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**COMPLEX AND LAGRANGIAN SURFACES OF THE** **COMPLEX PROJECTIVE PLANE VIA KÄHLERIAN**

**KILLING SPIN c SPINORS**

### Roger Nakad, Julien Roth

**To cite this version:**

### Roger Nakad, Julien Roth. COMPLEX AND LAGRANGIAN SURFACES OF THE COMPLEX

### PROJECTIVE PLANE VIA KÄHLERIAN KILLING SPIN c SPINORS. Journal of Geometry and

### Physics, Elsevier, 2017, 116, pp.316-329. �hal-01338103�

PROJECTIVE PLANE VIA K ¨AHLERIAN KILLING SPIN^{c}SPINORS

ROGER NAKAD AND JULIEN ROTH

ABSTRACT. The complex projective spaceCP^{2} of complex dimension2has aSpin^{c}
structure carrying K¨ahlerian Killing spinors. The restriction of one of these K¨ahlerian
Killing spinors to a Lagrangian or complex surfaceM^{2} characterizes the isometric im-
mersion ofMintoCP^{2}.

1. INTRODUCTION

A classical problem in Riemannian geometry is to know when a Riemannian manifold
(M^{n}, g)can be isometrically immersed into a fixed Riemannian manifold( ¯M^{n+p},g). The¯
case of space formsR^{n+1},S^{n+1}andH^{n+1}is well-known. In fact, the Gauss, Codazzi and
Ricci equations are necessary and sufficient conditions. In other ambient spaces, the Gauss,
Codazzi and Ricci equations are necessary but not sufficient in general. Some additional
conditions may be required like for the case of complex space forms, products, warped
products or 3-dimensional homogeneous space for instance (see [8, 9, 19, 22, 30, 33]).

In low dimensions, especially for surfaces, another necessary and sufficient condition is
now well-known, namely the existence of a special spinor field calledgeneralized Killing
spinor field(see [10, 27, 21, 23]). These results are the geometrically invariant versions of
previous work on the spinorial Weierstrass representation by R. Kusner and N. Schmidt,
B. Konoplechenko, I. Taimanov and many others (see [20, 18, 35]). This representation
was expressed by T. Friedrich [10]) for surfaces in R^{3} and then extended to other 3-
dimensonal (pseudo-)Riemannian manifolds ([27, 33, 32, 24]) as well as for hypersurfaces
of 4-dimensional space forms and products [23] or hypersurfaces of 2-dimensional com-
plex space forms by the mean ofSpin^{c}spinors [29].

More precisely, the restrictionϕof a parallel spinor field onR^{n+1}to an oriented Riemann-
ian hypersurfaceM^{n}is a solution of a generalized Killing equation

∇Xϕ=−1

2A(X))ϕ, (1)

where“·”and∇are respectively the Clifford multiplication and the spin connection on
M^{n}, andAis the Weingarten tensor of the immersion. Conversely, T. Friedrich proved in
[10] that, in the two dimensional case, if there exists a generalized Killing spinor field sat-
isfying Equation (1), whereAis an arbitrary field of symmetric endomorphisms ofT M,
then A satisfies the Codazzi and Gauss equations of hypersurface theory and is conse-
quently the Weingarten tensor of a local isometric immersion ofM intoR^{3}. Moreover, in
this case, the solutionϕof the generalized Killing equation is equivalently a solution of the
Dirac equation

Dϕ=Hϕ, (2)

where|ϕ|is constant andH is a real-valued function (which is the mean curvature of the
immersion inR^{3}).

2010Mathematics Subject Classification. 53C27, 53C40, 53D12, 53C25.

Key words and phrases. Spin^{c}structures, K¨ahlerian Killing Spin^{c}spinors, isometric immersions, complex
and Lagrangian surfaces, Dirac operator.

1

More recently, this approach was adapted by the second author, P. Bayard and M.A. Lawn in
codimension two, namely, for surfaces in Riemannian 4-dimensional real space forms [4],
and then generalized in the pseudo-Riemannian setting [3, 5] as well as for 4-dimensional
products [34]. As pointed out in [31], this approach coincides with the Weierstrass type
representation for surfaces inR^{4}introduced by Konopelchenko and Taimanov [18, 36].

The aim of the present article is to provide an analogue for the complex projective space
CP^{2}. The key point is that contrary to the case of hypersurfaces ofCP^{2}, which is consid-
erered in [29], the use ofSpin^{c}parallel spinors is not sufficient. Indeed, for both canonical
or anti-canonicalSpin^{c}structures, the parallel spinors are always in the positive half-part
of the spinor bundle. But, as proved in [4] or [34], a spinor with non-vanishing positive
and negative parts is required to get the integrability condition to get an immersion in the
desired target space. For this reason,Spin^{c}parallel spinors are not adapted to our problem.

Thus, we make use of real K¨ahlerian Killing spinors. Therefore, our argument holds also
for the complex projective space and not for the complex hyperbolic space, sinceCH^{2}does
not carry a real or imaginary K¨ahlerian Killing spinor.

We will focus on the case of complex and Lagrangian immersions intoCP^{2}. These two
cases and especially the Lagrangian case are of particular interest in the study of surfaces
inCP^{2}(see [14, 15, 37] and references therein for instance). Namely, we prove the two
following results.

The first theorem gives a spinorial characterization of complex surfaces in the complex
projective spaceCP^{2}.

Theorem 1.1. Let(M^{2}, g)be an oriented Riemannian surface andEan oriented vector
bundle of rank2 overM with scalar producth·,·iE and compatible connection∇^{E}. We
denote byΣ = ΣM⊗ΣEthe twisted spinor bundle. LetB:T M×T M −→Ea bilinear
symmetric map,j :T M −→T M a complex structure onM andt:E−→Ea complex
structure onE. Assume moreover thatt(B(X, Y)) = B(X, jY),∀X ∈ T M. Then, the
two following statements are equivalent

(1) There exists aSpin^{c}structure onΣM⊗ΣEwithΩ^{M}^{+E}(e1, e2) = 0and a spinor
fieldϕ∈Γ(ΣM⊗ΣE)

∇Xϕ = −1

2η(X)·ϕ−1

2X·ϕ+ i

2jX·ϕ, (3)

such thatϕ^{+}andϕ^{−}never vanish and whereηis given by
η(X) =

2

X

j=1

ej·B(ej, X).

(2) There exists a local isometriccompleximmersion of(M^{2}, g)intoCP^{2}withEas
normal bundle and second fundamental formBsuch the complex structure ofCP^{2}
overM is given byjandt(in the sense of Proposition 3.2).

The spinorsϕ^{+}andϕ^{−}denote respectiveley the positive and negative half part ofϕdefined
Section2. They are the projections ofϕon the eigensubspaces for the eigenvalues+1and

−1of the complex volume form andϕ¯is defined byϕ¯=ϕ_{+}−ϕ_{−}.

