Franki Dillen

Christine Scharlach

AND Kristof Schoels

Luc Vrancken

SPECIAL LAGRANGIAN 4-FOLDS WITH SO(2) ⋊ S 3 -SYMMETRY IN COMPLEX SPACE FORMS

 is incomplete. In this paper we give a complete classification of such submanifolds, and extend the classification to special Lagrangian submanifolds of arbitrary complex space forms with SO(2) ⋊ S 3 -symmetry.

Introduction

A space (N, J, g) is called a Hermitian manifold with complex structure J and Riemannian metric g, if g(JX, JY ) = g(X, Y ) for all X and Y . The (0, 2)-tensor ω(X, Y ) = g(X, JY ) is its symplectic form. If ω is closed, then (N, J, g) is said to be a Kähler manifold. In this case the Levi-Civita connection D of g satisfies Dω = 0 as well, see [START_REF] Nomizu | Foundations of Differential Geometry[END_REF]. A complex space form is a Kähler manifold for which the curvature tensor is given by ( 1)

R(X, Y )Z = ǫ (X ∧ Y + JX ∧ JY + 2g(X, JY )J) Z,
where ǫ is a real constant and X ∧ Y is defined as (X ∧ Y )Z = g(Y, Z)Xg(X, Z)Y.

Every complete, simply connected complex space form of dimension n with constant holomorphic sectional curvature 4ǫ is isometric to one of the following manifolds:

(1) the standard complex space C n when ǫ = 0, (2) the complex projective space CP n (4ǫ) when ǫ > 0, (3) the complex hyperbolic space CH n (4ǫ) when ǫ < 0. Because we consider submanifolds of a complex space form locally, we can restrict ourselves to those ambient spaces. By rescaling, we can even assume that ǫ = 0, 1, -1.

A Lagrangian submanifold M of a Kähler manifold (N, J, g) is a submanifold such that ω vanishes identically on M and the (real) dimension of M is half the (complex) dimension of N, see [START_REF] Audin | Lagrangian submanifolds[END_REF]. This implies that J induces an orthogonal isomorphism between the tangent and the normal bundle on the submanifold. The Gauss formula is given by

D X Y = ∇ X Y + h(X, Y ) = ∇ X Y + JA(X, Y ),
where A = -Jh defines a symmetric (1, 2)-tensor on the submanifold, and the Weingarten formula is given by D X (JY ) = J(∇ X Y ) -A(X, Y ). It is easy to see that the cubic form C, defined by C(X, Y, Z) = g(A(X, Y ), Z) 1 is totally symmetric. For Lagrangian submanifolds of complex space forms, the equations of Gauss and Codazzi simplify to

R(X, Y )Z = ǫ (X ∧ Y ) Z + [A X , A Y ] Z, (2)
∇A is symmetric. [START_REF] Chen | Interactions of Legendre curves and Lagrangian submanifolds[END_REF] The following theorem holds, see [START_REF] Chen | Interactions of Legendre curves and Lagrangian submanifolds[END_REF] and [START_REF] Chen | An exotic totally real minimal immersion of S 3 in CP 3 and its characterization[END_REF].

Theorem 1.1. Suppose (M n , g) is a Riemannian manifold equipped with a symmetric and g-symmetric (1, 2)-tensor A such that (2) and (3) are satisfied for some constant ǫ. Then for every point p ∈ M there exists a neighborhood U and a Lagrangian isometric immersion φ : U → N 2n (4ǫ) into the complex space form N 2n (4ǫ) such that g and JA are induced as first and second fundamental form. Such an immersion is unique up to isometries of the ambient space.

We focus on a particular form of A assuming that there is a pointwise G-symmetry of A (or equivalently of the cubic form C), where G is a subgroup of the special orthogonal group SO(n). We say that A has pointwise G-symmetry at p if for all tangent vectors X, Y in p, and all g ∈ G the relation A(gX, gY ) = gA(X, Y ) holds (or equivalently C(gX, gY, gZ) = C(X, Y, Z) for all X, Y, Z). Furthermore, we impose a minimality condition on A at p, so for every X at p, we assume that Tr(A X ) = 0. These manifolds are interesting, since in C n the minimal Lagrangian submanifolds are precisely the special Lagrangian submanifolds of C n as introduced by Harvey and Lawson [START_REF] Harvey | Calibrated geometries[END_REF]. If a special Lagrangian submanifold of C n has G-symmetry at every point, for the same group G, then a classification result for the dimension equal to 3 has been obtained by Bryant [START_REF] Bryant | Second order families of special Lagrangian 3-folds[END_REF]. An explicit classification for special (we also use the word "special" for "minimal" in case c = 0) Lagrangian submanifolds of complex space forms with pointwise symmetric cubic form is not yet done, but can be easily obtained from a similar classification for affine spheres in [START_REF] Vrancken | Special classes of 3-dimensional affine hyperpheres characterised by properties of their cubic form[END_REF].

In the present paper we consider the 4-dimensional case. In particular we consider special Lagrangian 4-folds in complex space forms with pointwise symmetry. The shape of the (1, 2)-tensor A, invariant under subgroups of SO(4), has been described by M. Ionel in [START_REF] Ionel | Second order families of special Lagrangian submanifolds in C 4[END_REF]. In the same article, the author classifies special Lagrangian 4-folds of C 4 according to their symmetry groups. However, the classification in case the symmetry group is given by SO(2) ⋊ S 3 in that article is incomplete; several possible subcases including the most general one is omitted. In the present article, we give a complete classification of all special Lagrangian 4-folds in any complex space form having this particular symmetry. This settles the problem for SO(2) ⋊ S 3 -symmetry for all ǫ. The classification for other symmetry groups remains open if ǫ = 0.

The SO(2) ⋊ S 3 -symmetry implies that A can be expressed as ( 4)

A(X 1 , X 1 ) = rX 1 , A(X 1 , X 2 ) = -rX 2 , A(X 1 , X 3 ) = 0, A(X 1 , X 4 ) = 0, A(X 2 , X 1 ) = -rX 2 , A(X 2 , X 2 ) = -rX 1 , A(X 2 , X 3 ) = 0, A(X 2 , X 4 ) = 0, A(X 3 , X 1 ) = 0, A(X 3 , X 2 ) = 0, A(X 3 , X 3 ) = 0, A(X 3 , X 4 ) = 0, A(X 4 , X 1 ) = 0, A(X 4 , X 2 ) = 0, A(X 4 , X 3 ) = 0, A(X 4 , X 4 ) = 0, in a well-chosen local orthonormal frame {X 1 , X 2 , X 3 , X 4 }.
In this expression r is a strictly positive function. The SO(2)-symmetry is given by the free rotation in the {X 3 , X 4 } plane and the S 3 -symmetry is essentially obtained by rotations over an angle 2π/3 in the {X 1 , X 2 } plane and reflections in the {X 1 , X 4 } plane. We can remark that the form of A is exactly that of Lagrangian submanifolds attaining equality in Chen's inequality, see [START_REF] Chen | Pseudo-Riemannian submanifolds, δ-invariants and applications[END_REF] and [START_REF] Chen | An exotic totally real minimal immersion of S 3 in CP 3 and its characterization[END_REF].

In order to list the different possible subcases, we introduce distributions

N 1 = span{X 1 , X 2 }, N + = span{X 1 , X 2 , [X 1 , X 2 ]}, N 2 = span{X 3 , X 4 }.
We will see that N 2 is always integrable. We obtain:

(1) If N 1 = N + , then the submanifold is a double warped product R × f R × g N 2 where N 2 is a minimal Lagrangian submanifold in an appropriate space form. (2) If N 1 N + and N + is integrable, then the submanifold is a single warped product R × f N 3 where N 3 is a special Lagrangian 3-fold with S 3 -symmetry in an appropriate space form. (3) If the smallest integrable distribution containing N 1 is T M, then for this final case, we do not obtain an explicit expression for the immersion, but we will rewrite the equations (7) to a system of partial differential equations in 2 coordinates out of 4 coordinates defined on the submanifold. Here, techniques will be used similar to those in [START_REF] Kriele | An extremal class of 3-dimensional hyperbolic affine spheres[END_REF]. When we consider the different cases, we will assume the defining conditions hold on an open neighborhood of the considered point.

Preliminaries

2.1. Complex space forms. We briefly recall the basic properties of C n and show how Lagrangian submanifolds of CP n and CH n can be lifted to subsets of C n+1 .

Consider the complex vector space C n . Its elements can be written as n-tuples of complex numbers, so they are given as

z = (z 1 , • • • , z n ) , z j = x j + iy j , x j , y j ∈ R.

Through the map

φ : C n → R 2n : (z 1 , • • • , z n ) → (x 1 , y 1 , • • • , x n , y n )
the space C n is a real 2n-dimensional manifold. The multiplication with the imaginary unit i translates to a linear map on R 2n given as

i (x 1 , y 1 , • • • , x n , y n ) = (-y 1 , x 1 , • • • , -y n , x n ) .
and its derivative J is given as

J∂ x k = ∂ y k , J∂ y k = -∂ x k .
This squares to -I and thus defines a complex structure on C n . On C n there is also a Hermitian form given by

s( z, w) = n j=1 z j wj = n j=1 (x j u j + y j v j ) -i n j=1 (x j v j -y j u j ).
The real part, which can be denoted as z, w defines the Euclidean scalar product on R 2n and induces a natural Riemannian metric on C n . We can see that J is an isometry and the induced Kähler form, which also coincides with the imaginary part of the Hermitian form, is closed. These structures make C n into a flat Kähler manifold.

The manifold CP n can be modeled as the quotient S 2n+1 /S 1 , where

S 2n+1 = {(z 0 , • • • , z n ) ∈ C n+1 | n i=0 |z i | 2 = 1}.
The equivalence is given by

z ∼ w ⇔ ∃φ ∈ R ∀j ∈ {0, • • • , n} : z j = e iφ w j .
So the unit sphere S 2n+1 is the preimage of the Hopf fibration

π : S 2n+1 → CP n : z → [ z] .
On S 2n+1 ⊂ C n+1 the complex structure J induces a contact structure and the standard metric on C n+1 induces a Riemannian metric. The metric on CP n that makes π a Riemannian submersion has constant holomorphic sectional curvature 4. An immersion φ : M → S 2n+1 is then said to be C-totally real or horizontal if iφ is orthogonal to the submanifold. It can be shown that every minimal C-totally real submanifold of S 2n+1 can be projected onto a special Lagrangian submanifold of CP n through π and conversely that a special Lagrangian submanifold in CP n has a 1-parameter family of mutually isometric horizontal lifts as a minimal C-totally real submanifold in S 2n+1 . So in order to classify special Lagrangian submanifolds in CP n , we can consider minimal C-totally real submanifolds in S 2n+1 ⊂ C n+1 , see [START_REF] Reckziegel | Horizontal lifts of isometric immersions into the bundle space of a pseudo-Riemannian submersion[END_REF]. For those submanifolds, the Gauss identity is given as

(5) D X Y = ∇ X Y + JA(X, Y ) -X, Y φ,
where D is the Levi Civita connection of C n+1 . Similarly, the space CH n can be modeled as H 2n+1 /S 1 , where

H 2n+1 = {(z 0 , • • • , z n ) ∈ C n+1 1 ||z 0 | 2 - n i=1 |z i | 2 = 1}.
The equivalence relationship determined by S 1 is the same as the one used in the projective space. The ambient space C n+1 1 is essentially the space C n+1 , but equipped with the scalar product

z, w 1 = ℜ n j=1 z j wj -z 0 w0 .
The complex structure is still obtained through multiplication with the imaginary unit i and induces a Kähler structure on C n+1 1 . This metric induces a Lorentzian metric on H 2n+1 and a metric of constant holomorphic sectional curvature -4 on CH n . Similar to the projective case C-totally real submanifolds φ : M → H 2n+1 can be defined having iφ as a normal. Each minimal C-totally real submanifold corresponds to the horizontal lift of a special Lagrangian submanifold of CH n . The Gauss identity is given as

(6) D X Y = ∇ X Y + JA(X, Y ) + X, Y φ,
where D is the Levi Civita connection of C n+1 1 .

