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Spinors and isometric immersions of surfaces in 4-dimensional products.

1 Introduction

In [4], Friedrich gave a spinorial characterization of surfaces in the Euclidean 3-space. Namely, he proved that the existence of a so-called generalized Killing spinor [psi] on a surface ([M.sup.2], g), that is

[nabla]x[psi] = A(X) x [psi],

where A is a symmetric (1,1)-tensor, is equivalent to the Gauss and Codazzi equations and therefore to an isometric immersion of ([M.sup.2], g) into [R.sup.3] with -2A as shape operator. Later on, Morel generalized in [9] this result for surfaces of the sphere [S.sup.3] and the hyperbolic space [H.sup.3] and we give in [12] an analogue for 3-dimensional homogeneous manifolds with 4-dimensional isometry group, as well as for surfaces into pseudo-Riemannian space forms [6] and Lorentzian products [13]. In a more recent work [2], we studied with Bayard and Lawn the spinorial characterization of surfaces into 4-dimensional space forms.

In this paper, we extend this spinorial characterization for surfaces in the product spaces [M.sup.3] (c) x R and [M.sup.2](c) x [R.sup.2], where [M.sup.n] (c) is the complete simply connected n-dimensional real space form of constant sectional curvature c [not equal to] 0. First we characterize immersions of surfaces into these product spaces by the existence of special spinor fields satisfying an appropriate generalized Killing-type equation, that is an equation involving the spinorial connection (see Theorem 3.1). Then, we show that this equation is equivalent to the corresponding Dirac equation with an additional condition on the norm of the spinor field (see Proposition 4.1 and Corollary 4.2).

2 Preliminaries

In this section of preliminaries, we will first recall some basics about surfaces into the product spaces [M.sup.2](c) x [R.sup.2] and [M.sup.3](c) x R. In particular, we will recall the compatibility equations assuring that a surface is isometrically immersed into one of these spaces. Then, we will give some facts about restrictions of spinors on a surface into a 4-dimensional space and deduce the particular spinor fields with which we will work in the sequel.

2.1 Compatibilty equations

Let ([M.sup.2], g) be a Riemannian surface isometrically immersed into the product space P = [M.sup.2] (c) x [R.sup.2] or [M.sup.3](c) x R, endowed with the product metric [??]. For more convinience, we will denote in the sequel all the metrics and also hermitian products on spinor bundles by the same classical notation (*, *) (no confusion is possible). We denote by F the product structure of P. The map F : TP [right arrow] TP is defined by F([X.sub.1] + [X.sub.2]) = [X.sub.1] - [X.sub.2], where [X.sub.1] belongs to the first factor (T[M.sup.2] (c) or T[M.sup.3] (c)) and [X.sub.2] belongs to the second factor (T[R.sup.2] or TR). Obviously, F satisfies

[F.sup.2] = Id (and F [not equal to] Id), (1)

[??](FX, Y) = [??](X, FY), (2)

[??] F = 0. (3)

Moreover, we recall that the curvature of (P, [??]) is given by

[??](X, Y)Z = - c/4[<Y,Z>X - <X,Z>Y + <FY,Z>FX - <FX,Z>FY + (Y, Z) FX - <X, Z> FY + <Y, FZ> X - <X, FZ> Y] (4)

This product structure F induces the existence of the following four operators

f : TM [right arrow] TM, h : TM [right arrow] NM, s : NM [right arrow] TM and t : NM [right arrow] NM

defined for any X [member of] TM and [xi] [member of] NM by

FX = fX + hX and F[xi] = s[xi] + t[xi]. (5)

From Equations (1) and (2), f and t are symmetric and we have the following relations between these four operators

[f.sup.2] X = X - shX, (6)

[t.sup.2] [xi] = [xi] - hs[xi], (7)

fs[xi] + st[xi] = 0, (8)

hfX + thX = 0, (9)

[??](hX, [xi]) = [??](X, s[xi]), (10)

for any X [member of] r(TM) and [xi] [member of] [GAMMA](NM). Moreover, from Equation (3), we have

([[nabla].sub.x]f)Y = AhyX + s(B(X, Y)), (11)

[[nabla].sup.[perpendicular to].sub.X] (hY) - h([[nabla].sub.x]Y) = t(B(X, Y)) - B(X,fY), (12)

[[nabla].sup.[perpendicular to]] (l[xi]) - t([[nabla].sup.[perpendicular to].sub.X][xi]) = -B(s[xi], X) - h([A.sub.[xi]]X), (13)

[nabla]x(s[xi]) - s([[nabla].sup.[perpendicular to].sub.X][xi]) = -f ([A.sub.[xi]]X) + [A.sub.t[xi]]X, (14)

where B : TM x TM [right arrow] NM is the second fundamental form and for any [xi] [member of] TM, [A.sub.[xi]] is the Weingarten operator associated with [xi] and defined by g([A.sub.[xi]]X, Y) = [??](B(X, Y), [xi]) for any vectors X, Y tangent to M.