The second theorem is the analogue of Theorem 1.1 for Lagrangian surfaces intoCP^{2}.
Theorem 1.2. Let(M^{2}, g)be an oriented Riemannian surface andEan oriented vector
bundle of rank2 overM with scalar producth·,·i_{E} and compatible connection∇^{E}. We
denote byΣ = ΣM⊗ΣEthe twisted spinor bundle. LetB:T M×T M −→Ea bilinear
symmetric map,h:T M −→Eands:E −→T Mthe dual map ofh. Assume moreover
thathandsare parallel,hs=−idE andAhYX +s(B(X, Y)) = 0, for allX ∈T M,
whereAν :T M −→T M if defined byg(AνX, Y) =hB(X, Y), νiEfor allX, Y ∈T M
andν ∈E. Then, the two following statements are equivalent

(1) There exists aSpin^{c} structure onΣM ⊗ΣE withΩ^{M}^{+N}(e_{1}, e_{2}) = −2and a
spinor fieldϕ∈Γ(ΣM ⊗ΣE)satisfying for allX ∈X(M)

∇Xϕ = −1

2η(X)·ϕ−1

2X·ϕ+ i

2hX·ϕ, (4)

such thatϕ^{+}andϕ^{−}never vanish and whereηis given by
η(X) =

2

X

j=1

ej·B(ej, X).

(2) There exists a local isometricLagrangianimmersion of(M^{2}, g)intoCP^{2}withE
as normal bundle and second fundamental formBsuch that overM the complex
structure ofCP^{2}is given byhands(in the sense of Proposition 3.2).

Note that in the statements of both theoremϕ^{+}andϕ^{−}denotes respectiveley the positive
and negative half part ofϕdefined Section . They are the projections ofϕon the eigensub-
spaces for the eigenvalues+1and−1of the complex volume form.

2. PRELIMINARIES ANDNOTATIONS

In this section, we briefly review some basic facts about K¨ahler geometry andSpin^{c}struc-
tures on manifolds and their submanifolds. For more details we refer to [28, 2, 6, 25, 26,
16, 17, 7, 1, 11, 12, 13].

2.1. Spin^{c} structures on K¨ahler-Einstein manifolds. Let(M^{n}, g)be ann-dimensional
closed RiemannianSpin^{c}manifold and denote byΣM its complex spinor bundle, which
has complex rank equal to2^{[}^{n}^{2}^{]}. The bundleΣMis endowed with a Clifford multiplication
denoted by “·” and a scalar product denoted byh·,·i.

Given a Spin^{c} structure on (M^{n}, g), one can check that the determinant line bundle
det(ΣM)has a root L of index 2^{[}^{n}^{2}^{]−1}. This line bundle L over M is called the aux-
iliary line bundle associated with theSpin^{c}structure. From a topological point of view, a
Riemannian manifold has aSpin^{c}structure if and only if there exists a complex line bundle
L(which will be the auxiliary line bundle) onM such that

ω2(M) = [c1(L)]mod 2,

whereω_{2}(M)is the second Steifel-Whitney class ofM andc_{1}(L)is the first Chern class
ofL. In the particular case, when the line bundle has a square root, i.e.,ω_{2}(M) = 0, the
manifold is called aSpinmanifold. In this case, we denote byΣ^{0}M the spinor bundle or
theSpinbundle. It can be chosen such that the auxiliary line bundle is trivial.

Locally, aSpin^{c} structure always exists. In fact, the square root of the auxiliary line bun-
dle LandΣ^{0}M always exist locally. But,ΣM = Σ^{0}M ⊗L^{1}^{2} is defined globally. This
essentially means that, while the spinor bundle andL^{1}^{2} may not exist globally, their tensor
product (the Spin^{c} bundle) is defined globally. Thus, the connection∇ onSSM is the
twisted connection of the one on the spinor bundle (induced by the Levi-Civita connection)
and a fixed connectionAonL. TheSpin^{c}Dirac operatorDacting on the space of sections
ofΣM is defined by the composition of the connection∇with the Clifford multiplication.

In local coordinates:

D=

n

X

j=1

e_{j}· ∇_{e}_{j},

where{e_{j}}_{j=1,...,n}is a local orthonormal basis ofT M.Dis a first-order elliptic operator
and is formally self-adjoint with respect to theL^{2}-scalar product.

We recall that the complex volume elementω_{C}= ı^{[}^{n+1}^{2} ^{]}e1∧. . .∧enacts as the identity on
the spinor bundle ifnis odd. Ifnis even,ω^{2}_{C}= 1. Thus, under the action of the complex

volume element, the spinor bundle decomposes into the eigenspacesΣ^{±}M corresponding
to the±1eigenspaces, thepositive(resp.negative) spinors.

Every spin manifold has a trivialSpin^{c} structure, by choosing the trivial line bundle with
the trivial connection whose curvatureF_{A} vanishes. Every K¨ahler manifold(M^{2m}, g, J)
has a canonicalSpin^{c} structure induced by the complex structure J. The complexified
tangent bundle decomposes intoT^{C}M =T1,0M ⊕T0,1M,thei-eigenbundle (resp.(−i)-
eigenbundle) of the complex linear extension ofJ. For any vector fieldX, we denote by
X^{±} := ^{1}_{2}(X ∓iJ X)its component in T1,0M, resp.T0,1M. The spinor bundle of the
canonicalSpin^{c}structure is defined by

ΣM = Λ^{0,∗}M = ⊕^{m}

r=0Λ^{r}(T_{0,1}^{∗} M),

and its auxiliary line bundle isL = (KM)^{−1} = Λ^{m}(T_{0,1}^{∗} M), whereKM = Λ^{m,0}M is
the canonical bundle ofM. The line bundleLhas a canonical holomorphic connection,
whose curvature form is given by−iρ, whereρis the Ricci form defined, for all vector
fieldsX andY, byρ(X, Y) = Ric(J X, Y)andRicdenotes the Ricci tensor. Similarly,
one defines the so called anti-canonicalSpin^{c} structure, whose spinor bundle is given by
Λ^{∗,0}M =⊕^{m}_{r=0}Λ^{r}(T_{1,0}^{∗} M)and the auxiliary line bundle byK_{M}. The spinor bundle of any
otherSpin^{c}structure onM can be written as:

ΣM = Λ^{0,∗}M⊗L,

whereL^{2}=K_{M}⊗LandLis the auxiliary line bundle associated with thisSpin^{c}structure.