2.2. Structure equations. We can return briefly to the equations ( 2) and (3). We can choose an orthogonal frame {X 1 , X 2 , X 3 , X 4 } corresponding to (4) and define the components Γ k ij and A k ij as

∇ X i X j = 4 k=1 Γ k ij X k , A(X i , X j ) = 4 k=1 A k ij X k .
Then the equations ( 2) and (3) can be rewritten as

X i Γ l jk -X j Γ l ik =ǫ δ jk δ l i -δ ik δ l j + A r jk A l ir -A r ik A l jr + Γ r ik Γ l jr -Γ r jk Γ l ir + Γ l rk Γ r ij -Γ r ji , (7)
X i A l jk -X j A l ik = Γ r ij -Γ r ji A l rk + Γ r ik A l jr -Γ r jk A l ir -Γ l ir A r jk + Γ l jr A r ik , (8)
where we have used the Einstein convention. We split the connection ∇ into its components and write

∇ X 1 X 1 = a 1 X 2 + a 2 X 3 + a 3 X 4 , ∇ X 1 X 2 = -a 1 X 1 + a 4 X 3 + a 5 X 4 , ∇ X 2 X 1 = b 1 X 2 + b 2 X 3 + b 3 X 4 , ∇ X 2 X 2 = -b 1 X 1 + b 4 X 3 + b 5 X 4 , ∇ X 3 X 1 = c 1 X 2 + c 2 X 3 + c 3 X 4 , ∇ X 3 X 2 = -c 1 X 1 + c 4 X 3 + c 5 X 4 , ∇ X 4 X 1 = d 1 X 2 + d 2 X 3 + d 3 X 4 , ∇ X 4 X 2 = -d 1 X 1 + d 4 X 3 + d 5 X 4 , ∇ X 1 X 3 = -a 2 X 1 -a 4 X 2 + a 6 X 4 , ∇ X 1 X 4 = -a 3 X 1 -a 5 X 2 -a 6 X 3 , ∇ X 2 X 3 = -b 2 X 1 -b 4 X 2 + b 6 X 4 , ∇ X 2 X 4 = -b 3 X 1 -b 5 X 2 -b 6 X 3 , ∇ X 3 X 3 = -c 2 X 1 -c 4 X 2 + c 6 X 4 , ∇ X 3 X 4 = -c 3 X 1 -c 5 X 2 -c 6 X 3 , ∇ X 4 X 3 = -d 2 X 1 -d 4 X 2 + d 6 X 4 , ∇ X 4 X 4 = -d 3 X 1 -d 5 X 2 -d 6 X 3 .
Equation [START_REF] Kriele | An extremal class of 3-dimensional hyperbolic affine spheres[END_REF] induces linear relations between the components, independent of the ambient space. The Gauss equations give further information about ∇ but use differential equations and depend on the ambient space form.

Lemma 2.1. On a special Lagrangian submanifold M having a local SO(2)⋊S 3 -symmetry there exists a frame corresponding to (4) such that:

(9) ∇ X 1 X 1 = a 1 X 2 + a 2 X 3 + a 3 X 4 , ∇ X 1 X 2 = -a 1 X 1 -b 2 X 3 , ∇ X 2 X 1 = b 1 X 2 + b 2 X 3 , ∇ X 2 X 2 = -b 1 X 1 + a 2 X 3 + a 3 X 4 , ∇ X 3 X 1 = b 2 3 X 2 , ∇ X 3 X 2 = -b 2 3 X 1 , ∇ X 4 X 1 = 0, ∇ X 4 X 2 = 0, ∇ X 1 X 3 = -a 2 X 1 + b 2 X 2 + a 6 X 4 , ∇ X 1 X 4 = -a 3 X 1 -a 6 X 3 , ∇ X 2 X 3 = -b 2 X 1 -a 2 X 2 + b 6 X 4 , ∇ X 2 X 4 = -a 3 X 2 -b 6 X 3 , ∇ X 3 X 3 = c 6 X 4 , ∇ X 3 X 4 = -c 6 X 3 , ∇ X 4 X 3 = d 6 X 4 , ∇ X 4 X 4 = -d 6 X 3 .
Furthermore, the derivatives of r are given by

(X 1 + iX 2 )(r) = 3ir(a 1 + ib 1 ), (10)
X 3 (r) = ra 2 , (11)
X 4 (r) = ra 3 . (12)
Proof. This is just a straightforward application of equation [START_REF] Kriele | An extremal class of 3-dimensional hyperbolic affine spheres[END_REF]. For instance

(∇ X 2 A) (X 1 , X 1 ) = X 2 (r)X 1 + 3rb 1 X 2 + rb 2 X 3 + rb 3 X 4 , (∇ X 1 A) (X 2 , X 1 ) = 3ra 1 X 1 -X 1 (r)X 2 -ra 4 X 3 -ra 5 X 4 .
Then the corresponding coordinates of both derivatives are the same. Finally we can set b 3 = 0, by rotating the distribution N 2 such that X 3 lies in the direction of

∇ X 1 X 2 , projected on N 2 .
It is interesting to note that N 2 is an integrable distribution. The distribution N 1 however is integrable if and only if b 2 = 0. Applying [START_REF] Ionel | Second order families of special Lagrangian submanifolds in C 4[END_REF], we obtain the following result.

Lemma 2.2. The equations (7) on our frame of choice induce a system of differential equations given by:

(X 1 + iX 2 )(a 2 -ib 2 ) = a 3 (a 6 + ib 6 ), ( 13
)
X 3 (a 2 + ib 2 ) = ǫ + a 2 3 + (a 2 + ib 2 ) 2 , (14) X 4 (a 2 + ib 2 ) = a 3 (a 2 + ib 2 ), ( 15
) (X 1 + iX 2 )(a 3 ) = -(a 2 -ib 2 )(a 6 + ib 6 ), (16) X 3 (a 3 ) = 0, (17) X 4 (a 3 ) = a 2 3 + ǫ, (18) X 1 (b 6 ) -X 2 (a 6 ) = -(a 1 a 6 + b 1 b 6 ), (19)
X 3 (a 6 + ib 6 ) = 5 3 ib 2 (a 6 + ib 6 ), (20)
X 4 (a 6 + ib 6 ) = 2a 3 (a 6 + ib 6 ), (21) X 1 (b 1 ) -X 2 (a 1 ) = 2r 2 -(ǫ + a 2 3 ) - 5 3 b 2 2 -a 2 2 -a 2 1 -b 2 1 , (22) X 2 (b 1 ) + X 1 (a 1 ) = - 2 3 a 2 b 2 , (23) 3X 3 (a 1 ) -X 1 (b 2 ) = 3a 1 a 2 -2b 1 b 2 , (24) 3X 3 (b 1 ) -X 2 (b 2 ) = 2b 2 a 1 + 3b 1 a 2 , (25) X 4 (a 1 + ib 1 ) = a 3 (a 1 + ib 1 ) + b 2 3 (a 6 + ib 6 ). (26)
Proof. This is also a straightforward application of [START_REF] Ionel | Second order families of special Lagrangian submanifolds in C 4[END_REF]. For example:

X 1 Γ 1 23 -X 2 Γ 1 13 = Γ r 13 Γ 1 2r -Γ r 23 Γ 1 1r + Γ 1 r3 Γ r 12 -Γ 1 r3 Γ r 21 = a 3 b 6 , X 1 Γ 2 23 -X 2 Γ 2 13 = Γ r 13 Γ 2 2r -Γ r 23 Γ 2 1r + Γ 2 r3 Γ r 12 -Γ 2 r3 Γ r 21 = -a 3 a 6 .
Combining both equations using the usual complex notations leads to [START_REF] Vrancken | Special classes of 3-dimensional affine hyperpheres characterised by properties of their cubic form[END_REF]. The other equations are obtained in a similar way.

2.3. Warped Products. In the analysis that follows, we will often encounter warped products of manifolds. When we consider a warped product of Riemannian manifolds (M 1 , g 1 ) and (M 2 , g 2 ) with warping function e f , where f :

M 1 → R, we get a Riemannian manifold (M 1 × M 2 , g f ) where M 1 × M 2
as a differentiable manifolds is the product of M 1 and M 2 and the metric g f is given as

g f (X, Y ) = g 1 (X 1 , Y 1 ) + e 2f g 2 (X 2 , Y 2 ),
where a vector field X is uniquely decomposed into a part X 1 tangent to M 1 and X 2 tangent to M 2 . We denote this warped product as M 1 × e f M 2 . The following result can be obtained, see [START_REF] Nölker | Isometric immersions of warped Products[END_REF].

Theorem 2.1. Consider a Riemannian manifold (M, g) with Levi-Civita connection ∇ and suppose that on a neighborhood of p ∈ M there are orthogonal distributions N 1 and

N 2 such that ∀X, Y ∈ N 1 (i.e. X and Y are sections of N 1 ) : ∇ X Y ∈ N 1 , ∀X, Y ∈ N 2 : ∇ X Y = ∇X Y + g(X, Y )H,
where ∇ is the projection of ∇ on N 2 and H ∈ N 1 . Then there exists a function f : M → R such that on a neighborhood of p, M can be written as

M 1 × e f M 2 , where M i is an integral manifold of N i .
If furthermore H = λH 0 , where H 0 = 1, and

X(λ) = 0 for every X ∈ N 2 , then f : M 1 → R and H = -grad e f .
The first part of the theorem constructs a twisted product, the second part reduces this to a warped product. This will be useful in choosing coordinates, since the product structures allows for coordinates to be chosen on each factor separately. In particular, if dim (N 1 ) = 1, then any non vanishing vector field in N 1 can be fixed as a useful coordinate vector field on M.

3.

Submanifolds in C 4 .

3.1.

The case where b 2 = 0. The assumption that X 3 lies along ∇ X 1 X 2 becomes redundant since the latter has no N 2 component. Instead, we can choose X 3 in the direction of ∇ X 1 X 1 , projected on N 2 . Hence without loss of generality we can assume that a 3 = 0. The equations [START_REF] Ionel | Second order families of special Lagrangian submanifolds in C 4[END_REF] show that in this case either a 2 = 0 or a 6 = b 6 = c 6 = d 6 = 0. First we will assume that a 2 = 0. Theorem 3.1. Consider M a special Lagrangian submanifold in C 4 having SO(2) ⋊ S 3symmetry and an orthogonal frame corresponding to (4). Suppose that N 1 is an integrable distribution and ∇ X 1 X 1 is nowhere contained within this distribution. Then M is locally congruent to

(27) F (t, s, u, v) = (t, sφ(u, v))
where φ is the horizontal lift of a special Lagrangian submanifold of CP 2 to the unit sphere in C 3 .

