Semi-slant submanifolds of a Sasakian manifold

SEMI–SLANT SUBMANIFOLDS OF A SASAKIAN MANIFOLD J.L. CABRERIZO, A. CARRIAZO, L. M. FERN´ ANDEZ and M. FERN´ ANDEZ (Sevilla)1 Abstract In this paper, we define and study both bi–slant and semi–slant submanifolds of an almost contact metric manifold and, in particular, of a Sasakian manifold. Key words: Sasakian manifold, slant submanifold, slant distribution, bi–slant submanifold, semi–slant submanifold. AMS 1991 Subject Classification: 53C15, 53C40. 0 Introduction. The differential geometry of slant submanifolds have shown an increasing development since B.-Y. Chen defined slant immersions in complex geometry as a natural generalization of both holomorphic immersions and totally real immersions (see [6]). Many authors have studied such slant immersions in almost Hermitian manifolds. In [8], A. Lotta has introduced the notion of slant immersion of a Riemannian manifold into an almost contact metric manifold. In [4], we have studied and characterized slant submanifolds of K–contact and Sasakian manifolds and we have given several examples of such immersions. We have also studied other properties of slant submanifolds in [5]. Moreover, in [3], we have presented existence and uniqueness theorems for slant immersions into Sasakian-space-forms, which are similar to that of B.-Y. Chen and L. Vrancken in complex geometry [7]. Recently, in [9] N. Papaghiuc has introduced a class of submanifolds in an almost Hermitian manifold, called the semi–slant submanifolds, such that the class of proper CR–submanifolds and the class of slant submanifolds appear as particular cases in the class of semi–slant submanifolds. The purpose of the present paper is to define and study a contact version of semi–slant submanifolds, so that both semi–invariant [1] and contact slant submanifolds to appear as particular cases of the introduced notion. In Section 1, we review basic formulas and definitions for almost contact metric manifolds and their submanifolds, which we shall use later. In Section 2, we recall the definitions and some properties given in [4, 8, 9]. We also study some relations 1The authors are partially supported by the PAI project (Junta de Andaluc´ıa, Spain 1998). 1 between slant submanifolds of an almost contact metric manifold and semi–slant submanifolds of an almost Hermitian manifold. In Section 3, we introduce the notion of slant distribution in contact geometry and we present a more general class of submanifolds: bi–slant submanifolds. Finally, we define and study semi–slant submanifolds in Section 4. 1 Preliminaries. Let ( f M, g) be an odd–dimensional Riemannian manifold and denote by Tf Mthe Lie algebra of vector fields in f M. Then f Mis said to be an almost contact metric manifold [2] if there exist on f Ma tensor φof type (1,1) and a global vector field ξ (structure vector field) such that, if ηis the dual 1–form of ξ, then φ2X=−X+η(X)ξ, g(X, ξ) = η(X), g(φX, φY ) = g(X, Y )−η(X)η(Y), for any X, Y ∈Tf M. Let Φ denote the 2–form in f Mgiven by Φ(X, Y ) = g(X, φY ) for all X, Y ∈Tf M. The 2–form Φ is called the fundamental 2–form in f Mand the manifold is said to be a contact metric manifold if Φ = dη. If ξis a Killing vector field with respect to g, the contact metric structure is called a K–contact structure. It is known that a contact metric manifold is K–contact if and only if f ∇Xξ=−φX, for any X∈Tf M, where f ∇denotes the Levi–Civita connection of f M. The almost contact structure of f Mis said to be normal if [φ, φ] + 2dη⊗ξ= 0, where [φ, φ] is the Nijenhuis torsion of φ. A Sasakian manifold is a normal contact metric manifold. Every Sasakian manifold is a K–contact manifold. It is easy to show that an almost contact metric manifold is a Sasakian manifold if and