The K¨ahler formΩ, defined asΩ(X, Y) =g(J X, Y), acts onΣM via Clifford multipli- cation and this action is locally given by:

Ω·ψ= 1 2

2m

X

j=1

e_{j}·J e_{j}·ψ,

for allψ∈Γ(ΣM), where{e1, . . . , e2m}is a local orthonormal basis ofTM. Under this action, the spinor bundle decomposes as follows:

ΣM = ⊕^{m}

r=0

Σ_{r}M, (5)

whereΣ_{r}M denotes the eigenbundle to the eigenvaluei(2r−m)ofΩ, of complex rank

m k

. It is easy to see thatΣ_{r}M ⊂Σ^{+}M(resp.Σ_{r}M ⊂Σ^{−}M) if and only ifris even (resp.

ris odd). Moreover, for anyX ∈Γ(T M)andϕ∈Γ(ΣrM), we haveX^{+}·ϕ∈Γ(Σr+1M)
andX^{−}·ϕ∈Γ(Σ_{r−1}M), with the conventionΣ_{−1}M = Σm+1M =M × {0}. Thus, for
anySpin^{c}structure, we haveΣrM = Λ^{0,r}M⊗Σ0M.Hence,(Σ0M)^{2}=KM⊗L,where
Lis the auxiliary line bundle associated with theSpin^{c} structure. For example, when the
manifold is spin, we have(Σ0M)^{2} = KM [38, 39]. For the canonicalSpin^{c} structure,
sinceL = (KM)^{−1}, it follows that Σ0M is trivial. This yields the existence of parallel
spinors (the constant functions) lying inΣ0M,cf.[40].

On a K¨ahler manifold(M, g, J)endowed with anySpin^{c}structure, a spinor of the form
ϕ_{r}+ϕ_{r+1} ∈ Γ(Σ_{r}M ⊕Σ_{r+1}M), for some0 ≤ r ≤ m, is called aK¨ahlerian Killing
Spin^{c} spinorif there exists a non-zero real constantα, such that the following equations
are satisfied, for all vector fieldsX,

( ∇_{X}ϕ_{r}=α X^{−}·ϕ_{r+1},

∇Xϕr+1=α X^{+}·ϕr. (6)
K¨ahlerian Killing spinors lying inΓ(Σ_{m}M ⊕Σ_{m+1}M) = Γ(Σ_{m}M)or inΓ(Σ_{−1}M ⊕
Σ_{0}M) = Γ(Σ_{0}M)are just parallel spinors. In [16], the authors gave examples ofSpin^{c}
structures on compact K¨ahler-Einstein manifolds of positive scalar curvature, which carry
K¨ahlerian KillingSpin^{c} spinors lying inΣrM ⊕Σr+1M, for r 6= ^{m±1}_{2} , in contrast to
the spin case, where K¨ahlerian Killing spinors may only exist formodd in the middle of
the decomposition (5). We briefly describe theseSpin^{c} structures here. If the first Chern

classc_{1}(K_{M})of the canonical bundle of the K¨ahlerM is a non-zero cohomology class,
the greatest numberp∈N^{∗}such that ^{1}_{p}c_{1}(K_{M})∈H^{2}(M,Z),is called theMaslov index
of the K¨ahler manifold. One can thus consider ap-th root of the canonical bundleK_{M},i.e.

a complex line bundleL, such thatL^{p}=K_{M}. In [16], O. Hijazi, S. Montiel and F. Urbano
proved the following:

Theorem 2.1(Theorem 14, [16]). LetM be a2m-dimensional K¨ahler-Einstein compact
manifold with scalar curvature4m(m+ 1)and indexp∈N^{∗}. For each0≤r≤m+ 1,
there exists on M a Spin^{c} structure with auxiliary line bundle given byL^{q}, where q =

p

m+1(2r−m−1)∈Z, and carrying a K¨ahlerian Killing spinorψ_{r−1}+ψ_{r}∈Γ(Σ_{r−1}M⊕
ΣrM),i.e.for allX ∈Γ(T M), it satisfies the first order system

(∇_{X}ψ_{r}=−X^{+}·ψ_{r−1},

∇Xψr−1=−X^{−}·ψr,

For example, ifM is the complex projective spaceCP^{m}of complex dimensionm, then
p = m+ 1andLis just the tautological line bundle. We fix 0 ≤ r ≤ m+ 1and we
endow CP^{m} with theSpin^{c} structure whose auxiliary line bundle is given byL^{q} where
q=_{m+1}^{p} (2r−m−1) = 2r−m−1∈Z. For thisSpin^{c}structure, the space of K¨ahlerian
Killing spinors inΓ(Σr−1M ⊕ΣrM)has dimension ^{m+1}_{r}

. In this example, forr = 0
(resp.r = m+ 1), we get the canonical (resp. anticanonical)Spin^{c} structure for which
K¨ahlerian Killing spinors are just parallel spinors.

2.2. Submanifolds of Spin^{c} manifolds. Let(M^{2}, g)be an oriented Riemannian surface,
with a givenSpin^{c}structure, andEan orientedSpin^{c}vector bundle of rank 2 onM with an
Hermitian producth·,·iEand a compatible connection∇^{E}. We consider the spinor bundle
ΣoverM twisted byEand defined byΣ = ΣM⊗ΣE,whereΣM andΣEare the spinor
bundles ofM andErespectively. We endowΣwith the spinorial connection∇defined by

∇=∇^{ΣM}⊗IdΣE+IdΣM⊗ ∇^{ΣE},

where∇^{ΣM}and∇^{ΣE}are respectively the spinorial connections onΣM andΣE. We also
define the Clifford product·by

X·ϕ= (X·_{M}α)⊗σ ifX∈Γ(T M)
X·ϕ=α⊗(X·_{E}σ) ifX ∈Γ(E)

for allϕ= α⊗σ ∈ΣM ⊗ΣE,where·_{M} and·_{E} denote the Clifford products onΣM
and onΣErespectively and whereσ=σ^{+}−σ^{−}for the natural decomposition ofΣE =
Σ^{+}E⊕Σ^{−}E. Here,Σ^{+}E andΣ^{−}E are the eigensubbundles (for the eigenvalue1 and

−1) ofΣEfor the action of the normal volume elementω_{⊥} =iξ1·Eξ2, where{ξ1, ξ2}
is a local orthonormal frame ofE. Note thatΣ^{+}M andΣ^{−}M are defined similarly by for
the tangent volume elementω=ie1·Me2. If{e1, e2}is an orthonormal basis ofT M, we
define the twisted Dirac operatorDonΓ(Σ)by

Dϕ=e1· ∇e_{1}ϕ+e2· ∇e_{2}ϕ.

We note thatΣis also naturally equipped with a hermitian scalar producth., .iwhich is
compatible with the connection ∇, and thus also with a compatible real scalar product
Reh., .i.We also note that the Clifford product·of vectors belonging toT M ⊕Eis anti-
hermitian with respect to this hermitian product. Finally, we stress that the four subbundles
Σ^{±±}:= Σ^{±}M⊗Σ^{±}Eare orthogonal with respect to the hermitian product. We will also
considerΣ^{+} = Σ^{++}⊕Σ^{−−}andΣ^{−} = Σ^{+−}⊕Σ^{−+}.Throughout the paper we will as-
sume that the hermitian product isC−linear w.r.t. the first entry, andC−antilinear w.r.t. the
second entry.