Proof. Taking into account every component that vanishes in [START_REF] Nölker | Isometric immersions of warped Products[END_REF], we find

∇ X 1 X 1 = a 1 X 2 + a 2 X 3 ∇ X 1 X 2 = -a 1 X 1 , ∇ X 2 X 1 = b 1 X 2 ∇ X 2 X 2 = -b 1 X 1 + a 2 X 3 , ∇ X 3 X 1 = 0 ∇ X 3 X 2 = 0, ∇ X 4 X 1 = 0 ∇ X 4 X 2 = 0, ∇ X 1 X 3 = -a 2 X 1 ∇ X 1 X 4 = 0, ∇ X 2 X 3 = -a 2 X 2 ∇ X 2 X 4 = 0, ∇ X 3 X 3 = 0 ∇ X 3 X 4 = 0, ∇ X 4 X 3 = 0 ∇ X 4 X 4 = 0.
We find that the distributions span{X 4 } and span{X 1 , X 2 , X 3 } satisfy the conditions for a warped product R × e f N 3 . But X 4 (f ) = 0, hence f is a constant. M is a standard Riemannian product R × N 3 and its immersion can be written, up to an isometry as

F (t, x) = (t, ψ(x)) , ψ : N 3 → C 4 .
The immersion ψ is contained in the subspace orthogonal to both X 4 and JX 4 , since they both are constant unit normals along N 3 . Now it is also obvious that span{X 3 } and N 1 satisfy the conditions for a warped product. So N 3 can be decomposed as R × e g N 2 and X 3 (g) = -a 2 . Then X 3 can be associated with a coordinate s on the manifold and it follows that

D X 3 X 3 = ∂ 2 F ∂s 2 = 0 ⇒ F = As + B.
Both A and B are independent of (s, t). Calculating ( 7), one has

X 3 (a 2 ) = ∂a 2 ∂s = a 2 2 .
The solution of this equation, after a translation of the s-coordinate is given as a 2 = -1 s . The derivatives of X 3 to X 1 and X 2 are

D X i X 3 = ∂F * X i ∂s = A * X i = 1 s X i = A * X i + B * X i s ⇒ B * = 0.
So B is a constant vector along the submanifold and vanishes when applying a translation. It is easy to see that X 3 = A and is orthogonal to X i = sA * (X i ), for i ∈ {1, 2}. Hence everywhere along A, the position vector is orthogonal to the tangent space. Thus A has constant length. Calculating the other covariant derivatives yields for example for i, j ∈ {1, 2} that

A * X i = F * X i s , D X i (A * X j ) = D X i (F * X j ) s = A * ∇X i X j + JA * (K(X i , X j )) - 1 s 2 δ ij φ. ( 28
)
Here ∇ is the connection restricted to N 2 . Combining this with the other equations in [START_REF] Ionel | Second order families of special Lagrangian submanifolds in C 4[END_REF], it follows that A is a C-totally real immersion in S 5 ⊂ C 3 . Furthermore, the components a 1 and b 1 have no other restrictions on them except satisfying the Gauss equations for a minimal C-totally real submanifold of S 5 . This proves the theorem.

The case a 2 = 0 was the only case that was studied in [START_REF] Ionel | Second order families of special Lagrangian submanifolds in C 4[END_REF]. We can quote the following result from [START_REF] Ionel | Second order families of special Lagrangian submanifolds in C 4[END_REF]. Theorem 3.2. Consider M a special Lagrangian submanifold in C 4 having a local SO(2)⋊ S 3 -symmetry group and an orthogonal frame corresponding to (4). Suppose that N 1 is an integrable distribution and ∇ X 1 X 1 is contained within this distribution. Then M is locally congruent to

(29) F (t, s, u, v) = (t, s, φ(u, v))
where φ : C → C 2 is a special Lagrangian surface.

Remark 3.1. As proved in [START_REF] Ionel | Second order families of special Lagrangian submanifolds in C 4[END_REF], a special Lagrangian surface in C 2 , with complex coordinates x 1 + iy 1 and x 2 + iy 2 is a holomorphic curve in C 2 with complex coordinates x 1ix 2 and y 1 + iy 2 , and conversely.

3.2.

The case where b 2 = 0. Now the distribution N 1 is no longer integrable. The simplest case one can hope for is that there is a 3-dimensional integrable distribution containing N 1 . Such a distribution should contain at least

X 3 since [X 1 , X 2 ] mod N 1 X 3 .
Using the fact that b 2 = 0, the equations ( 7) reduce ( 9) to (30)

∇ X 1 X 1 = a 1 X 2 + a 2 X 3 + a 3 X 4 ∇ X 1 X 2 = -a 1 X 1 -b 2 X 3 , ∇ X 2 X 1 = b 1 X 2 + b 2 X 3 ∇ X 2 X 2 = -b 1 X 1 + a 2 X 3 + a 3 X 4 , ∇ X 3 X 1 = b 2 3 X 2 ∇ X 3 X 2 = -b 2 3 X 1 , ∇ X 4 X 1 = 0 ∇ X 4 X 2 = 0, ∇ X 1 X 3 = -a 2 X 1 + b 2 X 2 + a 6 X 4 ∇ X 1 X 4 = -a 3 X 1 -a 6 X 3 , ∇ X 2 X 3 = -b 2 X 1 -a 2 X 2 + b 6 X 4 ∇ X 2 X 4 = -a 3 X 2 -b 6 X 3 , ∇ X 3 X 3 = a 3 X 4 ∇ X 3 X 4 = -a 3 X 3 , ∇ X 4 X 3 = 0 ∇ X 4 X 4 = 0.
It is apparent that the condition that N + is integrable is given by a 6 + ib 6 = 0. We consider this case first.

Theorem 3.3. Suppose M is a special Lagrangian submanifold in C 4 with SO(2) ⋊ S 3symmetry, such that N 1 is not an integrable distribution, but N + is. Then the submanifold, up to isometry, can be given locally by either

(31) F (t, s, u, v) = (t, φ(s, u, v)) ,
where φ is a special Lagrangian submanifold with S 3 -symmetry in C 3 , or

(32) F (t, s, u, v) = tφ(s, u, v)
where φ is the horizontal lift of a special Lagrangian submanifold with local S 3 -symmetry in CP 3 to the unit sphere in C 4 .

Proof. We find according to (30) and ( 7) that span{X 4 } and N + satisfy the conditions for a warped product. So M can be decomposed as R × e f N 3 , where X 4 (f ) = -a 3 . We can solve

X 4 (a 3 ) = ∂a 3 ∂t = a 2 3 .
This equation has 2 possible solutions. First, we assume a 3 = 0. In this case M is simply the manifold R × N 3 . Hence the immersion, up to isometry, can be given as

F (t, s, u, v) = (t, φ(s, u, v)),
where φ is a 3-fold immersed in the subspace C 3 orthogonal to X 4 and JX 4 . Similar calculations as in (28) show that this can be any special Lagrangian submanifold in C 3 , given the presence of an S 3 -symmetry in the second fundamental form.

The second solution, after a translation of t, is given by a 3 = -1 t . The calculations are similar to the case where b 2 = 0 and a 2 = 0. This gives the required result.

The last case in C 4 is the one where there is no integrable distribution containing N 1 other than the whole tangent bundle. In this case, we can no longer rely on an obvious warped product structure. We can attempt to introduce a set of independent coordinates and reduce (7) to a system of PDE's on C 4 using as little functions as possible. We now use [START_REF] Vrancken | Special classes of 3-dimensional affine hyperpheres characterised by properties of their cubic form[END_REF] to (26) to construct a coordinate frame from

{X 1 , X 2 , X 3 , X 4 }. Since N 2 is integrable, it is a good idea to choose X 4 = T and µX 3 = S. Requiring that [S, T ] = 0 implies that [µX 3 , X 4 ] = µ [X 3 , X 4 ] -X 4 (µ)X 3 = -(µa 3 + X 4 (µ)) X 3 = 0.
We can find such a µ by taking µ

= 1 √ |ǫ+a 2 3 |
. The equation a 2 3 + ǫ = 0 implies that a 3 is a constant and hence (a 2ib 2 )(a 6 + ib 6 ) = 0. This will correspond to the integrability of either N 1 or N + . Therefore µ is well defined.

Vector fields U and V can be sought such that every couple out of {S, T, U, V } commutes. Such an attempt can be made, writing

(33) U + iV = (ρ 1 -iρ 2 ) (X 1 + iX 2 ) + (α 1 + iβ 1 )S + (α 2 + iβ 2 )T
We rename the following expressions:

ρ = ρ 1 -iρ 2 , γ j = α j + iβ j j ∈ {1, 2}.
After calculating the Lie brackets of these four vector fields, the following conditions on the introduced functions make the vector fields commute:

X 4 (ρ) = -a 3 ρ, (34) X 3 (ρ) = -a 2 + 2 3 ib 2 ρ, (35) (X 1 -iX 2 )(ρ) = (b 1 + ia 1 ) ρ, (36)
X 4 (γ 1 ) = - 1 µ (a 6 + ib 6 ) + a 3 γ 1 , (37)
X 3 (γ 1 ) = 1 µ 2 (X 1 + iX 2 )(µ) + γ 1 a 2 + 2 3 ib 2 , (38) X 2 (α 1 ) -X 1 (β 1 ) = a 1 α 1 + b 1 β 1 - 2 µ b 2 , (39) X 4 (γ 2 ) = a 3 γ 2 , ( 40
)
X 3 (γ 2 ) = (a 6 + ib 6 ) + a 2 + 2 3 ib 2 γ 2 , (41)
X 2 (α 2 ) -X 1 (β 2 ) = a 1 α 2 + b 1 β 2 . ( 42
)
The following result can be obtained. Lemma 3.1. Suppose f and g are real valued functions on the manifold satisfying

S(f ) = 0, T (f ) = -1, S(g) = -1, T (g) = 0,
and defining

X 1 (f ) = α 2 , X 2 (f ) = β 2 , X 1 (g) = α 1 , X 2 (g) = β 1 ,
then the functions α i and β i obtained this way satisfy the conditions (37) to (42).

It is interesting to see that this way the vector fields

Ũ = X 1 + α 1 S + α 2 T, Ṽ = X 2 + β 1 S + β 2 T, satisfy Ũ (f ) = Ũ(g) = Ṽ (f ) = Ṽ (g) = 0
. Furthermore Ũ and Ṽ are independent of oneanother and they span the distribution which is the intersection of the kernel of d f and d g. Note that this distribution is indeed 2-dimensional since both forms have a hyperplane as a kernel and these kernels do not coincide, since the 1-forms are linearly independent. Using the dimension theorem, they have a 2-dimensional intersection. Construction (33) is just a complex rotation of these two vector fields in that distribution. This way, it is clear that f and g serve as coordinates s and t conjugate to S and T .

Proof. Apply the relation

[X i , X j ] (f ) = X i X j (f ) -X j X i (f ) = ∇ X i X j (f ) -∇ X j X i (f )
on both functions, using (30).

A suitable function for f is easily found, since S(a 3 ) = 0. Let f be a function of a 3 , then

X 4 (f ) = f ′ (ǫ + a 2 3 ) = -1 ⇔ f ′ = - 1 ǫ + a 2 3 .
Hence f can be given by

f = - 1 ǫ + a 2 3 d a 3 .
This also determines γ 2 completely, since using (13) yields

γ 2 = (X 1 + iX 2 )(f ) = f ′ (X 1 + iX 2 )(a 3 ) = a 2 a 6 + b 2 b 6 ǫ + a 2 3 + i a 2 b 6 -b 2 a 6 ǫ + a 2 3 .
As for the function g, the complex valued function z = µ(a 2 + ib 2 ) can be considered and calculations show

X 4 (z) = -µa 3 (a 2 + ib 2 ) + µa 3 (a 2 + ib 2 ) = 0, S(z) = µ 2 ǫ + a 2 3 + (a 2 + ib 2 ) 2 = sg ǫ + a 2 3 + z 2 .
Rewriting ǫ = sg(ǫ + a 2 3 ), we find that z is useful as long as z 2 + ǫ = 0. When ǫ = +1, this occurs when a 2 = 0 and

|b 2 | = ǫ + a 2 3 . When ǫ = -1, this occurs when |a 2 | = |ǫ + a 2 3 | and b 2 = 0, resulting in N 1 being integrable.
First we assume that z 2 = -ǫ. Then the function g can be calculated as the real part of a function G of z given by

S(G) = (ǫ + z 2 )G ′ = -1 ⇔ G ′ = - 1 ǫ + z 2 .
A function ρ still has to be constructed satisfying (34) to (36). Define a function H as

H = ρ 3 r(z 2 + ǫ) |ǫ + a 2 3 |.
This function is a constant on the submanifold and can be taken to be equal to 1. This defines a function ρ satisfying the necessary conditions.