only if (f ∇Xφ)Y=g(X, Y )ξ−η(Y)X, (1.1) for any X, Y ∈Tf M. Now, let Mbe a submanifold immersed in f M. We also denote by gthe induced metric on M. Let TM be the Lie algebra of vector fields in Mand T⊥Mthe set of all vector fields normal to M. Denote by ∇the Levi–Civita connection of M. Then, the Gauss – Weingarten formulas are given by f ∇XY=∇XY+σ(X, Y ),f ∇XV=−AVX+DXV, for any X, Y ∈TM and any V∈T⊥M, where Dis the connection in the normal bundle, σis the second fundamental form of Mand AVis the Weingarten endomorphism associated with V. The second fundamental form σand the shape operator Aare related by g(AVX, Y ) = g(σ(X, Y ), V ). 2 For any X∈TM, we write φX =TX +NX, where TX is the tangential component of φX and NX is the normal component of φX. Similarly, for any V∈T⊥M, we have φV =tV +nV , where tV (resp. nV ) is the tangential component (resp. normal component) of φV . The submanifold Mis said to be invariant if Nis identically zero, that is, φX ∈TM, for any X∈TM. On the other hand, Mis said to be an anti–invariant submanifold if Tis identically zero, that is, φX ∈T⊥M, for any X∈TM. 2 Slant Immersions. Let Mbe a Riemannian manifold, isometrically immersed in an almost contact metric manifold ( f M, φ, ξ, η, g). From now on, we suppose that the structure vector field ξis tangent to M. Hence, if we denote by Dthe orthogonal distribution to ξ in TM, we can consider the orthogonal direct decomposition TM =D ⊕ < ξ >. For each nonzero vector Xtangent to Mat x, such that Xis not proportional to ξx, we denote by θ(X) the angle between φX and TxM. Then, Mis said to be slant ([8]) if the angle θ(X) is a constant, which is independent of the choice of x∈Mand X∈TxM−< ξx>. The angle θof a slant immersion is called the slant angle of the immersion. Invariant and anti–invariant immersions are slant immersions with slant angle θ= 0 and θ=π/2 respectively. A slant immersion which is not invariant nor anti–invariant is called a proper slant immersion. An useful characterization of slant submanifolds in almost contact metric manifolds is given by the following theorem: Theorem 2.1.– [4] Let Mbe a submanifold of an almost contact metric manifold f Msuch that ξ∈TM. Then, Mis slant if and only if there exists a constant λ∈[0,1] such that T2=−λI +λη ⊗ξ. Furthermore, in such case, if θis the slant angle of M, then, λ= cos2θ. Another important notion about slant immersions is that of semi–slant submanifolds of an almost Hermitian manifold, introduced by N. Papaghiuc in [9]. Given a submanifold S, isometrically immersed in an almost Hermitian manifold ( e S, J, g1), Papaghiuc says a differentiable distribution Don Sto be a slant distribution if for any nonzero vector X∈Dx;x∈S, the angle between JX and the vector space Dxis constant, i.e., it is independent of the choice of x∈Sand of X∈Dx. This constant angle is called the slant angle of the slant distribution D. Now, Sis said to be a semi–slant submanifold if there exist on Stwo differentiable orthogonal distributions D1and D2such that TM =D1⊕D2,D1is a complex distribution (i.e. J(D1) = D1) and D2is a slant distribution with the slant angle θ6= 0. In particular, if dim D1= 0 and θ6=π/2, then Papaghiuc obtain the proper 3 slant submanifolds of almost Hermitian manifolds, introduced by B.Y. Chen (see [6]). In fact, by studying two classic examples, we can find some relations between slant submanifolds of almost contact metric manifolds and semi–slant submanifolds of almost Hermitian manifolds. Let ( f M, φ, ξ, η, g) be an almost contact metric manifold. We consider the manifold f M×R. We denote by (X, f d dt ) a vectorial field of f M×R, where Xis tangent to f M,tis the coordinate of Rand fis a differentiable function on f M×R. We define on this manifold the almost complex structure Jgiven by: J(X, f d dt)=(φX −fξ, η(X)d dt).