Let(Mf^{4},eg)be a RiemannianSpin^{c}manifold and(M^{2}, g)an oriented surface isometrically
immersed intoMf. We denote byN Mthe normal bundle ofMintoMf. AsMis an oriented

surface, it is alsoSpin^{c}. We denote byiFe(resp.iF) the curvature2-form of the auxiliary
line bundleL^{M}^{f}(resp.L) associated with theSpin^{c} structure onMf(resp.M). Since the
manifoldsM andMfareSpin^{c}, there exists aSpin^{c} structure on the bundleN M whose
auxiliary line bundle LN is given by LN := (L)^{−1}⊗L^{M}^{f}_{|}_{M}. We denote by ΣN the
Spin^{c}bundle ofN M and letΣ = ΣM⊗ΣN the spinor bundle overM twisted byN M
constructed as above with the associated connection and Clifford multiplication. It is a
classical fact that the spinor bundle ofMfoverM,ΣMf_{|M} identifies withΣ. Moreover the
connections on each bundle are linked by the so-called spinorial Gauss formula: for any
ϕ∈Γ(Σ)and anyX ∈T M,

∇eXϕ=∇Xϕ+1 2

X

j=1,2

ej·II(X, ej)·ϕ (7)

whereII is the second fundamental form,∇e is the spinorial connection ofΣMfand∇is the spinoral connection ofΣdefined as above and{e1, e2}is a local orthonormal frame of T M. Here·is the Clifford product onΣMfwhich identifies with the Clifford mulitplication onΣ.

3. IMMERSED SURFACES INTO THE COMPLEX PROJECTIVE SPACE

In this section, we will give the basic facts about immersed surfaces in the complex projec- tive plane and in particular derive a sequence of necessary and sufficent conditions for the existence of such immersions.

3.1. Compatibility equations. Let (M^{2}, g)be a Riemannian surface isometrically im-
mersed itn the2-dimensional complex projective space of constant holomorphic sectional
curvature4c > 0. We denote by∇the Levi-Civita connection of(M^{2}, g),egthe Fubini-
Study metric ofCP^{2}(4c)and∇e its Levi-Civita connection. First of all, we recall that the
curvature tensor ofCP^{2}(4c)is given by

R(X, Y, Z, We ) = c

"

hX, Wi hY, Zi − hX, Zi hY, Wi+hJ X, Wi hJ Y, Zi

− hJ X, Zi hJ Y, Wi+ 2hX, J Yi hJ Z, Wi

#

. (8)

The complex structureJinduces the existence of the following four operators j:T M −→T M, h:T M −→N M, s:N M −→T M andt:N M −→N M defined for anyX ∈T Mandξ∈N M by

J X =jX+hX and J ξ=sξ+tξ. (9)
From the fact thatJ^{2}=−IdandJis antisymmetric, we get thatjandtare antisymmetric
and we have the following relations between these four operators

j^{2}X =−X−shX, (10)

t^{2}ξ=−ξ−hsξ, (11)

jsξ+stξ= 0, (12)

hjX+thX= 0, (13)

eg(hX, ξ) =−eg(X, sξ), (14)

for anyX∈Γ(T M)andξ∈Γ(N M). Moreover, from the fact thatJ is parallel, we have

(∇Xj)Y =AhYX+s(B(X, Y)), (15)

∇^{⊥}_{X}(hY)−h(∇XY) =t(B(X, Y))−B(X, jY), (16)

∇^{⊥}(tξ)−t(∇^{⊥}_{X}ξ) =−B(sξ, X)−h(AξX), (17)

∇X(sξ)−s(∇^{⊥}_{X}ξ) =−j(AξX) +AtξX, (18)
whereB :T M×T M −→N Mis the second fundamental form and for anyξ∈T M,A_{ξ}
is the Weingarten operator associated withξand defined byeg(AξX, Y) =eg(B(X, Y), ξ)
for any vectorsX, Y tangent toM. Finally, from (8), we deduce that the Gauss, Codazzi
and Ricci equations are respectively given by

R(X, Y)Z = c

hY, ZiX− hX, ZiY +hjY, ZijX− hjX, ZijY +2hX, jYijZ

+A_{B(Y,Z)}X−A_{B(X,Z)}Y,

(∇_{X}B)(Y, Z)−(∇_{Y}B)(X, Z) = c

hjY, ZihX− hjX, ZihY + 2hjX, YihZ

,

R^{⊥}(X, Y)ξ = c

hhY, ξihX− hhX, ξihY + 2hjX, Yitξ

+B(A_{ξ}Y, X)−B(A_{ξ}X, Y).

In the local orthonormal frames{e1, e2}and{ν1, ν2}, these equations become (forc= 1):

K_{M} = 1+< B_{22}, B_{11}>−|B12|^{2}+ 3< je_{1}, e_{2}>^{2}, (19)
K_{N} =−<[S_{ν}_{1}, S_{ν}_{2}](e_{1}), e_{2}>+h_{21}h_{12}−h_{11}h_{22}+ 2j_{12}t_{12}, (20)
(∇e_{1}B)(e2, ek)−(∇e_{2}B)(e1, ek) =j2khe1−j1khe2+ 2j12hek (21)
It is clear that equations (10) to (21) are necessary condition for surfaces in CP^{2}(4c).

Conversely, given(M^{2}, g)a Reimannian surface,E a2-dimensional vector bundle over
M endowed with a scalar productg and a compatible connection∇^{E}. Letj : T M −→

T M, h:T M −→E, s:E−→T M andt:E−→Efour tensors.

Definition 3.1. We say that(M, g, E, g,∇^{E}, B, j, h, s, t)satisfies the compatibility equa-
tions forCP^{2}(4c)ifjandtare antisymmetric the Gauss, Codazzi and Ricci equations(19)
(20) (21)and equations(10)-(17)are fulfilled.

Now, we can state the following classical Fundamental Theorem for surfaces ofCP^{2},
which can be found for instance as a special case of the general result of P. Piccione and D.

Tausk [30].

Proposition 3.2. If(M, g, E, g,∇^{E}, B, j, h, s, t)satisfies the compatibility equations for
CP^{2}(4c)then, there exists an isometric immersion Φ : M −→ CP^{2}(4c)such that the
normal bundle ofM for this immersion is isomorphic toEand such that the second fun-
damental formII and the normal connection∇^{⊥}are given byBand∇^{E}. Precisely, there
exists a vector bundle isometryΦ :e E−→N M so that

II =Φe◦B,

∇^{⊥}Φ =e Φ∇e ^{E}.
Moreover, we have

J(Φ_{∗}X) = Φ_{∗}(jX) +Φ(hX),e
J(eΦξ) = Φ_{∗}(sX) +Φ(tξ),e

whereJ is the canonical complex structure ofCP^{2}(4c)and this isometric immersion is
unique up to an isometry ofCP^{2}(4c).