Using the Frobenius theorem in [START_REF] Nomizu | Foundations of Differential Geometry[END_REF], a coordinate frame on the submanifold is given by

X 4 = T, X 3 = 1 µ S, X 1 + iX 2 = U + iV ρ -γ 1 S -γ 2 T.
We can describe the dependence of a 6 + ib 6 on (s, t) by writing

a 6 + ib 6 = k 3 + ik 4 ρ |a 2 3 + ǫ| z2 + ǫ -1 2 .
The functions k 3 and k 4 depend solely on (u, v). This expression is obtained from (20) and ( 21). The rest of the equations in ( 7) can be rewritten and solved. Applying our method for ǫ = 0, we find after a translation of the coordinates that

a 3 = - 1 t , x = sin(2s) cos(2s) + cosh(2k 1 )
⇒ a 2 = -sin(2s) t(cos(2s) + cosh(2k 1 )) , y = sinh(2k 1 ) cos(2s) + cosh(2k 1 )

⇒ b 2 = -sinh(2k 1 ) t(cos(2s) + cosh(2k 1 )) , r = e k 2 t cos(2s) + cosh(2k 1 ) .

Here the functions k 1 and k 2 depend solely on (u, v). Then we can use [START_REF] Vrancken | Special classes of 3-dimensional affine hyperpheres characterised by properties of their cubic form[END_REF] to find an expression for γ 1 in terms of the coordinates. Equation ( 10) can be used to find an expression for a 1 and b 1 in terms of the coordinates. We obtain

γ 1 = (k 3 + ik 4 ) cos(s -ik 1 ) + t ∂k 1 ∂v -i ∂k 1 ∂u tρ , a 1 = 2 2 3 e 2 3 k 2 3t 3 (cos(2s) + cosh(2k 1 )) 2 t (cos(2s) + cosh(2k 1 ))(ρ 1 ∂k 2 ∂v + ρ 2 ∂k 2 ∂u ) + sin(2s)(ρ 1 ∂k 1 ∂u -ρ 2 ∂k 1 ∂v ) -sinh(2k 1 )(ρ 1 ∂k 1 ∂v + ρ 2 ∂k 1 ∂u ) + sinh(2k 1 ) (cos(s) cosh(k 1 )(k 4 ρ 2 -k 3 ρ 1 ) + sin(s) sinh(k 1 )(k 4 ρ 1 + k 3 ρ 2 )) , b 1 = 2 2 3 e 2 3 k 2 3t 3 (cos(2s) + cosh(2k 1 )) 2 t (cos(2s) + cosh(2k 1 ))(ρ 2 ∂k 2 ∂v -ρ 1 ∂k 2 ∂u ) + sin(2s)(ρ 2 ∂k 1 ∂u + ρ 1 ∂k 1 ∂v ) + sinh(2k 1 )(ρ 1 ∂k 1 ∂u -ρ 2 ∂k 1 ∂v ) + sinh(2k 1 ) (sin(s) sinh(k 1 )(k 4 ρ 2 -k 3 ρ 1 ) -cos(s) cosh(k 1 )(k 4 ρ 1 + k 3 ρ 2 )) .
Now every function on the submanifold is expressed in terms of (s, t, u, v), possibly indirectly through {k 1 , k 2 , k 3 , k 4 }. Demanding that the other Gauss equations are satisfied gives partial differential equations for k i , given by

∂k 4 ∂u - ∂k 3 ∂v = 2 tanh(k 1 ) k 3 ∂k 1 ∂v -k 4 ∂k 1 ∂u , ∂k 4 ∂v + ∂k 3 ∂u = -2 coth(k 1 ) k 3 ∂k 1 ∂u + k 4 ∂k 1 ∂v , ∆k 2 = 3 * 2 1 3 e -2k 2
3 -e 2k 2 + cosh(2k 1 ) ,

∆k 1 = -2 1 3 e -2k 2 3 sinh(2k 1 ). ( 43
)
Now we return to the case where -1 = z 2 and ǫ = 1. We assume first that ǫ isn't specified. In this case a 2 = 0, b 2 = ± ǫ + a 2 3 and S(z) = 0, so z is insufficient to construct the function g. Equations ( 7) are reduced to

(X 1 -iX 2 )(a 6 + ib 6 ) = -i(b 1 + ia 1 )(a 6 + ib 6 ), (44) X 3 (a 6 + ib 6 ) = ±i 5 3 ǫ + a 2 3 (a 6 + ib 6 ), (45) X 4 (a 6 + ib 6 ) = 2a 3 (a 6 + ib 6 ), (46) X 1 (b 1 ) -X 2 (a 1 ) = 2r 2 - 8 3 (ǫ + a 2 3 ) -a 2 1 -b 2 1 , (47) X 1 (a 1 ) + X 2 (b 1 ) = 0, (48) X 4 (a 1 + ib 1 ) = a 3 (a 1 + ib 1 ) ± ǫ + a 2 3 (a 6 + ib 6 ) 3 , (49)
X 3 (a 1 + ib 1 ) = i 3 a 3 (a 6 + ib 6 ) ± 2 ǫ + a 2 3 (a 1 + ib 1 ) . (50)
The first equation is obtained from applying integrability on a 3 . Now we define

w = a 6 + ib 6 ǫ + a 2 3
, which after derivation gives

X 4 (w) = -2a 3 (a 6 + ib 6 ) ǫ + a 2 3 + 2a 3 a 6 + ib 6 ǫ + a 2 3 = 0, S(w) = ±i 5 3 w.
The resulting differential equation for a function G of w will be

S(G) = ±G ′ i 5 3 w = -1 ⇔ G ′ = ±i 3 5w .
The solution is that G is a logarithm of w. We find that H defined by H = w 2 (ǫ + a 2 3 ) 2 ρ 5 r is a constant and hence can be used to express ρ. We can thus solve w as w = e k 1 ±i 5 3 s = e k 1 cos( 5 3 s) ± i sin( 5 3 s) .

Applying ( 18),( 46),( 45),( 12) and ( 11) when ǫ = 0 yields

a 3 = - 1 t , a 6 = e k 1 cos( 5 3 s) t 2 , b 6 = ±e k 1 sin( 5 3 s) t 2 , r = e k 2 t .
The equation ( 10) now gives a 1 + ib 1 immediately without going through γ 1 because of [START_REF] Nomizu | Foundations of Differential Geometry[END_REF]. The final unknown, γ 1 can then be determined using (44). When we pick b 2 = a 3 , we obtain

γ 1+ = -5e k 1 + 5si 3 + e 2k 1 +k 2 5 + 2si 3 t ( ∂k 2 ∂v -3 ∂k 1 ∂v ) -i( ∂k 2 ∂u -3 ∂k 1 ∂u ) 5t 2 , a 1+ = e k 1 cos( 5s 3 ) + e 2k 1 +k 2 5 t cos( 2s 3 ) ∂k 2 ∂v + sin( 2s 3 ) ∂k 2 ∂u 3t 2 , b 1+ = e k 1 sin( 5s 3 ) -e 2k 1 +k 2 5 t cos( 2s 3 ) ∂k 2 ∂u -sin( 2s 3 ) ∂k 2 ∂v 3t 2 ,
and for b 2 = -a 3 we obtain

γ 1-= -5e k 1 -5si 3 + e 2k 1 +k 2 5 -2si 3 t (3 ∂k 1 ∂v -∂k 2 ∂v ) -i(3 ∂k 1 ∂u -∂k 2 ∂u ) 5t 2 , a 1-= -e k 1 cos( 5s 3 ) + e 2k 1 +k 2 5 t cos( 2s 3 ) ∂k 2 ∂v -sin( 2s 3 ) ∂k 2 ∂u 3t 2 , b 1-= e k 1 sin( 5s 3 ) -e 2k 1 +k 2 5 t cos( 2s 3 ) ∂k 2 ∂u + sin( 2s 3 ) ∂k 2 ∂v 3t 2 .
Equations ( 47) and (39) result in restrictions on the functions k 1 and k 2 of (u, v) given by

∆k 2 = e -2 5 (2k 1 +k 2 ) (8 -6e 2k 2 ), ∆k 1 = e -2 5 (2k 1 +k 2 ) (6 -2e 2k 2 ). (51)
These equations are valid for both b 2 = ±a 3 . Using the constructed functions, the rest of the Gauss equations don't impose further conditions. We can summarize this result in the following theorem.

Theorem 3.4. Each special Lagrangian submanifold of C 4 with SO(2) ⋊ S 3 -symmetry where the only integral distribution containing N 1 is the tangent bundle, can be constructed in the way above using either functions {k 1 , k 2 , k 3 , k 4 } subject to (43) or functions {k 1 , k 2 } subject to (51). Conversely, each such a construction results in such a submanifold, unique up to local isometry.

In the upcoming sections we will consider the construction for ǫ = ±1.

4. Submanifolds in CP 4 .

4.1.

The case where b 2 = 0. This means that both N 1 and N 2 are integrable distributions. We can assume a 3 = 0. However, the Gauss equation

(52) X 3 (a 2 ) = 1 + a 2 2
no longer allows for a 2 being a constant. The following result is obtained:

Theorem 4.1. Suppose M is a special Lagrangian submanifold in CP 4 having SO(2)⋊S 3symmetry. Suppose N 1 is integrable. Then M can be lifted horizontally to a submanifold in the unit sphere of C 5 through F and this lift is congruent to

(53) F (t, s, u, v) = (φ(u, v) cos(s), sin(s) cos(t), sin(s) sin(t)) ,
where φ is the horizontal lift of a special Lagrangian submanifold in CP 2 to the unit sphere in C 3 .