(2.1) Then, it is well known that ( f M×R, J, g1) is an almost Hermitian manifold [2], where g1denotes the product metric: g1((X, f d dt),(Y, h d dt)) = g(X, Y ) + fh. The following result show how to obtain semi–slant submanifolds of f M×Rfrom slant submanifolds of f M. Theorem 2.2.– Let Mbe a non-invariant slant submanifold of an almost contact metric manifold f M. Then, M×Ris a semi–slant submanifold of f M×R, with complex distribution D1=<(ξ, 0),(0, d/dt)>and slant distribution D2={(X, 0) / X ∈ D}, respectively. Proof: It is clear that distributions D1and D2are perpendicular and that T(M× R) = D1⊕D2. Moreover, D1is a complex distribution, given that, by virtue of (2.1), J(ξ, 0) = (0, d/dt) and so J(0, d/dt) = −(ξ, 0). Finally, it is easy to see that D2is a slant distribution, in the sense of [9]. 2 Remark 2.3.– We have excluded the invariant case in the above theorem since, in [9], the slant distribution of a semi–slant submanifold must have a non-zero angle. Nevertheless, it is easy to prove that, if Mis an invariant submanifold of f M, then M×Ris an invariant submanifold of f M×R. Remark 2.4.– In [8, p. 193], Lotta has proved that, if M×Ris slant in f M×R, then Mmust be an invariant submanifold of f Mand so, M×Ris a complex submanifold of f M×R. Note that there is not a contradiction between this result and Theorem 2.2, because the semi–slant submanifold M×Ris slant if and only if it is a complex submanifold, since J(ξ, 0) = (0, d/dt). Note also that, by using the notion of semi–slant submanifold of an almost Hermitian manifold, we have obtained a more general result. 4 The second classic example was given by Tashiro in [10]. Let e Sbe an almost Hermitian manifold with almost complex structure J. Let f M ,→e Sbe an orientable hypersurface, isometrically immersed in e S. Denote by gboth the metric of e Sand the induced one in f M. Let Cbe the unit normal to f M. Then, ξ=−JC is tangent to M. We define φand ηby JX =φX +η(X)C, for any Xtangent to f M. It is easy to see that (φ, ξ, η, g) is an almost contact metric structure on f M. Let S ,→e Sbe an immersion such that Cand ξare tangent to S. Denote by D1the distribution in TS spanned by Cand ξ,D1=< C, ξ >, and by D2, the orthogonal distribution in TS. Suppose that there are an orientable hypersurface M ,→S, normal to C, and an immersion M ,→f Msuch that the following diagram is conmutative: f M ,→e S ↑ ↑ M ,→S . We can state the following result: Theorem 2.5.– In the above conditions, Mis a slant submanifold of f M, with slant angle θ6= 0, if and only if Sis a semi–slant submanifold of e S, with complex distribution D1and θ–slant distribution D2. Proof: Given that ξ=−JC, it is clear that D1is a complex distribution. Thus, it is enough to prove that Mis slant if and only if D2is a slant distribution on S, but this is easy to see since φand Jare equal on D2.2 Remark 2.6.– It is clear that Mis invariant if and only if Sis a complex submanifold. We can easily obtain an example of the above situation: Example 2.7.– Let R6be the Euclidean space of dimension 6, with the standard metric and the almost complex structure given by J(∂/∂xi) = ∂/∂yi, for any i= 1,2,3, where (xi, yi) denote the cartesian coordinates. Let R5,→R6be the usual immersion. Then, C=∂/∂y3is the unit normal to R5and so, ξ=−JC =∂/∂x3. Now, for any θ6= 0, we can consider the immersions: ϕ1:R4→R6: (u, v, t, s)7→ (ucos θ, u sin θ, t, v, 0, s), ϕ2:R3→R5: (u, v, t)7→ (ucos θ, u sin θ, t, v, 0). Then, it is easy to show that all conditions of Theorem 2.5 are satisfied, where the immersion R3,→R4is the usual one. In fact, we can directly prove that ϕ1 5 is a semi–slant immersion, with complex distribution D1=< ∂/∂x3, ∂/∂y3>and θ–slant distribution D2=<cos θ ∂/∂x1+ sin θ ∂/∂x2, ∂/∂y1>. On the other hand, ϕ2is a θ–slant immersion, where R5has the almost contact metric structure induced by the described almost Hermitian structure on R6. Hence, it is clear that semi–slant submanifolds are interesting from a geometrical point of view. In the following sections, we are going to introduce the contact versions of slant distributions and semi–slant immersions. 