3.2. Special cases. Two special cases are of particular interest and have been widely stud- ied, the complex and Lagrangian surfaces.

A surface(M^{2}, g)of CP^{2}(4c)is saidcomplexif the tangent bundle of M is stable by
the complex structure ofCP^{2}(4c), that is,J(T M) =T M. Note that we have necessarily
J(N M) =N M. Hence, in that case, with the above notations, we haveh= 0,s= 0and
sojandtare respectively almost complex structures onT MandN M. The compatibility
equations for complex surfaces ofCP^{2}(4c)becomes

h= 0, s= 0, j^{2}=−idT M, t^{2}=−idE

∇j= 0, ∇^{⊥}t= 0

t(B(X, Y))−B(X, jY) = 0, ∀X∈T M
KM = 4+< B22, B11>−|B12|^{2}
KN =−<[Sν1, Sν2](e1), e2>+2
(∇e1B)(e_{2}, e_{k})−(∇e2B)(e_{1}, e_{k}) = 0

(22)

A surface(M^{2}, g)ofCP^{2}(4c)is said totally real ifJ(T M)is transversal toT M. In the
particular case whenJ(T M) = N M, we say that(M^{2}, g)isLagrangian. In that case,
we havej = 0andt = 0. Hence, the compatibility equations for Lagranigan surfaces of
CP^{2}(4c)are

j= 0, t= 0, sh=−idT M, hs=−idE

∇s= 0, ∇^{⊥}h= 0

AhYX+s(B(X, Y)) = 0, ∀X∈T M
KM = 1+< B22, B11>−|B12|^{2}

KN =−<[Sν_{1}, Sν_{2}](e1), e2>+h21h12−h11h22

(∇e_{1}B)(e2, ek)−(∇e_{2}B)(e1, ek) = 0

(23)

4. RESTRICTION OF AK ¨AHLERIANKILLINGSPIN^{c}SPINOR AND CURVATURE
COMPUTATION

We consider a specialSpin^{c} structure onCP^{2}carrying a (real) K¨ahlerian Killing spinor
ϕ. For example, on can takeq = −1 and hencer = 1. For this structure, the curvature
of the line bundle is given byFA(X, Y) = −2ig(J X, Y). The spinorϕ = ϕ0+ϕ1 ∈
Γ(Σ0CP^{2}⊕Σ1CP^{2})satisfies the following:

(

∇eXϕ0=−X^{−}·ϕ1,

∇eXϕ1=−X^{+}·ϕ0,
Thus, we have

∇Xϕ=−1

2X·ϕ+ i

2J X·ϕ,

whereϕ =ϕ_{0}−ϕ_{1}is the conjugate ofϕfor the action of the complex volume element
ω_{4}^{C} = −e_{1}·e_{2}·e_{3}·e_{4}. Indeed,Σ_{0}CP^{2} ⊂ Σ^{+}CP^{2}andΣ_{1}CP^{2} = Σ^{−}CP^{2}. Note also
that such as spinor is of constant norm and each partϕ0andϕ1does not have any zeros.

Indeed, for instance, ifϕ0 vanishes at one point, then it must vanish everywhere (as it is
obtained by parallel transport) andϕ1is as a parallel spinor which is not the case for this
Spin^{c}structure.

Now, letMbe a surface ofCP^{2}with normal bundle denoted byN M. By the identification
of the Clifford multiplications and theSpin^{c}Gauss formula, we have

∇_{X}ϕ=−1

2η(X)·ϕ−1

2X·ϕ+ i

2J X·ϕ.

In intrinsic terms, it can be written as

∇Xϕ = −1

2η(X)·ϕ−1

2X·ϕ+ i

2jX·ϕ+ i

2hX·ϕ, (24)

whereηis given by

η(X) =

2

X

j=1

e_{j}·B(e_{j}, X). (25)

HereB is the second fundamental form of the immersion, and the operatorsjandhare those introduces in Section 3. We deduce immediately that

∇Xϕ=−1

2η(X)·ϕ+1

2X·ϕ− i

2jX·ϕ−i 2hX·ϕ.

Now, let us go back to an instrinsic setting by considering(M^{2}, g)an oriented Riemannian
surface andEan oriented vector bundle of rank2overM with scalar producth·,·iE and
compatible connection∇^{E}. We denote byΣ = ΣM ⊗ΣEthe twisted spinor bundle. Let
B :T M×T M −→Ea bilinear symmetric map andj:T M −→T M, h:T M −→E
two tensors. We assume that the spinor field ϕ ∈ Γ(Σ)satisfies Equation (24). We will
compute the spinorial curvature for this spinor fieldϕ. For this, let{e_{1}, e_{2}} be a normal
local orthonormal frame ofT Mand{ν1, ν2}a local orthonormal frame ofE. We have