Proof. Equations ( 7) reduce ∇ to

∇ X 1 X 1 = a 1 X 2 + a 2 X 3 ∇ X 1 X 2 = -a 1 X 1 , ∇ X 2 X 1 = b 1 X 2 ∇ X 2 X 2 = -b 1 X 1 + a 2 X 3 , ∇ X 3 X 1 = 0 ∇ X 3 X 2 = 0, ∇ X 4 X 1 = 0 ∇ X 4 X 2 = 0, ∇ X 1 X 3 = -a 2 X 1 ∇ X 1 X 4 = 0, ∇ X 2 X 3 = -a 2 X 2 ∇ X 2 X 4 = 0, ∇ X 3 X 3 = 0 ∇ X 3 X 4 = 0, ∇ X 4 X 3 = X 4 a 2 ∇ X 4 X 4 = -X 3
a 2 . The distributions N 1 and N 2 satisfy the conditions for a warped product N 2 × e f N 1 . Furthermore, the distributions span{X 3 } and span{X 4 } satisfy those of a warped product and we can write M = R × e g R × e f N 1 . The functions f and g depend solely on the parameter corresponding to X 3 and are given by X 3 (f ) = -a 2 and X 3 (g) = 1 a 2 . We can assume X 3 = ∂ ∂s on the submanifold. We can also find a function µ(s) such that µX 4 = ∂ ∂t . To find a suitable µ, we solve

[X 3 , µX 4 ] = X 3 (µ) - µ a 2 X 4 = µ ′ (1 + a 2 2 ) - µ a 2 X 4 = 0. The function µ = a 2 √ 1+a 2 2
satisfies this equation. We can find a 2 (s) by solving

∂a 2 ∂s = 1 + a 2 2 ⇒ a 2 = tan(s).
Hence µ(s) = sin(s) and we calculate for i ∈ {1, 2} that

D ∂ ∂s ∂ ∂s = ∂ 2 F ∂s 2 = -F ⇒ F = A cos(s) + B sin(s), D ∂ ∂t ∂ ∂s = ∂ 2 F ∂t∂s = - ∂A ∂t sin(s) + ∂B ∂t cos(s) = cot(s) ∂F ∂t = cos(s) 2 sin(s) ∂A ∂t + cos(s) ∂B ∂t ⇒ ∂A ∂t = 0, D X i ∂ ∂s = ∂F * X i ∂s = -A * X i sin(s) + B * X i cos(s) = -tan(s)X i = -A * X i sin(s) - sin(s) 2 cos(s) B * X i ⇒ B * X i = 0.
So A is the immersion of N 1 and B is a curve tangent to X 4 . Because F lies in the unit sphere, one has

F, F = cos(s) 2 A, A + sin(s) 2 B, B + sin(2s) A, B
which implies that A and B have both unit length and are orthogonal. We can also calculate

D ∂ ∂t ∂ ∂t = -cos(s) sin(s) ∂F ∂s -sin(s) 2 F = -sin(s)B = ∂ 2 F ∂t 2 = sin(s) ∂ 2 B ∂t 2 ⇒ B = B 1 cos(t) + B 2 sin(t).
Vector fields B 1 and B 2 are constant, normalized and orthogonal. This follows from the fact that B, B = 1. Finally similar to (28), A can be shown to be any special Lagrangian submanifold in CP 2 lifted to the unit sphere in C 3 orthogonal to B 1 and B 2 directions. Fixing B 1 and B 2 by an isometry leads to (53).

4.2.

The case where b 2 = 0. When N + is integrable, so when a 6 = b 6 = 0, the equations for ∇ are given by (30). We have: Theorem 4.2. Suppose M is a special Lagrangian submanifold in CP 4 having a local SO(2) ⋊ S 3 symmetry. Suppose N + is integrable. Then M can be lifted horizontally to a submanifold in the unit sphere of C 5 through F and is locally isometric to

(54) F (t, s, u, v) = (φ(s, u, v) cos(t), sin(t)) ,
where φ is the horizontal lift of a special Lagrangian submanifold in CP 3 with S 3 -symmetry to the unit sphere in C 4 .

Proof. The manifold is a warped product R × e f N 3 . Solving the Gauss equation

X 4 (a 3 ) = ∂a 3 ∂t = 1 + a 2 3
yields a 3 = tan(t). For i ∈ {1, 2, 3} this implies

D X 4 X 4 = ∂ 2 F ∂t 2 = -F ⇒ F = A cos(t) + B sin(t), D X i X 4 = -A * X i sin(t) + B * X i cos(t) = -tan(t)X i = -A * X i sin(t) -B * X i sin(t) 2 cos(t) ⇒ B * = 0.
Thus B is a constant vector field along the submanifold and A is an immersion of a 3-fold N 3 . Using the fact that F is of unit length, A and B are orthogonal and of unit length. Using calculations similar to (28) A is a C-totally real submanifold in S 7 having local S 3 -symmetry, where S 7 lies in the subspace orthogonal to B and JB. Applying a suitable isometry results in (54).

The method to solve the case where the only integrable distribution containing N 1 is the tangent bundle, has been analyzed earlier for a non-specific complex space form. We can now fill in ǫ = 1 and we find for z 2 = -1 that

a 3 = tan(t), a 2 = sin(2s) cos(t)(cos(2s) + cosh(2k 1 )) , b 2 = sinh(2k 1 ) cos(t)(cos(2s) + cosh(2k 1 )) , r = e k 2 cos(t) cos(2s) + cosh(2k 1 ) , a 6 + ib 6 = k 3 + ik 4 ρ 1 + a 2 3 1 + z2 -1 2 ,
where the functions k i depend only on (u, v). Solving for [START_REF] Ionel | Second order families of special Lagrangian submanifolds in C 4[END_REF], we obtain furthermore that

γ 1 = -tan(t)(k 3 + ik 4 ) cos(s -ik 1 ) + ∂k 1 ∂v -i ∂k 1 ∂u ρ , a 1 = 2 2 3 e 2 3 k 2 3 cos(t) 2 (cos(2s) + cosh(2k 1 )) 2 (cos(2s) + cosh(2k 1 ))(ρ 1 ∂k 2 ∂v + ρ 2 ∂k 2 ∂u ) + sin(2s)(ρ 1 ∂k 1 ∂u -ρ 2 ∂k 1 ∂v ) -sinh(2k 1 )(ρ 1 ∂k 1 ∂v + ρ 2 ∂k 1 ∂u ) -tan(t) sinh(2k 1 ) (cos(s) cosh(k 1 )(k 4 ρ 2 -k 3 ρ 1 ) + sin(s) sinh(k 1 )(k 4 ρ 1 + k 3 ρ 2 )) , b 1 = 2 2 3 e 2 3 k 2 3 cos(t) 2 (cos(2s) + cosh(2k 1 )) 2 (cos(2s) + cosh(2k 1 ))(ρ 2 ∂k 2 ∂v -ρ 1 ∂k 2 ∂u ) + sin(2s)(ρ 2 ∂k 1 ∂u + ρ 1 ∂k 1 ∂v ) + sinh(2k 1 )(ρ 1 ∂k 1 ∂u -ρ 2 ∂k 1 ∂v ) -sinh(2k 1 ) tan(t) (sin(s) sinh(k 1 )(k 4 ρ 2 -k 3 ρ 1 ) -cos(s) cosh(k 1 )(k 4 ρ 1 + k 3 ρ 2 )) .
The other equations in [START_REF] Ionel | Second order families of special Lagrangian submanifolds in C 4[END_REF] impose restrictions on {k 1 , k 2 , k 3 , k 4 } given by

∂k 4 ∂u - ∂k 3 ∂v = 2 tanh(k 1 ) k 3 ∂k 1 ∂v -k 4 ∂k 1 ∂u , ∂k 4 ∂v + ∂k 3 ∂u = -2 coth(k 1 ) k 3 ∂k 1 ∂u + k 4 ∂k 1 ∂v , ∆k 1 = e -2k 2 3 sinh(2k 1 ) 2 -2 4 3 + e 2k 2 3 (k 2 3 + k 2 4 ) , ∆k 2 = 3 * 2 1 3 e -2k 2 3 cosh(2k 1 ) -e 2k 2 . (55)
When a 2 = 0 and b 2 = ± 1 + a 2 3 , we find

a 6 = e k 1 cos( 5 3 s) cos(t) 2 ; b 6 = ± e k 1 sin( 5 3 s) cos(t) 2 ; r = e k 2 cos(t)
.

Furthermore, we obtain for b 2 = 1 + a 2 3 that and for b 2 = -1 + a 2 3 we obtain Solving the last equations in [START_REF] Ionel | Second order families of special Lagrangian submanifolds in C 4[END_REF] implies restrictions on the functions k 1 (u, v) and k 2 (u, v) given by ∆k 1 = 2e -2(2k 1 +k 2 ) 5

γ 1+ = -5e k 1 + 5si 3 tan(t) + e 2k 1 +k 2 5 + 2si 3 ( ∂k 2 ∂v -3 ∂k 1 ∂v ) -i( ∂k 2 ∂u -3 ∂k 1 ∂u ) 5 cos(t) , a 1+ = e k 1 cos( 5s
γ 1-= -5e k 1 -5si 3 tan(t) + e 2k 1 +k 2 5 -2si 3 (3 ∂k 1 ∂v -∂k 2 ∂v ) -i(3
3e 2k 1e 2k 2 ∆k 2 = e -2(2k 1 +k 2 ) 5

8e 2k 1 -6e 2k 2 .

(56)

These equations are valid for both b 2 = ± 1 + a 2 3 . We summarize this in the following theorem.

Theorem 4.3. Each special Lagrangian submanifold of CP 4 with SO(2) ⋊ S 3 -symmetry where the only integral distribution containing N 1 is the tangent bundle, can be constructed in the way above using either functions {k 1 , k 2 , k 3 , k 4 } subject to (55) or functions {k 1 , k 2 } subject to (56). Conversely, each such a construction results in such a submanifold, unique up to local isometry.

5.

Submanifolds in CH 4 .

5.1.

The case where b 2 = 0. This is the case where N 1 is an integrable distribution. We assume that a 3 = 0. Similar to the case in CP 4 we have that M is a double warped product R × e g R × e f N 2 . the function a 2 depends only on the coordinate s and is given by ∂a 2 ∂s = a 2 2 -1.

This equation has 3 possible solutions, depending on the initial conditions. For a 2 (0) = 1, it is a constant. For a 2 (0) > 1 it is given as a 2 =coth(s). Finally for a 2 (0) < 1, it is given as a 2 (s) =tanh(s). The connection ∇ is given by

∇ X 1 X 1 = a 1 X 2 + a 2 X 3 ∇ X 1 X 2 = -a 1 X 1 , ∇ X 2 X 1 = b 1 X 2 ∇ X 2 X 2 = -b 1 X 1 + a 2 X 3 , ∇ X 3 X 1 = 0 ∇ X 3 X 2 = 0, ∇ X 4 X 1 = 0 ∇ X 4 X 2 = 0, ∇ X 1 X 3 = -a 2 X 1 ∇ X 1 X 4 = 0, ∇ X 2 X 3 = -a 2 X 2 ∇ X 2 X 4 = 0, ∇ X 3 X 3 = 0 ∇ X 3 X 4 = 0, ∇ X 4 X 3 = -X 4 a 2 ∇ X 4 X 4 = X 3
a 2 . We have the following result.

Theorem 5.1. Suppose M is a special Lagrangian submanifold in CH 4 having a local SO(2) ⋊ S 3 -symmetry. Suppose N 1 is integrable. Then M can be lifted horizontally to a submanifold in H 9 through F and is locally isometric to either (57) F (t, s, u, v) = (sin(t) sinh(s), cos(t) sinh(s), φ(u, v) cosh(s)) ,

where φ is the horizontal lift of a special Lagrangian submanifold of CH 2 to H 5 in case a 2 2 < 1, or (58) F (t, s, u, v) = (φ(u, v) sinh(s), cos(t) cosh(s), sin(t) cosh(s)) ,

where φ is the horizontal lift of a special Lagrangian submanifold of CP 2 to S 5 in case

a 2 2 > 1, or (59)
F (t, s, u, v) = (φ(u, v), t) e -s , - 1 2 e -s , (φ(u, v), t) 2 + if (u, v) e -s + e s ,
where φ is a special Lagrangian surface in C 2 and f is the integral of the differential form

2 2 i=1
x i d y iy i d x i on C 2 in case a 2 2 = 1. Proof. We can check similar to the case in CP 4 that M = R × e g R × e f ×N 2 , where f and g are functions on the first factor, determined by X 3 (g) = -1 a 2 and X 3 (f ) = -a 2 . We can treat the cases separately for each solution to a 2 (s).