Finally, from (4), we deduce that the Gauss, Codazzi and Ricci equations are respectively given by

R(X, Y)Z = c/4 [<Y,Z>X- <X,Z>Y + <fY,Z> fX- <fX,Z> fY <Y, Z> fX - <X, Z> fY + <Y, fZ> X - <X, fZ> Y + [A.sub.B(Y,Z)]X - [A.sub.B(X,Z)]Y, (15)

([nabla]xB)(y,Z)-([[nabla].sub.y]B)(X,Z) = c/4 [<fY,Z> hX - <fX,Z> hY + <Y, Z> hX - <X, Z> hY], (16)

[R.sup.[perpendicular to]] (X, Y)[xi] = c/4 [<hY,[xi]>hX - <hX,Z>hY] + B([A.sub.[xi]]Y,X)-B([A.sub.[xi]]X,Y). (17)

Conversely, let ([M.sup.2], g) a Riemannian surface endowed with a rank 2 vector bundle E endowed with a metric and a compatible connection [[nabla].sup.[perpendicular to]]. Assume that there exist some tensors f, h, s, t and B satisfying Equations (6)-(13) (note that (14) is not required since it is the dual equation of (12)) and the Gauss-Codazzi-Ricci equations (15)-(17). Moreover we define the operator F : TM [direct sum] E [right arrow] TM [direct sum] E by relations (5). If in addition the operator F satisfy that the ranks of the maps F+Id/2 and F-Id/2 are 2 and 2 (resp. 3 and 1), then Kowalczyk [5] and De Lira-Tojeiro-Vitorio [8] proved independently that there exists a local isometric immersion from (M, g) into [M.sup.2] (c) x [R.sup.2] (resp. [M.sup.3](c) x R) with E as normal bundle, B as second fundamental form and such that the product structure of [M.sup.2] (c) x [R.sup.2] (resp. [M.sup.3] (c) x R) coincide with F over M. This immersion becomes global if M is simply connected. Note that this was previously proven in a more abstract way by Piccione and Tausk [11].

2.2 Spinors on surfaces of P

For details about the recalls of this section, the reader can refer to [1] for instance. Let ([M.sup.2], g) be an oriented Riemannian surface, with a given spin structure, and E an oriented and spin vector bundle of rank 2 on M. We consider the spinor bundle [SIGMA] over M twisted by E and defined by

[SIGMA] = [SIGMA]M [cross product] [SIGMA]E,

where [SIGMA]M and [SIGMA]E are the spinor bundles of M and E respectively. We endow [SIGMA] with the spinorial connection [nabla] defined by

V = [[nabla].sup.[SIGMA]M] [cross product] [Id.sub.[SIGMA]E] + [Id.sub.[SIGMA]M] [cross product] [[nabla.sup.[SIGMA]E].

We also define the Clifford product * by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all [phi] = [alpha] [cross product] [sigma] [member of] [SIGMA]M [cross product] [SIGMA]E, where [*.sub.M] and [*.sub.E] denote the Clifford products on [SIGMA]M and on [SIGMA]E respectively and where [bar.[sigma]] = -[[sigma].sup.+] [[sigma].sup.-] for the natural decomposition of [SIGMA]E = [[SIGMA].sup.+]E [direct sum] [[SIGMA].sup.-]E. Here, [[SIGMA].sup.+]E and [[SIGMA].sup.-]E are the eigensubbundles (for the eigenvalue 1 and -1) of [SIGMA]E for the action of the normal volume element [[omega].sub.[perpendicular] to] = i[[xi].sub.1] x [[xi].sub.2], where {[[xi].sub.1], [[xi].sub.2]} is a local orthonormal frame of E. Note that [[SIGMA].sup.+]M and [[SIGMA].sup.-] are defined similarly by for the tangent volume element [omega] = [ie.sub.1] [*.sub.M] [e.sub.2]. We finally define the twisted Dirac operator D on [GAMMA]([SIGMA]) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where {[e.sub.1], [e.sub.2]} is an orthonormal basis of TM.,

We note that [SIGMA] is also naturally equipped with a hermitian scalar product (.,.) which is compatible with the connection [nabla], since so are [SIGMA]M and [SIGMA]E, and thus also with a compatible real scalar product Re(.,.). We also note that the Clifford product * of vectors belonging to TM [direct sum] E is antihermitian with respect to this hermitian product. Finally, we stress that the four subbundles [[SIGMA].sup.[+ or -][+ or -]] := [[SIGMA].sup.[+ or -]]M 0 [[SIGMA].sup.[+ or -]]E are orthogonal with respect to the hermitian product. We will also consider [[SIGMA].sup.+] = [[SIGMA].sup.++] [direct sum] [[SIGMA].sup.--] and [[SIGMA].sup.-] = [[SIGMA].sup.+-] [direct sum] [[SIGMA].sup.-+]. Throughout the paper we will assume that the hermitian product is C-linear w.r.t. the first entry, and C-antilinear w.r.t. the second entry.

Now, let (P, [??]) be a 4-dimensional spin manifold. It is a well-known fact that there is an identification between the spinor bundle [SIGMA][P.suB.|M] of P over M, and the spinor bundle of M twisted by the normal bundle [SIGMA] := [SIGMA]M [cross product] [SIGMA]E. Moreover, we have the spinorial Gauss formula: for any [phi] [member of] [GAMMA](E) and any X [member of] TM,

[[??].sup.X[phi]] = [[nabla].sub.X[phi]] + [1/2] [summation over (j=1,2)] [e.sub.j] x B(X,[e.sub.j]) x [phi] (18)

where [??] is the spinorial connection of [SIGMA]P and [nabla] is the spinoral connection of [SIGMA] defined as above and {[e.sub.1], [e.sub.2]} is a local orthonormal frame of TM. We will also use this notation and {[[xi].sub.1], [[xi].sub.2]} for a local orthonormal frame of E. Here x is the Clifford product on P.