∇e_{1}∇e_{2}ϕ = −1

2∇e_{1}(η(e2))·ϕ+1

4η(e2)·η(e1)·ϕ+1

4η(e2)·e1·ϕ

−i

4η(e2)·j(e1)·ϕ− i

4η(e2)·h(e1)·ϕ+1

4e2·η(e1)·ϕ−1

2∇e_{1}e2·ϕ
+1

4e2·e1·ϕ− i

4e2·j(e1)·ϕ− i

4e2·h(e1)·ϕ +i

2∇_{e}_{1}(j(e_{2}))·ϕ− i

4j(e_{2})·η(e_{1})·ϕ+i

4j(e_{2})·e_{1}·ϕ
+1

4j(e_{2})·j(e_{1})·ϕ+1

4j(e_{2})·h(e_{1})·ϕ+ i

2∇^{⊥}_{e}_{1}(h(e_{2}))·ϕ

−i

4h(e2)·η(e1)·ϕ+i

4h(e2)·e1·ϕ+1

4h(e2)·j(e1)·ϕ +1

4h(e2)·h(e1)·ϕ

= −1

2∇e_{1}(η(e2))·ϕ+1

4η(e2)·η(e1)·ϕ+1

4e2·e1·ϕ

−1

2∇e1e2·ϕ+1

4η(e2)·e1·ϕ+1

4e2·η(e1)·ϕ +i

2∇e_{1}(j(e2))·ϕ+ i

2∇^{⊥}_{e}_{1}(h(e2))·ϕ

−i

4e2·j(e1)·ϕ+ i

4j(e2)·e1·ϕ

−i

4e_{2}·h(e_{1})·ϕ+ i

4h(e_{2})·e_{1}·ϕ
+1

4j(e2)·je1·ϕ +1

4h(e2)·he1·ϕ +1

4 j(e2)·h(e1)·ϕ+h(e2)·je1·ϕ

−i

4 η(e_{2})·h(e_{1})·ϕ+h(e_{2})·η(e_{1})·ϕ

−i

4 η(e_{2})·j(e_{1})·ϕ+j(e_{2})·η(e_{1})·ϕ

= I+II+III+IV +V +V I+V II+IIX+IX+X,

where

I = −1

2∇e_{1}(η(e2))·ϕ+1

4η(e2)·η(e1)·ϕ+1

4e2·e1·ϕ−1

2∇e_{1}e2·ϕ
II = +i

2∇e_{1}(j(e2))·ϕ+ i

2∇^{⊥}_{e}_{1}(h(e2))·ϕ

III = −i

4e2·j(e1)·ϕ+ i

4j(e2)·e1·ϕ

IV = −i

4e2·h(e1)·ϕ+ i

4h(e2)·e1·ϕ

V = +1

4j(e2)·je1·ϕ

V I = +1

4h(e_{2})·he_{1}·ϕ

V II = +1

4 j(e2)·h(e1)·ϕ+h(e2)·je1·ϕ

IIX = −i

4 η(e2)·h(e1)·ϕ+h(e2)·η(e1)·ϕ

IX = −i

4 η(e2)·j(e1)·ϕ+j(e2)·η(e1)·ϕ

X = 1

4η(e_{2})·e_{1}·ϕ+1

4e_{2}·η(e_{1})·ϕ
We point out that

∇[e_{1},e_{2}]ϕ=−1

2η([e1, e2])·ϕ−1

2[e1, e2]·ϕ

| {z }

I([ee _{1},e_{2}])

+i

2j([e1, e2])·ϕ+ i

2h([e1, e2])·ϕ

| {z }

fII([e1,e2])

.

Some terms are vanishing as shown in the following lemma.

Lemma 4.1. We have

X(e1, e2)−X(e2, e1) = 0 (26)
IV(e_{1}, e_{2})−IV(e_{2}, e_{1}) = 0 (27)
(II+IIX+IX)(e1, e2)−(II+IIX+IX)(e2, e1)−IIf([e1, e2]) = 0 (28)
Proof:

(1) Using the definition ofη, we get−^{1}_{2}B(e_{j}, X) =e_{j}·η(X)−η(X)·e_{j}. Hence
X(e1, e2)−X(e2, e1) =−1

8B(e2, e1) +1

8B(e1, e2) = 0.

(2)

IV(e1, e2)−IV(e2, e1) =

−i

4(e_{2}·h(e_{1})−h(e_{2})·e_{1})·ϕ+ i

4(e_{1}·h(e_{2})−h(e_{1})·e_{2})·ϕ

= i

4(2g(e2, h(e1))−2g(h(e2), e1))·ϕ

= 0, (29)

becauseXandh(X)are orthogonal forX ∈Γ(T M).

(3) First we have

II(e_{1}, e_{2})−II(e_{2}, e_{1})−IIf([e_{1}, e_{2}])

= i

2∇e_{1}(j(e2))·ϕ+ i
2∇^{⊥}_{e}

1(h(e2))·ϕ− i

2∇e_{2}(j(e1))·ϕ− i
2∇^{⊥}_{e}

2(h(e1))·ϕ

−i

2j([e_{1}, e_{2}])·ϕ− i

2h([e_{1}, e_{2}])·ϕ

= i 2

(∇e1j)e2·ϕ+ (∇e1h)e2·ϕ−(∇e2j)e1·ϕ−(∇e2h)e1·ϕ

= i 2

s(B(e1, e2))·ϕ+S_{h(e}_{2}_{)}e1·ϕ+t(B(e1, e2))·ϕ−B(e1, j(e2))·ϕ

−s(B(e1, e2))·ϕ−S_{h(e}_{1}_{)}e2·ϕ−t(B(e1, e2))·ϕ+B(e2, je1)·ϕ

= i 2

S_{h(e}_{2}_{)}e1·ϕ−S_{h(e}_{1}_{)}e2·ϕ−B(e1, j(e2))·ϕ+B(e2, je1)·ϕ

(30) Moreover, we calculate

−B(e_{1}, j(e_{2}))·ϕ+B(e_{2}, je_{1})·ϕ

= −g(j(e2), e1)B(e1, e1)·ϕ+g(e2, j(e1))B(e2, e2)·ϕ

= 2g(j(e1), e2)H·ϕ (31)

and

Sh(e_{2})e1·ϕ−Sh(e_{1})e2·ϕ

= −< S_{h(e}_{1}_{)}e2, e1> e1·ϕ−< S_{h(e}_{1}_{)}e2, e2> e2·ϕ
+< S_{h(e}_{2}_{)}e_{1}, e_{1}> e_{1}·ϕ+< S_{h(e}_{2}_{)}e_{1}, e_{2}> e_{2}·ϕ

= −< B(e2, e1), h(e1)> e1·ϕ−< B(e2, e2), h(e1)> e2·ϕ

+< B(e_{1}, e_{1}), h(e_{2})> e_{1}·ϕ+< B(e_{1}, e_{2}), h(e_{2})> e_{2}·ϕ (32)
In addition we have

IIX(e1, e2)−IIX(e2, e1)

= i 4

−e1·B(e1, e2)·h(e1)−e2·B(e2, e2)·h(e1)−h(e2)·e1·B(e1, e1)−h(e2)·e2·B(e1, e2) +e1·B(e1, e1)·h(e2) +e2·B(e1, e2)·h(e2) +h(e1)·e1·B(e1, e2) +h(e1)·e2·B(e2, e2)

·ϕ

= i 4

2< B(e1, e2), h(e1)> e1+ 2< B(e2, e2), h(e1)> e2

−2< B(e1, e2), h(e2)> e2−2< B(e1, e1), h(e2)> e1

·ϕ (33)

and

IX(e_{1}, e_{2})−IX(e_{2}, e_{1})

= −i

4η(e2)·j(e1)·ϕ− i

4j(e2)·η(e1)·ϕ+ i

4η(e1)·j(e2)·ϕ+ i

4j(e1)·η(e2)·ϕ

= i 4

−e_{1}·B(e_{1}, e_{2})·j(e_{1})· −e_{2}·B(e_{2}, e_{2})·j(e_{1})· −j(e_{2})·e_{1}·B(e_{1}, e_{1})·

−j(e2)·e2·B(e1, e2)·+e1·B(e1, e1)·j(e2)·+e2·B(e1, e2)·j(e2)· +j(e1)·e1·B(e2, e1)·+j(e1)·e2·B(e2, e2)·

ϕ

= i 4

e_{1}·j(e_{1})·B(e_{1}, e_{1}) +e_{2}·j(e_{1})·B(e_{2}, e_{2})−j(e_{2})·e_{1}·B(e_{1}, e_{1})−j(e_{2})·e_{2}·B(e_{1}, e_{2})

−e1·j(e2)·B(e1, e1)−e2·j(e2)·B(e1, e2) +j(e1)·e1·B(e1, e2) +j(e1)·e2·B(e2, e2)

·ϕ

= i

4 −2g(j(e_{1}), e_{2})B(e_{2}, e_{2}) + 2g(j(e_{2}, e_{1}))B(e_{1}, e_{1})

·ϕ

= −ig(j(e1, e2))H·ϕ. (34)

Now, replacing (31) and (32) in (30) and combining together with (33) and (34), we get the desired result.