3 Slant Distributions. In this section, we are going to give the notion of slant distribution on a submanifold of an almost contact metric manifold. By using this notion, we will define bi–slant submanifolds, which appear as a natural generalization of slant submanifolds. From now on, let Mbe a Riemannian manifold, isometrically immersed in an almost contact metric manifold ( f M, φ, ξ, η, g), such that ξ∈TM. We call a differentiable distribution Von Maslant distribution if for each x∈Mand each nonzero vector X∈ Vx, the angle θ0(X) between φX and the vectorial subspace Vx is a constant, which is independent of the choice of x∈Mand X∈ Vx. In this case, the constant angle θ0is called the slant angle of the distribution V(compare with the case of almost Hermitian manifolds). If we consider the distribution D, then we can state the following lemma: Lemma 3.1.– Let xbe a point of Mand X∈TxM. Then, θ(X) = θ0(X), where θ0(X)denotes the angle between φX and Dx. Proof: It is enough to prove that TX = ΠDX, for any X∈TM, where ΠD denotes the orthogonal proyection of φon D, but this is clear, since φξ = 0. 2 Hence, we can give a new notion of slant submanifold, equivalent to Lotta’s definition: Proposition 3.2.– The submanifold Mis slant if and only if Dis a slant distribution, with the same slant angle. Proof: By virtue of Lemma 3.1, it is obvious that if Mis a slant submanifold, then Dis a slant distribution with the same slant angle. Conversely, given X∈ TM−< ξ >, the angle θ(X) satisfies: cos θ(X) = g(TX, φX) |TX||φX|=|TX| q|X|2−η2(X).(3.1) 6 On the other hand, if we consider X−η(X)ξ∈ D, then we have: cos θ0(X−η(X)ξ) = |ΠD(X−η(X)ξ)| |X−η(X)ξ|.(3.2) But, q|X|2−η2(X) = |X−η(X)ξ|and TX = ΠD(X−η(X)ξ). Thus, (3.1) is equal to (3.2), which is a constant since Dis a slant distribution. 2 Now, we can define bi–slant submanifolds. We say that Mis a bi–slant submanifold of f Mif there exist two orthogonal distributions D1and D2on Msuch that: (i) TM admits the orthogonal direct decomposition: TM =D1⊕D2⊕< ξ >. (ii) For any i= 1,2, Diis a slant distribution with slant angle θi. Given a bi–slant submanifold M, we can write, for any X∈TM, X=P1X+P2X+η(X)ξ, (3.3) where PiXdenotes the component of Xin Di, for any i= 1,2. In particular, if X∈ Di, then we obtain X=PiX. If we define Ti=Pi◦T, then we have φX =T1X+T2X+NX, (3.4) for any X∈TM, given that η(φX) = 0. Let d1(resp. d2) denote the dimension of the distribution D1(resp. D2). By virtue of Proposition 3.2, if either d1or d2vanishes, the bi–slant submanifold is a slant submanifold. Thus, slant submanifolds (and, therefore, invariant and antiinvariant submanifolds) are particular cases of bi–slant submanifolds. Moreover, we easily find examples of non-trivial bi–slant submanifolds. Example 3.3.– For any θ1, θ2∈[0, π/2], x(u, v, w, s, t) = 2(u, 0, w, 0, v cos θ1, v sin θ1, s cos θ2, s sin θ2, t) defines a 5–dimensional bi–slant submanifold M, with slant angles θ1and θ2, in R9 with its usual Sasakian structure (φ0, ξ, η, g) (see [4]). In fact, it is easy to see that e1= 2( ∂ ∂x1+y1∂ ∂z), e2= cos θ1(2 ∂ ∂y1) + sin θ1(2 ∂ ∂y2), e3= 2( ∂ ∂x3+y3∂ ∂z), e4= cos θ2(2 ∂ ∂y3) + sin θ2(2 ∂ ∂y4), e5= 2 ∂ ∂z =ξ, (3.5) form a local orthonormal frame of TM. We define the distributions D1=< e1, e2> and D2=< e3, e4>. 