Assume a 2 =tanh(s), then it is easy to see that ∂ ∂t =sinh(s)X 4 commutes with ∂ ∂s . The Gauss identity now implies for i ∈ {1, 2} that

D ∂ ∂s ∂ ∂s = ∂ 2 F ∂s 2 = F ⇒ F = A sinh(s) + B cosh(s), D ∂ ∂T ∂ ∂s = ∂ 2 F ∂t∂s = ∂A ∂t cosh(s) + ∂B ∂t sinh(s) = coth(s) ∂F ∂t = ∂A ∂t cosh(s) + ∂B ∂t cosh(s) 2 sinh(s) ⇒ ∂B ∂t = 0, D X i ∂ ∂s = ∂F * X i ∂s = A * X i cosh(s) + B * X i sinh(s) = tanh(s)X i = A * X i sinh(s) 2 cosh(s) + B * X i sinh(s) ⇒ A * X i = 0.
Using the fact that F, F 1 = -1, we get that B, B 1 = -A, A 1 = -1 and A, B 1 = 0. Furthermore, we find

D ∂ ∂t ∂ ∂t = ∂ 2 F ∂t 2 = ∂ 2 A ∂t 2 sinh(s) = -cosh(s) sinh(s) ∂F ∂s + sinh(s) 2 F = -A sinh(s) ⇒ A = A 1 cos(t) + A 2 sin(t).
Because A has unit length, so do A 1 and A 2 and they are both orthogonal. Calculations similar to (28) show that B can be taken as the horizontal lift of any special Lagrangian submanifold in CH 2 and applying a suitable isometry gives (57).

For a 2 =coth(s) calculations similar to the previous case result in (58). Finally we assume a 2 = 1. Then the vector field given by ∂ ∂t = e -s X 4 commutes with ∂ ∂s . We can calculate for i ∈ {1, 2} that

D ∂ ∂s ∂ ∂s = ∂ 2 F ∂s 2 = F ⇒ F = Ae s + Be -s , D ∂ ∂t ∂ ∂s = ∂ 2 F ∂t∂s = ∂A ∂t e s - ∂B ∂t e -s = - ∂F ∂t = - ∂A ∂t e s - ∂B ∂t e -s ⇒ ∂A ∂t = 0, D X i ∂ ∂s = ∂F * X i ∂s = A * X i e s -B * X i e -s = -F * X i = -A * X i e s + B * X i e -s ⇒ A * = 0.
Using the fact that F, F 1 = -1, we obtain that A and B are vector fields with 0 length and they satisfy A, B 1 = -1 2 . Further calculations show

D ∂ ∂t ∂ ∂t = ∂ 2 F ∂t 2 = e -s ∂ 2 B ∂t 2 = e -2s ∂F ∂s + F = 2Ae -s ⇒ B = At 2 + B 1 t + B 2 , D X i ∂ ∂t = ∂F * X i ∂t = B 1 * X i e -s = 0 ⇒ B 1 * = 0.
We can conclude that F has the form

F = At 2 + B 1 t + φ e -s + Ae s
Here, φ is an immersion of a 2-fold in C 5 1 tangent to N 1 . Calculating the scalar products of B and A, we get

A, B 1 = t A, B 1 1 + A, φ 1 = - 1 2 ⇒ A, B 1 1 = 0 A, φ 1 = - 1 2 , B, B 1 = t 2 ( B 1 , B 1 1 -1) + 2t B 1 , φ 1 + φ, φ 1 = 0 ⇒ B 1 , B 1 1 = 1 B 1 , φ 1 = 0 φ, φ 1 = 0. (60)
We can shift to a different standard basis of C 5 1 such that z, w = ℜ 3 j=1 z j wj + z 4 w5 + z 5 w4 .

In this case the constant light-like vector A and time-like B 1 , after applying a suitable isometry, can be chosen to be A = (0, 0, 0, 0, 1) , B 1 = (0, 0, 1, 0, 0) .

We can write φ = (φ 1 , φ 2 , φ 3 , φ 4 , φ 5 ) where φ j = x j + iy j . Then (60) implies

x 4 = - 1 2 , x 3 = 0, x 5 -2y 4 y 5 = 3 j=1 |φ j | 2 .
We can use the fact that both F and iF are orthogonal to the tangent space in C 5 1 and this results in

φ 4 = - 1 2 , φ 3 = 0, d y 5 = 2 2 i=1 (x i d y i -y i d x i ).
This last equation is integrable if and only its derivative equals 0 along the submanifold. But this derivative is nothing more than a multiple of the Kähler form on C 2 spanned by the first 2 complex coordinates. In other words, for such a submanifold to exist, the projection of φ onto the first 2 coordinates should be a Lagrangian submanifold in C 2 .

Calculating the Gauss identity on D X i X j we find that the metric on this immersion is given by

φ * X i , φ * X j = e 2s δ ij ,
where a, b is the standard scalar product on C 2 and φ here is the restriction to the first 2 complex coordinates. Because F * X i , F * X j = δ ij and because φ 3 * = 0 and φ 4 * = 0 this condition is included in the warped product structure. Using calculations like (28) we conclude that (φ 1 , φ 2 ) can be any special Lagrangian 2-fold in C 2 . The result is summarized in (59).

5.2.

The case where b 2 = 0. First we assume that N + is an integrable distribution. This is equivalent to a 6 + ib 6 = 0. The connection is given by (30), resulting in a warped product structure R × e f N 3 . The equation

X 4 (a 3 ) = ∂a 3 ∂t = a 2 3 -1
has a solution given as |a 3 | = 1, a 3 =tanh(t) or a 3 =coth(t), depending on the initial value of a 3 . Using an analysis similar to the case of CP 4 and the case above gives the following result. x i d y iy i d x i .

Finally we assume that there is no integrable distribution that contains N 1 except for the tangent bundle. We return to the analysis as done for C 4 , but set ǫ = -1. The result will depend on the initial value of a 3 . First assume that a 2 3 < 1, then ǫ = -1. We find functions {k 1 , k 2 , k 3 , k 4 } of (u, v) such that a 3 =tanh(t), a 2 = -sinh(2s) cosh(t) (cosh(2s) + cos(2k 1 )) , b 2 = -sin(2k 1 ) cosh(t) (cosh(2s) + cos(2k 1 )) , r = e k 2 cosh(t) cosh(2s) + cos(2k 1 ) ,

a 6 + ib 6 = k 3 + ik 4 ρ 1 -a 2 3 1 -z2 -1 2 .
Using [START_REF] Ionel | Second order families of special Lagrangian submanifolds in C 4[END_REF] as earlier, we obtain a 1 , b 1 , γ 1 as γ 1 = tanh(t)(k 3 + ik 4 ) cosh(sik 1 ) + (64)

Then we set a 2 3 > 1 and assume z 2 = -1. We then find (66)

a 3 = -coth(t

4 3 e -2 3 k 2 + k 2 3 + k 2 4 ,∆k 2 = - 3 * 2 1 3 e -2 3 k 2 e

 4242312 2k 2 + cos(2k 1 ) .

  Theorem 5.2. Suppose M is a special Lagrangian submanifold in CH 4 having a local SO(2) ⋊ S 3 -symmetry. Suppose N 1 is non-integrable, but is contained in the integrable N + distribution. Then M can be lifted horizontally to a submanifold in H 9 through F and is locally isometric to either where φ is a special Lagrangian submanifold with a local S 3 -symmetry in C 3 and f is the integral of the differential form

	3
	(61) where φ is the horizontal lift of a special Lagrangian submanifold of CH 3 with a local F (t, s, u, v) = (sinh(t), φ(s, u, v) cosh(t)) , S 3 -symmetry to H 7 in case a 2 3 < 1, or (62) F (t, s, u, v) = (φ(s, u, v) sinh(t), cosh(t)) , where φ is the horizontal lift of a special Lagrangian submanifold of CP 3 with a local S 2 i=1

3 -symmetry to S 7 in case a 2 3 > 1, or (63) F (t, s, u, v) = φ(s, u, v)e -t , -e -t /2, φ(s, u, v) 2 + if (s, u, v) e -t + e t ,

  ∂k 1 ∂vi ∂k 1 ) tanh(t) (cosh(s) cos(k 1 )(k 4 ρ 2k 3 ρ 1 )sinh(s) sin(k 1 )(k 4 ρ 1 + k 3 ρ 2 )) ,The other equations in[START_REF] Ionel | Second order families of special Lagrangian submanifolds in C 4[END_REF] put restrictions on {k 1 , k 2 , k 3 , k 4 } given by

			∂k 4 ∂u	-	∂k 3 ∂v	= 2 tan(k 1 ) k 4	∂k 1 ∂u	-k 3	∂k 1 ∂v	,
			∂k 4 ∂v	+	∂k 3 ∂u	= -2 cot(k 1 ) k 3	∂k 1 ∂u	+ k 4	∂k 1 ∂v	,
					∆k 1 =	sin(2k 1 ) 2	2
					ρ					∂u	,
	a 1 =	2 3 cosh(t) 2 (cosh(2s) + cos(2k 1 )) 2 (cosh(2s) + cos(2k 1 ))(ρ 1 2 3 e 2 3 k 2	∂k 2 ∂v	+ ρ 2	∂k 2 ∂u	)
		+ sinh(2s)(ρ 2	∂k 1 ∂v	-ρ 1	∂k 1 ∂u	) + sin(2k 1 )(ρ 1	∂k 1 ∂v	+ ρ 2	∂k 1 ∂u	)
	+ sin(2k 1 b 1 = 3 cosh(t) 2 (cosh(2s) + cos(2k 1 )) 2 (cosh(2s) + cos(2k 1 ))(ρ 2 2 2 3 e 2 3 k 2	∂k 2 ∂v	-ρ 1	∂k 2 ∂u	)
		-sinh(2s)(ρ 2	∂k 1 ∂u	+ ρ 1	∂k 1 ∂v	) -sin(2k 1 )(ρ 1	∂k 1 ∂u	-ρ 2	∂k 1 ∂v	)

sin(2k 1 ) tanh(t) (sinh(s) sin(k 1 )(k 4 ρ 2k 3 ρ 1 ) + cosh(s) cos(k 1 )(k 4 ρ 1 + k 3 ρ 2 )) .

  ), + ik 4 ) cos(sik 1 ) + ∂k 1 ∂vi ∂k 1 ) coth(t) (cos(s) cosh(k 1 )(k 4 ρ 2k 3 ρ 1 ) + sin(s) sinh(k 1 )(k 4 ρ 1 + k 3 ρ 2 )) , The functions {k 1 , k 2 , k 3 , k 4 } have to satisfy -2 3 k 2 cosh(2k 1 )e 2k 2 .Solving the other Gauss equations results in the relations ∆k 1 = e -2 5 (2k 1 +k 2 ) 6 + 2e 2k 1 -2e 2k 2 , ∆k 2 = e -2 5 (2k 1 +k 2 ) 8 + e 2k 1 -6e 2k 2 .