From now on, we will take P = [M.sup.2] (c) x [R.sup.2] or [M.sup.3](c) x R. By restriction of a parallel spinor of the Euclidean space [R.sup.5] if c > 0 or the Lorentzian space [R.sup.4,1] if c < 0, we obtain on P a spinor field [phi] satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [alpha] [member of] C so that 4[[alpha].sup.2] = c. In other words, for any X [member of] [GAMMA](TP), we have

[[??].sub.x[phi]] = [alpha]/2 (X + FX) x [phi].

Hence, by the spinorial Gauss formula (18), the restriction of [phi] on M satisfies

[[nabla].sub.x[phi]] = [alpha]/2(X + fX + hX) x [phi] + [zeta](X) x [phi], (19)

where [zeta](X) = - [1/2] [2.summation over (j=1)] [e.sub.j] x B ([e.sub.j], X).

3 Main result

Now, we have the ingredients to state the the main result of this note.

Theorem 3.1. Let c [member of] R, c [not equal to] 0 and a [member of] C such that 4[[alpha].sup.2] = c. Let ([M.sup.2], g) be an oriented Riemannian surface and E an oriented and spin vector bundle of rank 2 over M with scalar product (*,*)E and compatible connection [[nabla].sup.E]. We denote by [SIGMA] = [SIGMA]M [cross product] [SIGMA]E the twisted spinor bundle. Let B : TM x TM [right arrow] E a bilinear symmetric map and

f : TM [right arrow] TM, h : TM [right arrow] E, s : E [right arrow] TM and t: E [right arrow] E

satisfying Equations (6)-(13). Moreover we assume that the rank of the maps F+Id/2 and F-Id/ 2is 2 and 2 (resp. 3 and 1), where F : TM [direct sum] E [right arrow] TM [direct sum] E is defined from f,h,s and t by relations (5). Then, the two following statements are equivalent

1. There exists a local isometric immersion of ([M.sup.2], g) into P = [M.sup.2] (c) x [R.sup.2] (resp. [M.sup.3] (c) x R) with E as normal bundle and second fundamental form B such that over M the product structure is given by f, h, t and s.

2. There exists a spinor field [phi] in [SIGMA] satisfying for all X [member of] [??](M)

[[nabla].sub.X[phi]] = [alpha]/2 (X + fX + hX) x [phi] + [zeta](X) x [phi],

such that [[phi].sup.+] and [[phi].sup.-] vanish nowhere.

Remark 3.2. The case c = 0, that is, for isometric immersions in [R.sup.4], has been treated in [2]. In that case, the operators f, h, s, t are not considered since we do not need the product structure of [R.sup.4] = [R.sup.2] x [R.sup.2].

Proof: First, we remark that the fact that (1) implies (2) has been proved in the discussion of Section 2. The work consists in proving that (2) implies (1). The computations are in the same spirit as in [2], with some technical difficulties due to the terms arising from the product structure. We will emphasize on these differences. We have to compute the spinorial curvature of the particular spinor [phi]. For this, let us compute R([e.sub.1], [e.sub.2])[[phi], where ([e.sub.1], [e.sub.2]) is a local orthonormal frame of TM. We also denote by ([e.sub.3], [e.sub.4]) a local orthonormal frame of E. Then, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where we denote [d.sup.[nabla]][zeta](X, Y) = [[nabla].sub.X]([zeta](Y)) - [[nabla].sub.Y]([zeta](X)) - [zeta]([X, Y]). First, by a straightforward computation, we see that the term

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

vanishes. Moreover, by Equations (11) and (12) and the fact that the Levi-civita is torsion-free, the term

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

also vanishes. Hence, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

But, as computed in [2] (Lemma 3.3), we have

R[([e.sub.1], [e.sub.2]).sub.[phi]] = - 1/2 [Ke.sub.1] x [e.sub.2] x [phi] - 1/2 [K.sub.N][e.sub.3] x [e.sub.4] x [phi], (20)

[d.sup.[nabla][zeta](X,Y) = 1/2 [2.summation over (j=1)] [e.sub.j] x (([[bar.[nabla]].sub.x]B)(Y/[e.sub.j]) -([[bar.[nabla]].sub.Y]B)(X/[e.sub.j]), (21)

where [bar.[nabla]] stands for the natural connection on T*M [cross product] T*M [cross product] E, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, we have

G x [phi] + R x [phi] + C x [phi] = 0,

where G, R and C are the 2-forms defined by

G = [K + <B([e.sub.1],[e.sub.1]),B([e.sub.2],[e.sub.2])> - [absolute value of B([e.sub.1],[e.sub.2])] + [[alpha].sup.2] (1 - [<[fe.sub.1], [e.sub.2]>.sup.2] + <f[e.sub.1], [e.sub.1]> <f[e.sub.2], [e.sub.2]>)] [e.sub.1] x [e.sub.2],

where K is the Gauss curvature of (M, g),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [K.sub.E] = <[R.sup.E] ([e.sub.1], [e.sub.2])[e.sub.3], [e.sub.4]>) is the curvature of the bundle E, and

C = 2[d.sup.[nabla]][zeta]([e.sub.1], [e.sub.2]) + [[alpha].sup.2] (f[e.sub.2] x h[e.sub.1] - [fe.sub.1] x [he.sub.2] + [e.sub.2] * [he.sub.1] - [e.sub.1] x [he.sub.2]).