7 Then, it is clear that TM =D1⊕D2⊕< ξ >. It can be easily proved that Di is a slant distribution with slant angle θi, for any i= 1,2. In particular, if we consider θ1=θ2=θin Example 3.3, it results that Mis a θ–slant submanifold. Nevertheless, this is not a general fact, as we can see in the following example: Example 3.4.– For any θ1∈[0, π/2], we choose θ2∈(0, π/2] such that cos θ2= (cos θ1)/√2. Then, x(u, v, w, s, t) = 2(u, 0, w, u, v cos θ1, v sin θ1, s cos θ2, s sin θ2, t) defines a 5–dimensional bi–slant submanifold Min (R9, φ0, ξ, η, g), with both slant angles equal to θ2, such that it is not a slant submanifold. In fact, we can choose a local orthonormal frame {e1, . . . , e5}of TM such that e1= 1/√2{2(∂/∂x1+y1∂/∂z) + 2(∂/∂x4+y4∂/∂z)}and e2, . . . , e5are given by (3.5). We define D1=< e1, e2>and D2=< e3, e4>. It is easy to see that both D1 and D2are slant distributions with the same slant angle θ2. Nevertheless, we can obtain that Mis not slant since θ26= 0. However, we can prove, by a direct calculation, the following result: Proposition 3.5.– Let Mbe a bi–slant submanifold with angles θ1=θ2=θ. If g(φX, Y ) = 0, for any X∈ D1and any Y∈ D2, then Mis slant with angle θ. Remark 3.6.– Note that Example 3.3 satisfies the aditional condition of Proposition 3.5. Bi–slant submanifolds are also a generalization of semi–invariant submanifolds, introduced by Bejancu and Papaghiuc in [1]. In that paper, Mis said to be a semi– invariant submanifold if there exist two orthogonal distributions Dand D⊥on M, such that: (i) TM =D⊕D⊥⊕< ξ >. (ii) The distribution Dis invariant, i.e., φD=D. (iii) The distribution D⊥is anti-invariant, i.e., φD⊥⊂T⊥M. We are now going to show that semi–invariant submanifolds are bi–slant submanifolds, with angles θ1= 0 and θ2=π/2. We first need the following two lemmas: Lemma 3.7.– Suppose that there exist two orthogonal distributions D1and D2on Msuch that TM =D1⊕D2⊕< ξ >. Then, D1is invariant if and only if it is slant with angle θ1= 0. Proof: It is clear that, if D1is invariant, then it is slant with zero angle. The converse is easy to prove, by taking into account (3.4) and that, if D1is a slant 8 distribution with zero slant angle, then |T1X|=|X|, for any X∈ D1.2 Lemma 3.8.– In the above conditions, if D1is invariant, then TX =T2Xfor any X∈ D2. Proof: For any X∈ D1and Y∈ D2, we have g(TX, Y ) = −g(X, φY ) = 0, since D1is invariant. Thus, T1X= 0 and the result holds. 2 Proposition 3.9.– The submanifold Mis semi–invariant if and only if Mis bi– slant with angles θ1= 0 and θ2=π/2. Proof: The direct implication is immediate. The converse follows directly from Lemmas 3.7 and 3.8. 2 Remark 3.10.– In Proposition 3.9, we have proved that, if D2is an anti-invariant distribution, then it is slant with angle π/2. This is not a general fact, as we can show by putting θ1=θ2=π/2 in Example 3.4. Thus, if we write D1=< e3, e4> and D2=< e1, e2>, we obtain a submanifold Mendowed with two orthogonal distributions such that TM =D1⊕D2⊕< ξ >,D2is slant with angle π/2 but it is not anti-invariant, because φ0e1is not normal to M. We finish this section by stablishing a general theorem for bi–slant submanifolds, which we shall use later. Note that this result is an obvious generalization of the direct implication of Theorem 2.1, and it can be proved by following the same steps. Theorem 3.11.