				∂k 4 ∂u	-	∂k 3 ∂v	= 2 tanh(k 1 ) k 3	∂k 1 ∂v	-k 4	∂k 1 ∂u	,
	∂k 4 ∂v 3 e (65) + ∂k 3 ∂u = -2 coth(k 1 ) k 3 ∆k 1 = -sinh(2k 1 ) 2 1 3 e -2 ∂k 1 ∂u 3 k 2 + + k 4 k 2 ∂k 1 ∂v 3 + k 2 , 4 2 ∆k 2 = 3 * 2 1	,
	Finally, assume a 2 3 > 1 and z 2 = -1. Then we find a 3 = -coth(t), a 6 = e k 1 cos( 5 3 s) sinh(t) 2 ,
	a 2 = b 2 = r = a 6 + ib 6 = 3 -1 that We obtain for b 2 = a 2 γ 1+ = 5e k 1 + 5si 3 coth(t) + e b 6 = ± sinh(t) (cos(2s) + cosh(2k 1 )) e k 1 sin( 5 3 s) sinh(t) 2 , sin(2s) r = e k 2 . sinh(t) sinh(2k 1 ) sinh(t) (cos(2s) + cosh(2k 1 )) e k 2 sinh(t) cos(2s) + cosh(2k 1 ) 2k 1 +k 2 5 + 2si 3 ( ∂k 2 ∂v -3 ∂k 1 ∂v ) -i( ∂k 2 , , , ∂u -3 ∂k 1 ∂u ) 5 sinh(t) k 3 + ik 4 ρ a 2 3 -1 1 + z2 -1 2 . a 1+ = -e k 1 cos( 5s 3 ) coth(t) + e 2k 1 +k 2 5 cos( 2s 3 ) ∂k 2 ∂v + sin( 2s 3 ) ∂k 2 ∂u 3 sinh(t) ,	,
	We obtain	b 1+ =	-e k 1 sin( 5s 3 ) coth(t) -e 3 sinh(t) 2k 1 +k 2 5 cos( 2s 3 ) ∂k 2 ∂u -sin( 2s 3 ) ∂k 2 ∂v	,
	γ 1 = a 1 = and for b 2 = -a 2 coth(t)(k 3 ∂u ρ 2 2 3 e 2 3 k 2 3 sinh(t) 2 (cos(2s) + cosh(2k 1 )) 2 (cos(2s) + cosh(2k 1 ))(ρ 1 , 3 -1 we obtain γ 1-= 5e k 1 -5si 3 coth(t) + e 2k 1 +k 2 5 -2si 3 (3 ∂k 1 ∂v -∂k 2 ∂v ) -i(3 ∂k 1 ∂k 2 ∂v + ρ 2 ∂u -∂k 2 ∂k 2 ∂u ∂u ) 5 sinh(t) + sin(2s)(ρ 1 ∂k 1 ∂u -ρ 2 ∂k 1 ∂v ) -sinh(2k 1 )(ρ 1 ∂k 1 ∂v + ρ 2 ∂k 1 ∂u a 1-= e k 1 cos( 5s 3 ) coth(t) + e 2k 1 +k 2 5 cos( 2s 3 ) ∂k 2 ∂v -sin( 2s 3 ) ∂k 2 ∂u , 3 sinh(t) ) + sinh(2k 1 b 1 = 2 2 3 e 2 3 k 2 3 sinh(t) 2 (cosh(2s) + cos(2k 1 )) 2 (cosh(2s) + cos(2k 1 ))(ρ 2 ∂k 2 ∂v -ρ 1 ∂u ∂k 2 b 1-= -e k 1 sin( 5s 3 ) coth(t) -e 2k 1 +k 2 5 cos( 2s 3 ) ∂k 2 ∂u + sin( 2s 3 ) ∂k 2 ∂v 3 sinh(t) .	) )	,
	+ sin(2s)(ρ 2	∂k 1 ∂u	+ ρ 1	∂k 1 ∂v	) + sinh(2k 1 )(ρ 1	∂k 1 ∂u	-ρ 2	∂k 1 ∂v	)

sinh(2k 1 ) coth(t) (sin(s) sinh(k 1 )(k 3 ρ 1k 4 ρ 2 ) + cos(s) cosh(k 1 )(k 4 ρ 1 + k 3 ρ 2 )) .

These equations are valid for both b 2 = ± a 2 3 -1. We can conclude with the following proposition.

Theorem 5.3. Each special Lagrangian submanifold of CH 4 with SO(2) ⋊ S 3 -symmetry where the only integral distribution containing N 1 is the whole tangent bundle, can be constructed in the way above using functions {k 1 , k 2 , k 3 , k 4 } subject to (64) in case a 2 3 (0) < 1, subject to (65) in case a 2 3 (0) > 1, or functions {k 1 , k 2 } subject to (66) when a 2 3 (0) > 1 and z 2 = -1. Conversely, each such a construction results in such a submanifold, unique up to local isometry.


Benraiss-Noailles Laïla
email: laila.benraiss-noailles@u-bordeaux.fr

Full

Olivier Herrbach
email: olivier.herrbach@u-bordeaux.fr

Catherine Viot
email: catherine.viot@univ-lyon1.fr

Digital-sector SMEs, the Eldorado of Generation Z?

Keywords:

It has always been more difficult for SMEs to attract candidates than larger organizations, whose arguments can be more convincing. This article compares the factors that influence the attractiveness of SMEs and large enterprises (LEs) in the digital sector, as perceived by members of Generation Z. To this end, a study was conducted with young people from Generation Z as future employees in the digital sector. The results show that attractiveness factors differ between LEs and SMEs. Digital-sector SMEs are seen as employers that pay lower wages than large companies but promote creativity and innovation. This study was conducted prior to the COVID crisis.

Introduction

Being perceived as an attractive employer by young graduates is a recurring problem for any company. Recruiting in a competitive environment is an even greater challenge in sectors, such as digital services, in which small and medium-sized enterprises (SMEs) are pitted against large enterprises (LEs). The difficulty SMEs have in attracting young graduates is confirmed by the quarterly barometer survey by the Rexecode Institute, which identifies recruitment as the main obstacle to development for 59% of SMEs. The arrival of Generation Z on the job market provides an opportunity to examine what makes a company attractive to them. Are they more attracted to SMEs than their elders? The objective of this research is therefore to further knowledge concerning the attractiveness of companies as perceived by members of Generation Z. The underlying research question can be stated as follows: "In a context of shortages and fierce competition, what factors could tempt members of Generation Z to join digital SMEs rather than LEs?" To answer this question, we compared the attractiveness factors of SMEs with those of LEs, based on the Employer Brand (EB) concept proposed by [START_REF] Berthon | Captivating company: Dimensions of attractiveness in employer branding[END_REF].

Organizational attractiveness in the age of Generation Z

While there is nothing new about adopting a generational rather than an age-based approach [START_REF] Mannheim | Le problème des générations (1 ère éd[END_REF]Attias-Donfut, 1991;[START_REF] Strauss | Generations: The History of America's Future, 1584 to 2069[END_REF], the emergence of Generation Y, followed more recently by its successor Generation Z, has led to a resurgence of interest in this strategy. The entry of generations Y and Z into adulthood poses questions for recruiters, insofar as each generation has its own characteristics, values and heroes (Attias-Donfut, 1991;[START_REF] Strauss | Generations: The History of America's Future, 1584 to 2069[END_REF], all of which are likely to influence their image of the ideal company [START_REF] Parker | On the cusp of adulthood and facing an uncertain future: What we know about Gen Z So Far[END_REF] and its associated perceived attractiveness. Organizational attractiveness, which [START_REF] Rampl | Employer brand trust and affect: Linking brand personality to employer brand attractiveness[END_REF] regard as the intention to apply for or accept a potential offer from a particular employer, is defined as a positive affective attitude toward an organization, which is considered desirable, with a view to entering into a relationship with that organization, by becoming part of its staff (Aiman-Smith et al. 2001, p. 221). While the link between the EB and organizational attractiveness has been investigated by numerous studies for Generation Y (authors, 2019;[START_REF] Dalmas | Quelles valeurs organisationnelles pour la génération Y ?[END_REF][START_REF] Petit | Enquête sur la génération Y, rapport n°3[END_REF][START_REF] Pichault | Pour en finir avec la génération Y,… Enquête sur une représentation managériale[END_REF][START_REF] Erickson | Gen Y in the workforce[END_REF][START_REF] Josiam | An Empirical Study ofthe Work Attitudes of Generation Y College Students in the USA: The Case of Hospitality and Merchandising Undergraduate Majors[END_REF][START_REF] Sullivan | Looking at gen Y shopping preferences and intentions: exploring the role of experience and apparel involvement[END_REF][START_REF] Yeaton | Recruiting and managing the "Why?" Generation Y[END_REF][START_REF] Pelton | Teaching Business Ethics: Why Gen Y ?[END_REF], little research has been conducted in relation to Generation Z [START_REF] Schroth | Are You Ready forGen Z in the Workplace?[END_REF]. Our study favors the adoption of a sociological conception of generations by considering the age, history and values of their members. Inspired by the work of [START_REF] Mannheim | Le problème des générations (1 ère éd[END_REF], Strauss andHowe (1991, 2007) define a generation according to four factors: age of birth, average age, historical markers defining a collective consciousness, and a specific identity factor (each generation has a specific dominant personality defined by its values, customs, attitudes and representations). Members of Generation Z were born into a totally different technological environment than their predecessors. According to Geck (2007), they are "ultra-connected, pragmatic, demanding and 'entrepreneurial' in their lives". 1 Generation Z is also described as utopian, quite disobedient, and requiring a new management style; its energy must be channeled toward positive ends, which will require the implementation of management models that are simultaneously structuring, educational and psychological. As proposed by the Pew Research Center, an independent American association, we consider 1996-1997 to be the pivotal period between Generation Y and Generation Z [START_REF] Parker | On the cusp of adulthood and facing an uncertain future: What we know about Gen Z So Far[END_REF].

Are SMEs low-profile employers? Their EB is often underdeveloped because the leaders of these firms tend to focus on the financial and commercial dimensions [START_REF] Purusottama | Employer Branding of Micro, Small and Medium Enterprises (MSMES) In Indonesia: An Exploration Research[END_REF]. For young graduates, several factors can skew the image of SMEs, such as education focusing primarily on LEs, or the greater media presence of Les (Chatelin-Ertur & Nicolas, 2012, p. 267). Students, who are unfamiliar with SMEs, may have a distorted perception of them and consider them to be less attractive than the LEs they think they know better. In 2018, more than half of all SMEs (52%) reported difficulties in recruiting highly skilled professionals. This observation is much more pronounced at the regional level in France, where nearly two-thirds (60%) of companies report such shortages, than in the Paris region (Ile-de-France), where 30% of SMEs share the same difficulties in recruiting this type of staff. The situation is particularly difficult for digital service companies (DSCs). According to the "Grand Angle ESN" survey, 80% of firms reported that they were struggling to attract talented candidates in 2018. They mentioned the scarcity of sought-after skills in a saturated market as the main reason for this problem. These recruitment problems are exacerbated by the national or international competition between DSCs and the difficulties experienced by firms in making themselves visible to candidates. Smaller DSCs are hit particularly hard, as they face competition from their counterparts with a national or even international dimension, which benefit from a high market profile and their ability to dedicate substantial resources and make convincing arguments in order to attract future graduates. Does the salvation of SMEs in the digital sector therefore depend on the development of strong EBs?