As proved in [2] (Lemma 3.4), if T is a 2-form such that T * [phi] = 0 with [[phi].sup.+] and [[phi].sup.-] nowhere vanishing, then T = 0. Moreover, since G belongs to [[LAMBDA].sup.2]M [cross product] 1, R belongs to 1 [cross product] [[LAMBDA].sup.2]E and C is of mixed type, that is, belongs to TM [cross product] E, then each of these three parts are zero. But G = 0 is nothing else but

K + <B([e.sub.1],[e.sub.1]), B([e.sub.2],[e.sub.2])> - [absolute value of B([e.sub.1],[e.sub.2])] = - c/4 (1 - [<f[e.sub.1], [e.sub.1]>.sup.2] + <f[e.sub.1],[e.sup.1]> <f[e.sub.2],[e.sub.2]>),

that is the Gauss equation. Similarly, R = 0 is equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

that is the Ricci equation. Finally C = 0, gives the Codazzi equations. Indeed, since

[d.sup.[nabla][zeta](X,Y) = 1/2 [2.summation over (j=1)] [e.sub.j] x (([[bar.[nabla]].sub.x]B)(Y/[e.sub.j]) - ([[bar.[nabla]].sub.Y]B)(X/[e.sub.j]),

Thus, from C = 0, we deduce for j = 1,2

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

which are the Codazzi equations. Since in addition, we have assumed Equations (6)-(12), by the theorem of Kowalczyk and De Lira-Tojeiro-Vitorio, we get that ([M.sup.2], g) is isometrically immersed into P with B as second fundamental form and f, h, s and t coming from the product structure F of P. This concludes the proof.

Remark 3.3. Note that in the proof, we only use Equations (11) and (12) in the computations. The other Equations (6)-(10) and (11)-(13) are only needed to apply the theorem of Kowalczyk and De Lira-Tojeiro-Vitori, as well as the hypothesis on the rank of the maps F+Id/2 and F-Id/2.

4 The Dirac equation

Let [phi] be a spinor field satisfying Equation (19), then it satisfies the following Dirac equation

[D.sub.[phi]] = [??] x [phi] - [alpha]/2 [(2 + tr(f))[phi] - [beta] - [beta] x [phi]], (24)

where is the 2-form defined by = [beta] [summation over (i=1,2)] [e.sub.i] x [he.sub.i] = [summation over (i=j,1)] [h.sub.iy][e.sub.i] x [[xi].sub.j], where

[h.sub.i,j] = <[he.sub.i], [[xi].sub.j]>.

As in [2], we will show that this equation with an appropiate condition on the norm of both [[phi].sup.+] and [[phi].sup.-] is equivalent to Equation (19), where the tensor B is expressed in terms of the spinor field [phi] and such that tr (B) = 2[??]. Moreover, from Equation (19) we deduce the following conditions on the derivatives of [[absolute value of [[phi].sup.+]].sup.2] and [[absolute value of [[phi].sup.-]].sup.2]. Indeed, after decomposition onto [[SIGMA].sup.+] and [[SIGMA].sup.-], (19) becomes

[[nabla].sub.x[phi].sup.[+ or -]] = [[alpha]/2] (X + fX + hX) x ([[phi].sup.[??]] + [eta] (X) x [[phi].sup.[+ or -]].

From this we deduce that

X([[absolute value of [[phi].sup.[+ or -]]].sup.2]) = Re <[alpha](X + fX + hX) * [[phi].sup.[??]], [[phi].sup.[+ or -]]> (25)

Now, let [phi] a spinor field solution of the Dirac equation (24) with [[phi].sup.+] and [[phi].sup.-] nowhere vanishing and satisying the norm condition (25), we set for any vector fields X and Y tangent to M and [xi] [member of] E

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

Finally, we set B = [B.sup.+] + [B.sup.-]. Then, we have the following

Proposition 4.1. Let [phi] [member of] [LAMBDA]([SIGMA]) satisfying the Dirac equation (24)

D[phi] = [??] x [phi] - [[alpha]/2] [(2 + tr (f))[phi] - [beta] x [phi]]

such that

X([[absolute value of [[phi].sup.[+ or -]]].sup.2]) = Re <[alpha](X + fX + hX) x [[phi].sup.[??]], [[phi].sup.[+ or -]]

then [phi] is solution of Equation (19)

[[nabla].sub.x[phi]] = [[alpha]/2] (X + fX + hX) x [phi] + [zeta](X) x [phi],

where [zeta] is defined by [zeta](X) = 1/2 [2.summation over (j=1)] [e.sub.j] x B([e.sub.j], X).

For the sake of clarity, the proof of this proposition will be given in the next section. Now, combining this proposition with Theorem 3.1, we get the following corollary.

