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Roger Nakad, Julien Roth

To cite this version:



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




ABSTRACT. The complex projective spaceCP2 of complex dimension2has aSpinc structure carrying K¨ahlerian Killing spinors. The restriction of one of these K¨ahlerian Killing spinors to a Lagrangian or complex surfaceM2 characterizes the isometric im- mersion ofMintoCP2.


A classical problem in Riemannian geometry is to know when a Riemannian manifold (Mn, g)can be isometrically immersed into a fixed Riemannian manifold( ¯Mn+p,g). The¯ case of space formsRn+1,Sn+1andHn+1is 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 R3 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 ofSpincspinors [29].

More precisely, the restrictionϕof a parallel spinor field onRn+1to an oriented Riemann- ian hypersurfaceMnis a solution of a generalized Killing equation


2A(X))ϕ, (1)

where“·”and∇are respectively the Clifford multiplication and the spin connection on Mn, 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 intoR3. 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 inR3).

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

Key words and phrases. Spincstructures, K¨ahlerian Killing Spincspinors, isometric immersions, complex and Lagrangian surfaces, Dirac operator.



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 inR4introduced by Konopelchenko and Taimanov [18, 36].

The aim of the present article is to provide an analogue for the complex projective space CP2. The key point is that contrary to the case of hypersurfaces ofCP2, which is consid- erered in [29], the use ofSpincparallel spinors is not sufficient. Indeed, for both canonical or anti-canonicalSpincstructures, 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,Spincparallel 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, sinceCH2does not carry a real or imaginary K¨ahlerian Killing spinor.

We will focus on the case of complex and Lagrangian immersions intoCP2. These two cases and especially the Lagrangian case are of particular interest in the study of surfaces inCP2(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 spaceCP2.

Theorem 1.1. Let(M2, 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 aSpincstructure onΣM⊗ΣEwithΩM+E(e1, e2) = 0and a spinor fieldϕ∈Γ(ΣM⊗ΣE)

Xϕ = −1


2X·ϕ+ i

2jX·ϕ, (3)

such thatϕ+andϕnever vanish and whereηis given by η(X) =




ej·B(ej, X).

(2) There exists a local isometriccompleximmersion of(M2, g)intoCP2withEas normal bundle and second fundamental formBsuch the complex structure ofCP2 overM is given byjandt(in the sense of Proposition 3.2).

The spinorsϕ+andϕdenote 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 intoCP2. Theorem 1.2. Let(M2, 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,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 aSpinc structure onΣM ⊗ΣE withΩM+N(e1, e2) = −2and a spinor fieldϕ∈Γ(ΣM ⊗ΣE)satisfying for allX ∈X(M)

Xϕ = −1


2X·ϕ+ i

2hX·ϕ, (4)

such thatϕ+andϕnever vanish and whereηis given by η(X) =




ej·B(ej, X).

(2) There exists a local isometricLagrangianimmersion of(M2, g)intoCP2withE as normal bundle and second fundamental formBsuch that overM the complex structure ofCP2is given byhands(in the sense of Proposition 3.2).

Note that in the statements of both theoremϕ+andϕdenotes 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.


In this section, we briefly review some basic facts about K¨ahler geometry andSpincstruc- 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. Spinc structures on K¨ahler-Einstein manifolds. Let(Mn, g)be ann-dimensional closed RiemannianSpincmanifold and denote byΣM its complex spinor bundle, which has complex rank equal to2[n2]. The bundleΣMis endowed with a Clifford multiplication denoted by “·” and a scalar product denoted byh·,·i.

Given a Spinc structure on (Mn, g), one can check that the determinant line bundle det(ΣM)has a root L of index 2[n2]−1. This line bundle L over M is called the aux- iliary line bundle associated with theSpincstructure. From a topological point of view, a Riemannian manifold has aSpincstructure 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,

whereω2(M)is the second Steifel-Whitney class ofM andc1(L)is the first Chern class ofL. In the particular case, when the line bundle has a square root, i.e.,ω2(M) = 0, the manifold is called aSpinmanifold. In this case, we denote byΣ0M the spinor bundle or theSpinbundle. It can be chosen such that the auxiliary line bundle is trivial.

Locally, aSpinc structure always exists. In fact, the square root of the auxiliary line bun- dle LandΣ0M always exist locally. But,ΣM = Σ0M ⊗L12 is defined globally. This essentially means that, while the spinor bundle andL12 may not exist globally, their tensor product (the Spinc 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. TheSpincDirac operatorDacting on the space of sections ofΣM is defined by the composition of the connection∇with the Clifford multiplication.

In local coordinates:





ej· ∇ej,

where{ej}j=1,...,nis a local orthonormal basis ofT M.Dis a first-order elliptic operator and is formally self-adjoint with respect to theL2-scalar product.

We recall that the complex volume elementωC= ı[n+12 ]e1∧. . .∧enacts as the identity on the spinor bundle ifnis odd. Ifnis even,ω2C= 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 trivialSpinc structure, by choosing the trivial line bundle with the trivial connection whose curvatureFA vanishes. Every K¨ahler manifold(M2m, g, J) has a canonicalSpinc structure induced by the complex structure J. The complexified tangent bundle decomposes intoTCM =T1,0M ⊕T0,1M,thei-eigenbundle (resp.(−i)- eigenbundle) of the complex linear extension ofJ. For any vector fieldX, we denote by X± := 12(X ∓iJ X)its component in T1,0M, resp.T0,1M. The spinor bundle of the canonicalSpincstructure is defined by

ΣM = Λ0,∗M = ⊕m

r=0Λr(T0,1 M),

and its auxiliary line bundle isL = (KM)−1 = Λm(T0,1 M), whereKM = Λm,0M 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-canonicalSpinc structure, whose spinor bundle is given by Λ∗,0M =⊕mr=0Λr(T1,0 M)and the auxiliary line bundle byKM. The spinor bundle of any otherSpincstructure onM can be written as:

ΣM = Λ0,∗M⊗L,

whereL2=KM⊗LandLis the auxiliary line bundle associated with thisSpincstructure.