Employer branding, a lifeline for SMEs in need of attractiveness

The employer brand (EB) is defined by [START_REF] Ambler | The employer brand[END_REF] as the set of functional, economic and psychological benefits inherent to a job and with which the company, as the employer, is identified. The functional benefits correspond to the attractiveness of the work [START_REF] Berthon | Captivating company: Dimensions of attractiveness in employer branding[END_REF], and the development activities. The economic benefits correspond to material and/or financial advantages, while the psychological benefits are essentially related to feelings of control and belonging (or to the social value associated with the work environment). In this way, the EB represents the company's qualities as perceived by current and potential employees [START_REF] Ambler | The employer brand[END_REF]). An attractive EB improves the quality of recruitments, as well as the loyalty of staff [START_REF] Ambler | The employer brand[END_REF][START_REF] Backhaus | Conceptualizing and researching employer branding[END_REF][START_REF] Davies | Employer branding and its influence on managers[END_REF][START_REF] Gaddam | Modeling employer Branding communication: the softer aspect of HR marketing management[END_REF][START_REF] Kapoor | Employer Branding: a study of its relevance in India[END_REF]. It therefore reduces the costs associated with staff turnover and the recruitment process [START_REF] Kapoor | Employer Branding: a study of its relevance in India[END_REF]. It improves employee engagement, satisfaction and motivation [START_REF] Ambler | The employer brand[END_REF][START_REF] Gaddam | Modeling employer Branding communication: the softer aspect of HR marketing management[END_REF][START_REF] Kapoor | Employer Branding: a study of its relevance in India[END_REF][START_REF] Mandhanya | Employer branding -A tool for talent management[END_REF][START_REF] Priyadarshi | Employer brand image as predictor of employee satisfaction, affective commitment & turnover[END_REF][START_REF] Roy | Identifying the dimensions of attractiveness of an employer branding: the Indian context[END_REF], which in turn make it a source of organizational performance and competitiveness [START_REF] Ambler | The employer brand[END_REF][START_REF] Collins | The relationship between early recruitment-related activities and the application decisions of new labour market entrants: a brand equity approach to recruitment[END_REF]Fulmer et al., 2000). The EB's role is to promote an attractive and distinctive corporate image in relation to the company's competitors [START_REF] Ambler | The employer brand[END_REF][START_REF] Backhaus | Conceptualizing and researching employer branding[END_REF][START_REF] Maxwell | Motivating employees to "live the brand": a comparative case study of employer brand attractiveness within the firm[END_REF][START_REF] Mandhanya | Employer branding -A tool for talent management[END_REF]. The EB's image is a key factor in attracting candidates. A company with a strong EB can attract a large number of high-quality candidates [START_REF] Cable | Establishing the dimensions, sources and value of job seekers' employer knowledge during recruitment[END_REF] and creates a desire to join the company [START_REF] Mandhanya | Employer branding -A tool for talent management[END_REF][START_REF] Collins | The relationship between early recruitment-related activities and the application decisions of new labour market entrants: a brand equity approach to recruitment[END_REF][START_REF] Agrawal | Effect of employer brand image on application intentions of B-school undergraduates[END_REF]. [START_REF] Collins | The relationship between early recruitment-related activities and the application decisions of new labour market entrants: a brand equity approach to recruitment[END_REF] found that the perceived attributes of the organization and attitudes toward the organization are positively related to the intention to apply and the decision to apply for a post. The literature provides various conceptualizations of employer branding, among which the one proposed by [START_REF] Berthon | Captivating company: Dimensions of attractiveness in employer branding[END_REF] is one of the most consensual (Table 1). An employer that offers a good working atmosphere and team spirit.

Economic value

Attributes of an employer that offers above-average pay, job security and promotion opportunities.

Development value

An employer that offers recognition, self-esteem and confidence, combined with work experience that can act as a stepping-stone to future employment. This is about the personal development of employees.

Application value

An employer that gives its employees the opportunity to apply what they have learned and to pass it on to others in an environment that is both customeroriented and socially responsible.

In accordance with the objective of this study, we set out to identify whether the EB as an organizational attractiveness factor influences the attractiveness of digital service SMEs and LEs to differing extents.

Figure 1. Causal model and hypotheses

Research Methodology

This study focuses on DSCs, 2 a sector that covers a wide variety of firms, from microenterprises to LEs. Whatever their size, DSCs experience difficulties in recruiting and retaining staff (turnover rate of around 20%).

The digital economy has gradually emerged as one of the Nouvelle-Aquitaine region's fields of excellence, leading to the creation of a competitiveness cluster called "Digital Aquitaine" in 2014. This manifestation of political intent has had many consequences, which include putting the digital technology labor market under significant pressure. As a result, Nouvelle-Aquitainebased DSCs -mostly SMEs and intermediate-sized enterprises -have found themselves in direct competition with large groups that have chosen to set up shop near Bordeaux (Betclic, Ubisoft, etc.). This regional and sectoral context is a textbook case of the talent war.

To measure the concepts, we used existing scales: the five-factor scale developed by Berthon, Ewing and Hah (2005, p. 159-160) to measure the employer brand equity and an overall measure of organizational attractiveness (Rampl & Ke nning, 2014) that we have adapted to DSCs. Each respondent rated the attractiveness of SMEs and LEs. Respondents rated their opinions on a five-point scale (from 1 = strongly disagree to 5 = strongly agree). "Sphinx" software was used to design and publish an online questionnaire intended to collect data from a sample of future graduates of training programs that were targeted by companies' hiring departments. 3 The initial sample consisted of 179 observations, from which the responses of students who were considering starting their own businesses and those who belonged to Generation Y were removed (giving a final sample of 145 observations). The average age of the respondents was 21 years and 7 months. Students with two and more years of higher education represented 57% of the sample. 73% of the respondents were male. In 2017, 73% of 2 A digital service company is any company specializing in the field of new technologies and computer science. 3 This study was conducted with the support of SYRPIN (regional professional society of IT and digital professionals).

Employer Brand H1 Digital SMEs attractiveness

Digital LEs attractiveness H2 students in these study programs were male.

Results

First, the structure and reliability of the scales were checked using principal component analysis (PCA), whose detailed results are not reproduced here.

• The five-factor structure of the Berthon et al. scale (2005) was verified. It explained 66% of the variance. The communalities were greater than 0.5. Cronbach's alpha ranged from 0.5 to 0.83. • The PCA shows that the organizational attractiveness of DSCs is measured by two factors: the attractiveness of SMEs (α = 0.87) and that of LEs (α = 0.77). The explained variance is 75%. Two regression models were then tested, the first for the attractiveness of SMEs and the second for the attractiveness of LEs. These models are based on the factor scores recorded during PCA. The "stepwise" method was used to optimize the regression models. The results show that the attractiveness of the EB for SMEs stems from two dimensions: economic value, which is negatively related to SME attractiveness (β = -0.25, t = -3.05; p = 0.003), and interest value, which exerts a positive influence (β = 0.2, t = 2.910; p = 0.004). Although the variance explained by the model is small (R 2 = 0.10), the model is significant (F = 7.77, p = 0.001). The impact of the three other dimensions on the attractiveness of SMEsnamely, application value, development value and social value -is not significant. The attractiveness of the EB of LEs is also a product of two dimensions. One of these dimensions is common to the attractiveness of SMEs, but it works in reverse. This time, economic value positively influences the attractiveness of large enterprises (β = 0.18, t = 2.15; p = 0.033). The second determinant is social value, which is negatively related to the attractiveness of large firms (β = -0.18; t = -2.13; p = 0.035). Although the variance explained by the model is small (R 2 = 0.06); the model is significant (F = 4.33, p = 0.015). The regression model of attractiveness vis-à-vis large firms shows a significant and negative impact of social value on the attractiveness of large firms. The impact of economic value on attractiveness is also differentiated according to the type of company. In contrast to SMEs, this significant impact is positive for large enterprises.

Discussion

The results of this research show a positive and significant relationship between the interest value dimension and the attractiveness of SMEs. This relationship reflects high expectations for a stimulating work environment and new practices that will use candidates' creativity to produce innovative, high-quality products and services. In the candidates' minds, this expectation is strongly linked to the size of the firm. This interest value does not impact the attractiveness of LEs. This result confirms the arguments put forward by [START_REF] Dabirian | Employer branding: Understanding employer attractiveness of IT companies[END_REF], who stated that certain candidates were concerned about LEs in the digital sector being highly bureaucratic and offering them repetitive tasks that would be out of step with their qualifications and without lasting interest, hence their preference for SMEs with less complex and more flexible hierarchical structures. SMEs in the digital sector would therefore benefit from conveying a clear EB message to candidates belonging to Generation Z, in which they explicitly state that creativity, innovation and responsiveness are among the foremost values upheld by the company. Another result relates to the fact that the relationship between social value and attractiveness is non-significant for SMEs, but negative for large enterprises. Indeed, young people who are attracted to large enterprises in the digital service sector are not expecting to find an employer that will offer them a good working atmosphere and team spirit. The explanation could come from a shared belief that in DSCs, the atmosphere and team spirit are guaranteed due to the prevailing "start-up" culture (this belief is often illustrated by the provision of recreational facilities such as table soccer for employees). In their research on the best employers in the digital sector, [START_REF] Dabirian | Employer branding: Understanding employer attractiveness of IT companies[END_REF] emphasize the importance of social value (collaborative work environment, relationships with colleagues), which stands out amongst the most important values perceived by employees in this sector. The authors point out that this value is not sufficiently promoted by SMEs and encourage these firms to make it a more central part of their EB communication. In this way, SMEs could make social value a key thrust of their Generation Z-oriented communication strategy. Their size, enabling the creation of stronger ties between employees, and the start-up spirit, could prove to be assets in conveying this message. The impact of economic value on attractiveness differs according the size of an enterprise. The effect is positive for LEs, which are likely to be perceived as more attractive due to their ability to offer newly recruited staff above-average salaries, numerous career opportunities, and ensure job security for future employees from Generation Z. Conversely, economic value has a negative influence on the attractiveness of SMEs in the digital sector. This effect reflects a conscious choice on the part of young respondents. Indeed, by choosing an SME, they do not expect to benefit from an above-average salary, nor from the job security and promotion opportunities that LEs in the sector might offer them. For [START_REF] Dabirian | Employer branding: Understanding employer attractiveness of IT companies[END_REF], economic value remains the primary attrition factor in DSCs. Our results show that development value does not influence the attractiveness of DSCs to members of Generation Z. One possible explanation is that future graduates are aware that their first experience, whether in an LE or an SME, is merely a stepping-stone toward a more advantageous future job: it is only a first step in their career. Regarding knowledge application value, both LEs and SMEs are seen as being likely to allow employees to put their knowledge into practice and pass it on to others. [START_REF] Dabirian | Employer branding: Understanding employer attractiveness of IT companies[END_REF] recommend that managers in DSCs of all sizes stay in tune with employees by monitoring changes of interest in these values. The results provide some guidelines for managerial actions towards future employers. For SMEs, factors related to a stimulating work environment and practices that encourage employees' creativity make the difference, rather than the promise of highly attractive remuneration and career prospects. SMEs would find it difficult to compete with national or international companies on wages. The future employer should thus position itself as a company that puts innovation at the center of its concerns, in products and services development as well as in the work organization and in the work environment. Given the high turnover rate (around 20%) in the digital sector, combining actions aimed at attracting young recruits with those aimed at retaining existing employees is now essential. If the company communicates different and inconsistent internal and external messages, the risk of brand fragmentation is high (Hatch & Schultz, 2009). In addition, negative perceptions of job reality, which may turn out to be different from the one that led to the decision to join the company, will hinder new recruits' identification with the organization (Moroko & Uncles, 2008) and damage the psychological contract [START_REF] Backhaus | Conceptualizing and researching employer branding[END_REF].

Limitations and research avenues

Our sample size is limited, and we collected data in a specific geographical area. This sample size did not enable the performance of multi-group analyses in order to test the moderating effects of gender, familiarity with SMEs and LEs, and engagement vis-à-vis the firm's service category (authors, 2019).

Table 1 -Dimensions of the Berthon, Ewing and Hah (2005) scale Dimension Meaning

 1

Interest value Stimulating work environment; new practices that use employees' creativity to produce innovative and high-quality products and services. Social value

La génération Z expliquée aux DRH -BNP Paribas -The Boson Project,