Corollary 4.2. Let c [member of] R, c [not equal to] 0 and [alpha] [member of] C such that 4[[alpha].sup.2] = c. Let ([M.sup.2], g) be an oriented Riemannian surface and E an oriented and spin vector bundle of rank 2 over M with scalar product [<*, *>.sub.E] and compatible connection [[nabla].sup.E]. We denote by [SIGMA] = [SIGMA]M [cross product] [SIGMA]E the twisted spinor bundle. Let f, h, s and s be some maps

f : TM [right arrow] TM, h : TM [right arrow] E, s : E [right arrow] TM and t : E [right arrow] E

satisfying Equations (6)-(10). Moreover we assume that the rank of the maps and F-Id/2 are 2 and 2 (resp. 3 and 1), where F : TM [direct sum] E [right arrow] TM [direct sum] E is defined by relations (5). Then, the two following statements are equivalent

1. There exists an isometric immersion of ([M.sup.2], g) into [M.sup.2] (c) x [R.sup.2] (resp. [M.sup.3](c) x R) with E as normal bundle and mean curvature H such that over M the product strcuture is given by f, h, t and s.

2. There exists a spinor field [phi] in E solution of the Dirac equation

D[phi] = [??] x [phi] - [alpha]/2 [(2 + tr(f))[phi] - [beta] x [phi]]

such that [[phi].sup.+] and [[phi].sup.-] never vanish, satisfy the norm condition (25) and such that the maps f, h, s, t and the tensor B defined by (26) and (27) satisfy relations (11)-(13).

5 Proof of Proposition 4.1

First, we decompose the Dirac equation (24) on the four spinor subbundles [[SIGMA].sup.++], [[SIGMA].sup.--], [[SIGMA].sup.+-] and [[SIGMA].sup.-+]. We get the following four equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, we fix a point p [member of] M, and consider [e.sub.3] a unit vector in [E.sub.p] so that the mean curvature vector is given by [??] = [absolute value of [??]] [e.sub.3] at p. We complete [e.sub.3] by [e.sub.4] to get a positively oriented and orthonormal frame of [E.sub.p]. First, we assume that [[phi].sup.--], [[phi].sup.++], [[phi].sup.+-] and [[phi].sup.-+] do not vanish at p. It is easy to see that

{[e.sub.1] x [e.sub.3] x [[phi].sup.--]/[absolute value of [[phi].sup.--]], [e.sub.2] x [e.sub.3] x [[phi].sup.--]/ [absolute value of [[phi].sup.--]]}

is an orthonormal frame of [[SIGMA].sup.++] for the real scalar product Re <*, *>. Indeed, we have

Re<[e.sub.1] x [e.sub.3] x [[phi].sup.--],[e.sub.2] x [e.sub.3] x [[phi].sup.--]> = Re <[[phi].sup.--], [e.sub.3] x [e.sub.1] x [e.sub.2] x [e.sub.3] x [[phi].sup.--]) = Re (i[[absolute value of [[phi].sup.--]].sup.2]) = 0.

Of course, by the same argument,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are orthonormal frames of [[SIGMA].sup.--], [[SIGMA].sup.+-] and [[SIGMA].sup.-+] respectively. We define the following bilinear forms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We have this first lemma:

Lemma 5.1. We have

1. tr([F.sub.++]) = -[absolute value of [??]][[absolute value of [[phi].sup.--]].sup.2] + 1/2 Re([alpha](2 + tr(f))[[phi.sup.]-+] + [alpha][beta] x [[phi].sup.+-], [e.sub.3] x [[phi].sup.--]>,

2. tr([F.sub.--]) = -[absolute value of [??]][[absolute value of [[phi].sup.++]].sup.2] + 1/2 Re([alpha](2 + tr(f))[[phi.sup.]+-] + [alpha][beta] x [[phi].sup.-+], [e.sub.3] x [[phi].sup.++]>,

3. tr([F.sub.+-]) = -[absolute value of [??]][[absolute value of [[phi].sup.-+]].sup.2] + 1/2 Re([alpha](2 + tr(f))[[phi.sup.]++] + [alpha][beta] x [[phi].sup.--], [e.sub.3] x [[phi].sup.-+]>,

4. tr([F.sub.-+]) = -[absolute value of [??]][[absolute value of [[phi].sup.+-]].sup.2] + 1/2 Re([alpha](2 + tr(f))[[phi.sup.]--] + [alpha][beta] x [[phi].sup.++], [e.sub.3] x [[phi].sup.+-]>,

Proof: We only compute the trace of [F.sub.++], the computations for the three other forms [F.sub.--], [F.sub.+-] and [F.sub.-+] are the same. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since D[[phi].sup.++] = [??] x [[phi].sup.--] - [alpha]/2 (2 + tr(f))[[phi].sup.++] + [alpha]/2 [beta] x [[phi].sup.+-], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This concludes the proof.

Now, we have this second lemma which gives the defect of symmetry:

Lemma 5.2. We have

1. [F.sub.++]([e.sub.1],[e.sub.2]) = [F.sub.++]([e.sub.2],[e.sub.1]) - 1/2 Re<(2 + tr(f))[[phi].sup.-+] - [alpha][beta] x [[phi].sup.+-], [e.sub.4] x [[phi].sup.--]>,

2. [F.sub.--]([e.sub.1],[e.sub.2]) = [F.sub.--]([e.sub.2],[e.sub.1]) - 1/2 Re<(2 + tr(f))[[phi].sup.+-] - [alpha][beta] x [[phi].sup.-+], [e.sub.4] x [[phi].sup.++]>,