– Let Mbe a bi–slant submanifold of an almost contact metric manifold f M. Denote by D1and D2the slant distributions of M, with slant angles θ1and θ2, respectively. Then, given i= 1,2, for any X∈ Di: T2 iX=−cos2θiX. (3.6) 4 Semi–Slant Submanifolds. Now, we can introduce the notion of semi–slant submanifolds in contact geometry. We say that Mis a semi–slant submanifold of f Mif there exist two orthogonal distribution D1and D2on Msuch that: (i) TM admits the orthogonal direct decomposition TM =D1⊕D2⊕< ξ >. (ii) The distribution D1is an invariant distribution, i.e., φ(D1) = D1. (iii) The distribution D2is slant with angle θ6= 0. In this case, we call the angle θthe slant angle of submanifold M. By virtue of Lemma 3.7, the invariant distribution of a semi–slant submanifold is a slant distribution with zero angle. Thus, it is obvious that, in fact, semi–slant submanifolds are 9 for any X∈TM and any Z∈ D2. Thus, we can deduce from (4.30) that ∇XY∈ D1⊕< ξ >, for any X∈TM and any Y∈ D1. Therefore, by applying Lemma 4.1, (4.4), (4.9) and (4.29), we can compute (∇XφP1)Y=P1ANP2YX+g(P1X, Y )−η(Y)P1X, (4.31) for any X, Y ∈TM. On the other hand, condition (ii), (4.5), (4.10) and (4.30) imply P1ANP2YX= 0,(4.32) for any X, Y ∈TM. Finally, equation (4.16) follows from (4.2), (4.31), (4.32) and condition (ii). 2 The following corollary appears directly from equation (4.28): Corollary 4.13.– If Mis a proper semi–slant submanifold of a Sasakian manifold f M, satisfaying (4.16), then, for any X, Y ∈TM: (∇XφP1)Y=g(P1X, Y )ξ−η(Y)P1X. On the other hand, we can also emphasize the integrability conditions obtained in the proof of Theorem 4.12: Corollary 4.14.– If Mis a proper semi–slant submanifold of a Sasakian manifolf f Msatisfaying (4.16), then ∇XY∈ D1⊕< ξ >, ∇XZ∈ D2⊕< ξ >, (4.33) for any X∈TM,Y∈ D1and Z∈ D2. In particular, distributions D1⊕< ξ > and D2⊕< ξ > are integrable. Remark 4.15.– Given that distributions D1and D2are orthogonal, we can prove that both equations in (4.33) are equivalent. We have implicitly used this fact in the proof of Theorem 4.12. Moreover, if dim D2= 2, these equations are equivalent to the conditions of above theorem, as we are going to show in Theorem 4.16. This last theorem is a clear generalization of Theorem 5.1 of [4]. Theorem 4.16.– Let Mbe a proper semi–slant submanifold, with angle θ, of a Sasakian manifold f M. Suppose dim D2= 2. Then, the following statements are equivalent: (i) Msatisfies (4.16). (ii) (∇XTP2)Y= cos2θ(g(P2X, Y )ξ−η(Y)P2X), for any X, Y ∈TM. (iii) ∇XY∈ D1⊕< ξ >, for any X∈TM and any Y∈ D1. (iv) ∇XY∈ D2⊕< ξ >, for any X∈TM and any Y∈ D2. Proof: By virtue of Theorem 4.12, Corollary 4.14 and Remark 4.15, it is enough to prove that (iv) implies (ii). This proof is similar to that of [4, Theorem 5.1]. 2 16 References [1] A. Bejancu and N. Papaghiuc. Semi-invariant submanifolds of a Sasakian manifold. An. st. Univ. Iasi, tom. XXVII, s. I. a (f1) (1981), 163–170. [2] D. E. Blair. Contact Manifolds in Riemannian Geometry, volume 509 of Lecture Notes in Mathematics. Springer-Verlag, New York, 1976. [3] J.L. Cabrerizo, A. Carriazo, L.M. Fern´andez and M. Fern´andez. Existence and uniqueness theorem for slant immersions in Sasakian-space-forms. Submitted. [4] J.L. Cabrerizo, A. Carriazo, L.M. Fern´andez and M. Fern´andez. Slant submanifolds in Sasakian manifolds. Submitted. [5] J.L. Cabrerizo, A. Carriazo, L.M. Fern´andez and M. Fern´andez. Structure on a slant submanifold of a contact manifold. Submitted. [6] B. Y. Chen. Geometry of Slant Submanifolds. Katholieke Universiteit Leuven, Leuven, 1990. [7] B. Y. Chen and L. Vrancken. Existence and uniqueness theorem for slant immersions and its applications. Results in Math.,31 (1997), 28–39. [8] A. Lotta. Slant submanifolds in contact geometry. Bull. Math. Soc. Roumanie, 39 (1996), 183–198. [9] N. Papaghiuc. Semi-slant submanifolds of a Kaehlerian manifold. An. st. Univ. Iasi, tom. XL, s. I. a (f1) (1994), 55–61. [10] Y. Tashiro. On contact structures of hypersurfaces in almost complex manifolds I. Tˆohoku Math. J.,15 (1963), 62–78. Jos´e L. Cabrerizo, Alfonso Carriazo, Luis M. Fern´andez and Manuel Fern´andez: Departamento de Geometr´ıa y Topolog´ıa. Facultad de Matem´aticas. Universidad de Sevilla. Apartado de Correos 1160. 41080–Sevilla. Spain. e–mail: [email protected] 17