Now, we have this second lemma.

Lemma 4.2. We have (1)

V(e_{1}, e_{2})−V(e_{2}, e_{1}) =−1

2hj(e_{2}), e_{1})i^{2}e_{1}·e_{2},
(2)

V I(e1, e2)−V I(e2, e1) = 1

2[hh(e2), ν1ihh(e1), ν2i − hh(e1), ν1ihh(e2), ν2i]ν1·ν2. (3)

III(e1, e2)−III(e2, e1) =ig(e2, je1)ϕ (35) (4)

V II(e1, e_{2})−V II(e_{2}, e_{1}) = 1

2 j_{21}h_{11}e_{1}·ν_{1}+j_{21}h_{12}e_{1}·ν_{2}+j_{21}h_{21}e_{2}·ν_{1}+j_{12}h_{22}e_{2}·ν_{2}

·ϕ(36) Proof.

(1) We denote byj_{kl} =g(je_{k}, e_{l}). Sincejis antisymmetric, we havej_{kl} =−jlkand
so

V(e_{1}, e_{2})−V(e_{2}, e_{1})

= 1

4 j(e2)·j(e1)−j(e1)·j(e2)

·ϕ

= 1

4(j21j12e1·e2−j12j21e2·e1)·ϕ

= 1

2j21j12e1·e2·ϕ

= −1

2g(j(e1), e2)^{2}e1·e2·ϕ (37)

(2)

V I(e_{1}, e_{2})−V I(e_{2}, e_{1})

= 1

4(h(e2)·h(e1)−h(e1)·h(e2))

= 1

4 −h21h11+h21h12ν1·ν2+h22h11ν2·ν1−h22h12+h11h21

−h11h22ν1·ν2−h12h21−h11h22ν1·ν2−h12h21ν2·ν1+h12h22

·ϕ

= 1

2[g(h(e2), ν1)g(h(e1), ν2)−g(h(e1), ν1)g(h(e2), ν2)]ν1·ν2·ϕ (3)

III(e_{1}, e_{2})−III(e_{2}, e_{1}) = −i

4 e_{2}·j(e_{1})−j(e_{2})·e_{1}−e_{1}·j(e_{2}) +j(e_{1})·e_{2}

·ϕ

= −i

4 −j_{12}+j_{21}+j_{21}−j_{12}

·ϕ

= ig(e2, j(e1))ϕ (38)

(4) We have

V II(e_{1}, e_{2})−V II(e_{2}, e_{1})

= 1

4 j(e2)·h(e1) +h(e2)·je1−j(e1)·h(e2)−h(e1)·je2

·ϕ

= 1

4 j21e1·(h11ν1+h12ν2) +j12(h21ν1+h22ν2)e2

−j12e2·(h21ν1+h22ν2)−j21(h11ν1+h12ν2)e1

·ϕ

= 1

2 j21h11e1·ν1+j21h12e1·ν2+j21h21e2·ν1+j12h22e2·ν2

·ϕ (39) Finally, we have this last lemma obtained by a straightforward calculation and using Lemma 3.3 of [4].

Lemma 4.3.

I(e1, e2)−I(e2, e1)−I(ee 1, e2)

= −1 2

2

X

j=1

e_{j}· (∇^{0}_{e}_{1}B)(e_{2}, e_{j}))−(∇^{0}_{e}_{2}B)(e_{1}, e_{j})

+1

2g([S_{ν}_{1}, S_{ν}_{2}])(e_{1}), e_{2})ν_{1}·ν_{2}·ϕ

−1

2e_{1}·e_{2}·ϕ, (40)

where∇^{0}is the natural connection onT^{∗}M⊗T^{∗}M⊗EandBkl=B(ek, el)

5. LAGRANGIAN CASE,PROOF OFTHEOREM1.2

Now, we have all the ingredients to prove Theorems 1.1 and 1.2. We begin by the Lagragian case. Assume thatj= 0andt= 0. So we have

Re_{1},e_{2}ϕ = 1

2KMe1·e2·ϕ−1

2KEν1·ν2·ϕ+ i

2α^{M+E}(e1, e2)ϕ, (41)
withα^{M}^{+E}(e1, e2) = 0becausej= 0. From the other hand, we have

Re_{1},e_{2}ϕ (42)

= −1 2

2

X

j=1

ej·((∇^{‘}_{e}_{1}B)(e2, ej)−(∇^{‘}_{e}_{2}B)(e1, ej))·ϕ

+1

2(|B12|^{2}−< B11, B22>)e1·e2·ϕ
+1

2 <[Sν_{1}, Sν_{2}](e1), e2> ν1·ν2·ϕ

−1

2e1·e2·ϕ+1

2(h21h12−h11h22)ν1·ν2·ϕ (43)
We get finally thatT·ϕ= 0, whereT ∈(Λ^{2}M⊗1⊕T M⊗E⊕1⊗Λ^{2}E). Thus, since
ϕ^{+}andϕ^{−}never vanish, we haveT = 0by [4][Lemma 3.4] and it gives

K_{M} =< B_{11}, B_{22}>−|B_{12}|^{2}+ 1,

KE=−<[Sν_{1}, Sν_{2}](e1), e2>−(h21h12−h22h11),
(∇e1B)(e_{2}, e_{j})−(∇e2B)(e_{1}, e_{j}) = 0,

which are Gauss, Ricci and Codazzi equations for a Lagrangian surface inCP^{2}and so the
conditions (23) are fulfuilled. Hence, by Proposition 3.2, we conclude that there exists a
Lagrangian isomertic immersion from(M, g)intoCP^{2}withEas normal bundle andBas
second fundamental form. This proves that assertion (2) of Theorem 1.2 implies assertion
(1). The converse is immediate by the discussions of Sections 3 and 4. Theorem 1.2 is
proven.

6. COMPLEX CASE,PROOF OFTHEOREM1.1

Assume thats= 0;h= 0soα^{M}^{+E}(e1, e2) =−2. We takeje1 =e2andtν1 =ν2, i.e,
g(je_{1}, e_{2}) =g(tν_{1}, ν_{2}) = 1.