3. [F.sub.+-]([e.sub.1],[e.sub.2]) = [F.sub.-+]([e.sub.2],[e.sub.1]) - 1/2 Re<(2 + tr(f))[[phi].sup.--] - [alpha][beta] x [[phi].sup.++], [e.sub.4] x [[phi].sup.-+]>,

4. [F.sub.-+]([e.sub.1],[e.sub.2]) = [F.sub.-+]([e.sub.2],[e.sub.1]) - 1/2 Re<(2 + tr(f))[[phi].sup.--] - [alpha][beta] x [[phi].sup.++], [e.sub.4] x [[phi].sup.-+]>,

Proof: As for the proof of the previous lemma, we only give the details for [F.sub.++]. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first term is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where we have use that [ie.sub.1] x [e.sub.2] x [[phi].sup.--] = -[[phi].sup.--], that is, [e.sup.1] x [e.sub.2] x [phi] = i[[phi].sup.--] and [??] = [absolute value of H] [e.sub.3]. Moreover, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Finally, since [[phi].sup.--] [member of] [[SIGMA].sup.+], we have [[phi].sub.4] x [[phi].sup.--] = [[phi].sup.--], which implies [e.sub.1] x [e.sub.2] x [e.sub.3] x [[phi].sup.--] = -[e.sub.4] x [[phi].sup.--] and we get

[F.sub.++]{[e.sub.1],[e.sub.2]) = [F.sub.++]{[e.sub.2],[e.sub.1]) - 1/2 Re <[alpha](2 + tr(f))[[phi].sup.-+] - [alpha][beta] x [[phi].sup.+-], [e.sub.4] x [[phi].sup.--]).

The proof is similar for the three other forms.

By analogous computations, we also get the following lemmas. We do not give the proof which is similar to the two previous ones.

Lemma 5.3. We have

1. tr([B.sub.++]) = - 1/2 Re<[alpha](2 + tr(f))[[phi].sup.-+] + [alpha][beta] x [[phi].sup.+-], [e.sub.3] x [[phi].sup.--],

2. tr([B.sub.--]) = - 1/2 Re<[alpha](2 + tr(f))[[phi].sup.+-] + [alpha][beta] x [[phi].sup.-+], [e.sub.3] x [[phi].sup.++],

3. tr([B.sub.+-]) = - 1/2 Re<[alpha](2 + tr(f))[[phi].sup.++] + [alpha][beta] x [[phi].sup.--], [e.sub.3] x [[phi].sup.-+],

4. tr([B.sub.-+]) = - 1/2 Re<[alpha](2 + tr(f))[[phi].sup.--] + [alpha][beta] x [[phi].sup.++], [e.sub.3] x [[phi].sup.+-],

Lemma 5.4. We have

1. [B.sub.++]([e.sub.1][e.sub.2]) = [B.sub.++]([e.sub.2],[e.sub.1]) + 1/2 Re<(2 + tr(f))[[phi].sup.-+] - [alpha][beta] x [[phi].sup.+-], [e.sub.4] x [[phi].sup.--]),

2. [B.sub.--]([e.sub.1][e.sub.2]) = [B.sub.--]([e.sub.2],[e.sub.1]) + 1/2 Re<(2 + tr(f))[[phi].sup.+-] - [alpha][beta] x [[phi].sup.-+], [e.sub.4] x [[phi].sup.++]),

3. [B.sub.+-]([e.sub.1][e.sub.2]) = [B.sub.+-]([e.sub.2],[e.sub.1]) + 1/2 Re<(2 + tr(f))[[phi].sup.++] - [alpha][beta] x [[phi].sup.--], [e.sub.4] x [[phi].sup.+-]),

4. [B.sub.-+]([e.sub.1][e.sub.2]) = [B.sub.++]([e.sub.2],[e.sub.1]) + 1/2 Re<(2 + tr(f))[[phi].sup.--] - [alpha][beta] x [[phi].sup.++], [e.sub.4] x [[phi].sup.-+]),

Now, we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[F.sub.+] = [A.sub.++]/[[absolute value of [[phi].sup.--]].sup.2] - [A.sub.--]/[[absolute value of [[phi].sup.++]].sup.2] and [F.sub.-] = [A.sub.+-]/[[absolute value of [[phi].sup.-+]].sup.2] - [A.sub.+-]/[[absolute value of [[phi].sup.+-]].sup.2]

From the last four lemmas we deduce immediately that [F.sup.+] and [F.sup.-] are symmetric and trace-free. Moreover, by a direct computation using the conditions (25) on the norms of [[phi].sup.+] and [[phi].sup.-], we get the following lemma:

Lemma 5.5. The symmetric operators [F.sup.+] and [F.sup.-] of TM associated to the bilinear forms [F.sup.+] and [F.sup.-], defined by

[F.sup.+](X) = [F.sub.+](X, [e.sub.1])[e.sub.1] + [F.sup.+](X, [e.sub.2])[e.sub.2] and [F.sup.-] (X) = [F.sub.-] (X, [e.sub.1])[e.sub.1] + [F.sub.-] (X, [e.sub.2])[e.sub.2]

for all X [member of] TM, satisfy

1. Re ([F.sup.+](X) x [e.sub.3] x [[phi].sup.--], [[phi].sup.++]) = 0,

2. Re ([F.sup.-] (X) x [e.sub.3] x [[phi].sup.-+], [[phi].sup.+-]) = 0.

Proof. First, we have

[A.sub.++](X, Y) = Re <[[nabla].sub.X[phi].sup.++] - [alpha](X + fX) x [[phi].sup.-+] + [alpha]hX x [[phi].sup.+-], Y x [e.sub.3] x [[phi].sup.--]>,

Since ([e.sub.1] x [e.sub.3] x [[phi].sup.--]/[absolute value of [[phi].sup.--]] [e.sub.2] x [e.sub.3] x [[phi].sup.-- ]/ [absolute value of [[phi].sup.--]]) is an orthonormal frame of [[SIGMA].sup.++], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Summing these two formulas imply that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By the condition (25) on the derivative of the norm of [[phi].sup.+], this last expression is zero. The proof of the second relation is similar.

