Theorem 4. Since this situation is excluded a priori, we may use Lemma 1 and conclude that there exists a minimal hypersurface parallel to S or its polar S ′ , and therefore whose normal congruence isS.
(The fact that we obtained two different expressions forh 112 andh 122 accounts for the Codazzi equation). Hence In the same way, we get G(2 H, Jdφ(e 2 )) = − 2(1 + ǫH 2 − ǫλ 2 )e 2 (H) + ǫ4Hλe 2 (λ) On the other hand, using the fact that K = H 2 + λ 2 , It follows that G(2 H, J.) = dβ, which is equivalent to 2 H = ǫJ∇β, the required formula. If ǫ = −1 we have, using the fact that λ = 0, Therefore, ifS is G-minimal, i.e. β is constant, we may use again Lemma 1 to conclude that there exists a minimal surface parallel to φ or N. Hence we have proved Corollary 3 in this complex diagonalizable case.

The non diagonalizable case
Here there exists a local frame (e 1 , e 2 ) on (M 2 , g) such that g = 0 1 1 0 and A = H 1 0 H .

Geometry of Lagrangian
which gives the claimed formula forḡ ′ . We now discuss the degeneracy and the signature ofḡ ′ , which depend on the type of the shape operator A: -The real diagonalizable case. Write g and A in canonical form, with (e 1 , e 2 ) an oriented, orthonormal local frame. It follows that J ′ e 1 = e 2 , J ′ e 2 = −ǫ ′ e 1 . We easily get We see in particular thatḡ ′ is degenerate at umbilic points and indefinite otherwise.
-The complex diagonalizable case. Write g and A in canonical form. It follows that J ′ e 1 = e 2 , J ′ e 2 = e 1 (here ǫ ′ = −1 since the metric g is indefinite). Henceḡ which shows that the metricḡ ′ is everywhere definite.
-The non diagonalizable case. Write g and A in canonical form. Since (e 1 , e 2 ) is a g-null basis, the complex structure is given by J ′ e 1 = e 1 , J ′ e 2 = −e 2 , so we getḡ which shows that the metricḡ ′ is everywhere degenerate. In particular, we don't need to take into consideration the case of A being non diagonalizable in the proofs of Corollaries 4 and 5.

The mean curvature vector and the proof of Corollary 4
The real diagonalizable case It has been seen in Section 4.2.2 that h ijj :=h(e i , e j , e j ) = −ǫǫ j e i (κ j ).
Hence κ 1 κ 2 = −ǫǫ ′ . Taking into account the vanishing of e 1 (κ 2 ) and e 2 (κ 1 ), it implies that both principal curvatures are constant, non vanishing and different of ±1. In particular S has parallel second fundamental form andS is totally geodesic.
We conclude that the metricḡ ′ (and thereforeḡ as well) is flat if and only if dκ 1 ∧ dκ 2 vanishes, i.e. S is Weingarten.
We conclude that the metricḡ ′ (and thereforeḡ as well) is flat if and only if dH ∧ dλ vanishes, i.e. S is Weingarten.

G-marginally trapped Lagrangian surfaces
We have seen in Section 4.2.3 that if the shape operator A of φ is not diagonalizable, thenḡ(e 1 , e 1 ) vanishes. If follows that dφ(e 1 ), and therefore Jdφ(e 1 ) as well, is a G-null vector. We have also seen that G(2 H, Jdφ(e 1 )) vanishes, so H, a vector of the plane NS spanned by Jdφ(e 1 ) and Jdφ(e 2 ), must be collinear to Jdφ(e 1 ). Hence it is a G-null vector as well.
Sinceḡ ′ (e 1 , e 1 ) andḡ ′ (e 1 , e 1 ) vanish, the pair (J ′ dφ(e 1 ), J ′ dφ(e 2 )) is a G-null basis of the normal space NS. Therefore, the mean curvature vector H ′ is G ′null if and only if it is collinear to one of the two vectors J ′ dφ(e i ), i.e. if and only if either e 1 (κ 2 ) or e 2 (κ 1 ) vanishes. This occurs at least in the following two cases: -If S is a tube, i.e. the set of equidistant points to a given curve of S 3 p,1 , then one of its principal curvatures is constant; -If S is a surface of revolution, i.e. a surface invariant by the action of a subgroup SO(2) or SO(1, 1) of SO(4−p, p), then both principal curvatures are constant along the orbits of the action, which are in addition tangent to one of the principal directions (cf [An]). Therefore, e 1 (κ 2 ) or e 2 (κ 1 ) vanishes.