We calculate and we get Re1,e2ϕ = −1

2KMe1·e2·ϕ−1

2KNν1·ν2·ϕ+ i

2α^{M}^{+N}(e1, e2)ϕ

= −1

2KMe1·e2·ϕ−1

2KNν1·ν2·ϕ−iϕ

= T·ϕ−iϕ, (44)

whereT is a2-form. From the other hand, we have
Re_{1},e_{2}ϕ = −e1·e2·ϕ+iϕ

−1 2

2

X

j=1

ej·((∇e_{1}B)(e2, ej)−(∇e_{2}B)(e1, ej))·ϕ

+1

2(|B12|^{2}−< B11, B22>)e1·e2·ϕ
+1

2 <[Sν_{1}, Sν_{2}](e1), e2> ν1·ν2·ϕ

= ˜T·ϕ+iϕ. (45)

Together, it givesT ·ϕ−T˜·ϕ−iϕ−iϕ= 0, which means thatT ·ϕ−iϕ−iϕ= 0, whereT is again a2form given by

T = −1

2KMe1·e2·ϕ−1

2KNν1·ν2·ϕ+e1·e2·ϕ

−1

2(|B12|^{2}−< B11, B22>)e1·e2·ϕ

−1

2 <[Sν_{1}, Sν_{2}](e1), e2> ν1·ν2·ϕ
+1

2

2

X

j=1

ej·((∇e_{1}B)(e2, ej)−(∇e_{2}B)(e1, ej))·ϕ (46)

We give now the following Lemma

Lemma 6.1. LetT be a2form, i.e,T ∈Λ^{2}M ⊗1⊕Λ^{1}M ⊗Λ^{1}E⊕1⊗Λ^{2}E. Assume
that

T ·ϕ−iϕ−iϕ= 0,

and writeT =T^{t}e1·e2+T^{n}ν1·ν2+T^{m},whereT^{m}∈Λ^{1}M⊗Λ^{1}E. Then,
T^{t}=−1, T^{n} = 0, and T^{m}= 0.

Proof.Letϕ=ϕ^{+}+ϕ^{−},with

ϕ^{+}=ϕ^{++}+ϕ^{−−},
ϕ^{−}=ϕ^{−+}+ϕ^{+−},
a solution of (24) withh= 0. This means that

∇_{X}ϕ^{++}=−1

2X·ϕ^{−+}− i

2jX·ϕ^{−+}

∇Xϕ^{+−}=−1

2X·ϕ^{−−}+ i

2jX·ϕ^{−−}

∇Xϕ^{−+}=−1

2X·ϕ^{++}+ i

2jX·ϕ^{++}

∇Xϕ^{−−}=−1

2X·ϕ^{+−}− i

2jX·ϕ^{+−}.

For a sake of simplicity, and without lost of generality, we can restrict onlyϕ^{+}=ϕ^{++}and
ϕ^{−}=ϕ^{−+}which which have no zeros by assumption. The equation

T ·ϕ−iϕ−iϕ= 0, becomes

T^{t}e_{1}·e2·(ϕ^{++}+ϕ^{−+})+(T^{n}+1)ν_{1}·ν2·(ϕ^{++}+ϕ^{−+})+T^{m}·(ϕ^{++}+ϕ^{−+}) =iϕ=i(ϕ^{++}−ϕ^{−+})
Taking the scalar product withϕ^{++}then withϕ^{−+}, we get

T^{t}+T^{n}+ 1 =−1,

−T^{t}+T^{n}+ 1 = 1,

which gives T^{n} = −1,T^{t} = −1 andT^{m} = 0. These are Gauss, Codazzi and Ricci
equations and so the conditions (22) are fulfilled. There are exactly the conditions ofr a
complex immersion. Hence, by Proposition 3.2, we conclude that there exists a complex
isometric immersion from(M, g)into CP^{2} withE as normal bundle andB as second
fundamental form. As for the Lagrangian case, this proves that assertion (2) of Theorem 1.1
implies assertion (1). Here again, the converse is immediate by the discussions of Sections
3 and 4, which concludes the proof of Theorem 1.1.

7. THEDIRAC EQUATION

Letϕbe a spinor field satisfying Equation (24), then it satisfies the following Dirac equation Dϕ=H~ ·ϕ−ϕ+ i

2β·ϕ, (47)

whereβis the 2-form defined byβ= X

i=1,2

e_{i}·hei=

2

X

i,j=1

h_{ij}e_{i}·ξj, whereh_{i,j}=hhei, ξ_{j}i.

As in [4] and [32], we will show that this equation with an appropiate condition on the norm
of bothϕ^{+}andϕ^{−}is equivalent to Equation (24), where the tensorBis expressed in terms
of the spinor fieldϕand such thattr(B) = 2H~. Moreover, from Equation (24) we deduce
the following conditions on the derivatives of|ϕ^{+}|^{2}and|ϕ^{−}|^{2}. Indeed, after decomposition
ontoΣ^{+}andΣ^{−}, (24) becomes

∇Xϕ^{±}=−1

2η(X)·ϕ^{±}−1

2X·ϕ^{∓}∓ i

2jXϕ^{∓}∓ i
2hXϕ^{∓}.
From this we deduce that

X(|ϕ^{±}|^{2}) = Re

−1

2X·ϕ^{∓}∓ i

2jX·ϕ^{∓}∓ i

2hX·ϕ^{∓}, ϕ^{±}

(48)
Now, letϕa spinor field solution of the Dirac equation (47) with ϕ^{+} andϕ^{−} nowhere
vanishing and satisying the norm condition (48), we set for any vector fields X andY
tangent toM andξ∈E

hB(X, Y), ξi = 1

|ϕ^{+}|^{2}Re

X· ∇Yϕ^{+}−1

2(X+ijX+ihX)·Y ·ϕ^{−}, ξ·ϕ^{+}

(49) + 1

|ϕ^{−}|^{2}Re

X· ∇Yϕ^{−}−1

2(X−ijX−ihX)·Y ·ϕ^{−}, ξ·ϕ^{+}

Then, we have the following

Proposition 7.1. Letϕ∈Γ(Σ)satisfying the Dirac equation(47) Dϕ=H~ ·ϕ−ϕ−β·ϕ such that

X(|ϕ^{±}|^{2}) = Re

−1

2X·ϕ^{∓}∓ i

2jX·ϕ^{∓}∓ i

2hX·ϕ^{∓}, ϕ^{±}

thenϕis solution of Equation(24)

∇Xϕ=−1

2η(X)·ϕ−1

2X·ϕ+ i

2jX·ϕ+ i 2hX·ϕ whereηis defined byη(X) =−1

2

2

X

j=1

ej·B(ej, X).

Moreover,Bis symmetric.

The proof of this proposition will not be given, since it is completely similar to the case of Riemannian products [32]. Now, combining this proposition with Theorems 1.1 and 1.2, we get the following corollaries. We have this first one for complex surfaces.

Corollary 7.2. Let(M^{2}, g)be an oriented Riemannian surface andEan oriented vector
bundle of rank2over M with scalar product< ·,· >E and compatible connection∇^{E}.
We denote byΣ = ΣM ⊗ΣE the twisted spinor bundle. Letj be a complex structure
onM andta complex structure onE. LetH~ be a section ofE. Then, the two following
statements are equivalent