Hence, the operators [F.sup.+] and [F.sup.-] are of rank at most [less than or equal to] 1. Since they are symmetric and trace-free, they vanish identically.

Using again that ([e.sub.1] x [e.sub.3] x [[phi].sup.--]/[absolute value of [[phi].sup.--]] [e.sub.2] x [e.sub.3] x [[phi].sup.--]/[absolute value of [[phi].sup.--]]) is an orthonormal frame of [[SIGMA].sup.++], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [F.sub.++] = [A.sub.++] - [B.sub.++] and denoting by [A.sub.++] and [B.sub.++] the operators of TM associated to [A.sub.++] and [B.sub.++] and defined by

[A.sub.++](X) = [A.sub.++](X,[e.sub.3])[e.sub.3] + [A.sub.++](X,[e.sub.2])[e.sub.2], [B.sup.++](X) = [B.sub.++](X,[e.sub.3])[e.sub.3] + [B.sub.++](X,[e.sub.2])[e.sub.2],

we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

Similarly, we denote by [A.sup.--] and [B.sup.--] the operators of TM associated to [A.sub.--] and [B.sub.--]. Thus, we have

[[nabla].sub.x[phi].sup.--] = 1/[[absolute value of [[phi].sup.++].sup.2] [[A.sup.--](X) x [e.sub.3] x [[phi].sup.++] - [B.sup.++](X) x [e.sub.3] x [[phi].sup.++]]. (29)

Moreover, we easily get

[B.sup.++](X) x [e.sub.3] x [[phi].sup.--] = - 1/2 [[absolute value of [[phi].sup.--].sup.2] ([alpha](X + fX) x [[phi].sup.-+] + [alpha]hX x [[phi].sup.+-])

and

[B.sup.--](X) x [e.sub.3] x [[phi].sup.++] = - 1/2 [[absolute value of [[phi].sup.++].sup.2] ([alpha](X + fX) x [[phi].sup.+-] + [alpha]hX x [[phi].sup.-+])

Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, we set [A.sup.+] = [A.sup.++] + [A.sup.--]. From the definition of [A.sub.++] and [A.sup.--] and since [F.sup.+] = 0, we have [A.sup.++]/[[absolute value of [[phi].sup.--]].sup.2] = [A.sup.--]/[[absolute value of [[phi].sup.++]].sup.2] hearing in mind that [[absolute value of [[phi].sup.+]].sup.2] = [[absolute value of [[phi].sup.++]].sup.2] + [[absolute value of [[phi].sup.--]].sup.2], we get finally

[A.sup.+]/[[absolute value of [[phi].sup.+]].sup.2] = [A.sup.++]/[[absolute value of [[phi].sup.+]].sup.2] = [A.sup.-- ]/ [[absolute value of [[phi].sup.++]].sup.2] (30)

So, we have

[nabla]x[[phi].sup.+] = 1/[[absolute value of [[phi].sup.+]].sup.2] [A.sup.+](X)-[e.sub.3] x [[phi].sup.+] + [alpha](X + fX + hX) x [[phi].sup.-]. (31)

Analogously, we set [A.sup.+-] and [A.sup.-+] the operators of TM associated to [A.sub.+-] and [A.sub.-+], and we denote [A.sup.-] = [A.sup.+-] + [A.sup.-+]. Using the fact that [F.sup.-] = 0 we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We now observe that formulas (31) and (32) also hold if [[phi].sup.++] or [[phi].sup.--], (resp. [[phi].sup.+-] or [[phi].sup.-+]) vanishes at p : indeed, assuming for instance that [[phi].sup.++] (p) = 0, and thus that [[phi].sup.--] (p) [not equal to] 0 since [[phi].sup.+] (p) [not equal to] 0, equation (28) holds, and, from the norm condition in (25), we have

Re <[[nabla].sub.xp.sup.--] - [alpha]/2 (X + fX) x [[phi].sup.+-] + [alpha]/2hX x [[phi].sup.-+], [[phi].sup.--]> = 0.

Since [[phi].sup.--]/[absolute value of [[phi].sup.--]], i [[phi].sup.--]/[absolute value of [[phi].sup.--]]) is an orthonormal basis of [[SIGMA].sup.--], we deduce that

[[nabla].sub.x[phi].sup.--] - [alpha]/2(X + fX) x [[phi].sup.+-] + [alpha]/2hX x [[phi].sup.-+] = i[delta](X) [[phi].sup.--]/[absolute value of [[phi].sup.--]]

for some real 1-form [delta]. Moreover, since [[phi].sup.++] = 0 at p, we have

[D.sub.[phi].sup.--] + [alpha](2 + tr(f))[[phi].sup.+-] + [alpha][beta] x [[phi].sup.-+] = 0,

which implies

([delta]([e.sub.1])[e.sub.1] + [delta]([e.sub.2])[e.sub.2]) x = [[phi].sup.--]/[absolute value of [[phi].sup.--]] = 0,

and thus that [delta] = 0. We thus get [[nabla].sub.x[phi].sup.--] = [alpha]/2 (X + fX) x [[phi].sup.+-] + [alpha]/2 hX x [[phi].sup.-+] instead of (29), which, together with (28), easily implies (31).

Now, we set

[[eta].sup.+](x) = [(1/[[absolute value of [[phi].sup.+]].sup.2] [A.sup.+](X) x [e.sub.3]).sup.+] and [[eta].sup.-](X) = [(1/[[absolute value of [[phi].sup.-]].sup.2] [A.sup.-](X) x [e.sub.3]).sup.-]

where, if [sigma] belongs to [Cl.sup.0] (TM [direct sum] [xi]), we denote by [[sigma].sup.+] := 1+[[omega].sub.4]/2. [sigma] and by [[sigma].sup.-] := 1-[[omega].sub.4]/2 [sigma] the parts of [sigma] acting on [[SIGMA].sup.+] and on [[SIGMA].sup.-] only, i.e., such that

[[sigma].sup.+] x [phi] = [sigma] x [[phi].sup.+] [member of] [[SIGMA].sup.+] and [[sigma].sup.-] x [phi] = [sigma] x [[phi].sup.-] [member of] [[SIGMA].sup.-].

Setting [eta] = [[eta].sup.+] + [[eta].sup.-] we thus get

[[nabla].sup.x[phi]] = [eta](X) x [phi] + [alpha]/2(X + fX + hX) x p. (33)

as claimed in Proposition 4.1.

Now, we will compute [eta] explicitely. For this, we set [A.sub.+] (X, Y) := ([A.sup.+] (X), Y) and [A.sub.-](X, Y) := ([A.sup.-](X), Y). Then, the form n is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with

[A.sub.+] (X, Y) = Re <[[nabla].sub.x[phi].sup.+] - [alpha]/2(X + fX + hX) x [[phi].sup.-], Y x [e.sub.3] x [[phi].sup.+]]

and

[A.sub.-] (X, Y) = Re <[[nabla].sub.x[phi].sup.-] - [alpha]/2(X + fX + hX) x [[phi].sup.+], Y x [e.sub.3] x [[phi].sup.-]]

Moreover, we set for any vectors X and Y tangent to M,

C(X, Y) := X * [eta](Y) - [eta](Y) * X.

C(X, Y) is a vector belonging to E which is such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all [xi] [member of] E.

Lemma 5.6. The operator B defined above is symmetric in X and Y.

Proof: The proof is analogous to the symmetry of [A.sub.++] proven above and uses the Dirac equations

[D.sub.[phi].sub.+] = [??] x [[phi].sup.+] - [alpha] [(2 + tr(f))[[phi].sup.-] - [beta] x [[phi].sup.-]]

and

[D.sub.[phi].sub.-] = [??] x [[phi].sup.-] - [alpha] [(2 + tr(f))[[phi].sup.+] - [beta] x [[phi].sup.+]]

Now, computing

<C(X, Y), [xi]) = 1/2 (<C(X, [xi]), [xi]> + (C(Y, X), [xi]>)

we finally obtain that C is in fact equal to the tensor B defined in the discussion of Section 4.

Since B([e.sub.j], X) = C([e.sub.j], X) = [e.sub.j] x [eta](X) - [eta] (X) x [e.sub.j], we obtain

[summation over (j=1,2)] [e.sub.j] x B([e.sub.j], X) = -2[eta](X) - [summation over (j=1,2)] [e.sub.j], x [eta](X) x [e.sub.j]. (34)

Writing [eta](X) in the form [summation over (k=1,2)] [e.sub.k] * [[eta].sub.k] for some vectors [[eta].sub.k] belonging to E, we easily get that [summation over (j=1,2)] [e.sub.j], x [eta] (X) x [e.sub.j], = 0. Indeed, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, from (34), we get

[eta](X) = - 1/2 [summation over (j=1,2)] [e.sub.j] x B([e.sub.j],X) = [zeta](X).

Therefore, Equation (33) becomes

[[nabla].sub.x[phi]] = [zeta](X) x [phi] + [alpha]/2(X + fX + hX) x [phi],

and the last claim in Proposition 4.1 is now proved.

Received by the editors in November 2013.

Communicated by F. Bourgeois.

2010 Mathematics Subject Classification : 53C27, 53C42.

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[3] P. Bayard, On the spinorial representation of spacelike surfaces into 4-dimensional Minkowski space, J. Geom. Phys. 74 (2013), 289-313.

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[8] J.H. Lira, R. Tojeiro & F. Vitorio, A Bonnet theorem for isometric immersions into products of space forms Arch. Math. (Basel) 95 (5) (2010), 469-479

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[10] R. Nakad & J. Roth, Hypersurfaces of Spinc Manifolds and Lawson Correspondence, Ann. Glob Anal. Geom 42 (3) (2012), 421-442.

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LAMA, Universite Paris-Est Marne-la-Vallee, Cite Descartes, Champs sur Marne, 77454 Marne-la-Vallee cedex 2, France

email:julien.roth@u-pem.fr
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