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




ej·J ej·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


ΣrM, (5)

whereΣrM denotes the eigenbundle to the eigenvaluei(2r−m)ofΩ, of complex rank

m k

. It is easy to see thatΣrM ⊂Σ+M(resp.ΣrM ⊂ΣM) 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(Σr−1M), with the conventionΣ−1M = Σm+1M =M × {0}. Thus, for anySpincstructure, we haveΣrM = Λ0,rM⊗Σ0M.Hence,(Σ0M)2=KM⊗L,where Lis the auxiliary line bundle associated with theSpinc structure. For example, when the manifold is spin, we have(Σ0M)2 = KM [38, 39]. For the canonicalSpinc 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 anySpincstructure, a spinor of the form ϕrr+1 ∈ Γ(ΣrM ⊕Σr+1M), for some0 ≤ r ≤ m, is called aK¨ahlerian Killing Spinc spinorif there exists a non-zero real constantα, such that the following equations are satisfied, for all vector fieldsX,

( ∇Xϕr=α X·ϕr+1,

Xϕr+1=α X+·ϕr. (6) K¨ahlerian Killing spinors lying inΓ(ΣmM ⊕Σm+1M) = Γ(ΣmM)or inΓ(Σ−1M ⊕ Σ0M) = Γ(Σ0M)are just parallel spinors. In [16], the authors gave examples ofSpinc structures on compact K¨ahler-Einstein manifolds of positive scalar curvature, which carry K¨ahlerian KillingSpinc spinors lying inΣrM ⊕Σr+1M, for r 6= m±12 , 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 theseSpinc structures here. If the first Chern


classc1(KM)of the canonical bundle of the K¨ahlerM is a non-zero cohomology class, the greatest numberp∈Nsuch that 1pc1(KM)∈H2(M,Z),is called theMaslov index of the K¨ahler manifold. One can thus consider ap-th root of the canonical bundleKM,i.e.

a complex line bundleL, such thatLp=KM. 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 Spinc structure with auxiliary line bundle given byLq, where q =


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



For example, ifM is the complex projective spaceCPmof complex dimensionm, then p = m+ 1andLis just the tautological line bundle. We fix 0 ≤ r ≤ m+ 1and we endow CPm with theSpinc structure whose auxiliary line bundle is given byLq where q=m+1p (2r−m−1) = 2r−m−1∈Z. For thisSpincstructure, the space of K¨ahlerian Killing spinors inΓ(Σr−1M ⊕ΣrM)has dimension m+1r

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

2.2. Submanifolds of Spinc manifolds. Let(M2, g)be an oriented Riemannian surface, with a givenSpincstructure, andEan orientedSpincvector 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∇ΣMand∇ΣEare respectively the spinorial connections onΣM andΣE. We also define the Clifford product·by

X·ϕ= (X·Mα)⊗σ ifX∈Γ(T M) X·ϕ=α⊗(X·Eσ) ifX ∈Γ(E)

for allϕ= α⊗σ ∈ΣM ⊗ΣE,where·M and·E denote the Clifford products onΣM and onΣErespectively and whereσ=σ+−σfor the natural decomposition ofΣE = Σ+E⊕ΣE. Here,Σ+E andΣE 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 andΣM 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· ∇e1ϕ+e2· ∇e2ϕ.

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(Mf4,eg)be a RiemannianSpincmanifold and(M2, g)an oriented surface isometrically immersed intoMf. We denote byN Mthe normal bundle ofMintoMf. AsMis an oriented


surface, it is alsoSpinc. We denote byiFe(resp.iF) the curvature2-form of the auxiliary line bundleLMf(resp.L) associated with theSpinc structure onMf(resp.M). Since the manifoldsM andMfareSpinc, there exists aSpinc structure on the bundleN M whose auxiliary line bundle LN is given by LN := (L)−1⊗LMf|M. We denote by ΣN the Spincbundle 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



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.


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 (M2, 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(M2, g),egthe Fubini- Study metric ofCP2(4c)and∇e its Levi-Civita connection. First of all, we recall that the curvature tensor ofCP2(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 thatJ2=−IdandJis antisymmetric, we get thatjandtare antisymmetric and we have the following relations between these four operators

j2X =−X−shX, (10)

t2ξ=−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(∇Xξ) =−B(sξ, X)−h(AξX), (17)

X(sξ)−s(∇Xξ) =−j(AξX) +AX, (18) whereB :T M×T M −→N Mis the second fundamental form and for anyξ∈T M,Aξ 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


(∇XB)(Y, Z)−(∇YB)(X, Z) = c

hjY, ZihX− hjX, ZihY + 2hjX, YihZ


R(X, Y)ξ = c

hhY, ξihX− hhX, ξihY + 2hjX, Yitξ

+B(AξY, X)−B(AξX, Y).

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

KM = 1+< B22, B11>−|B12|2+ 3< je1, e2>2, (19) KN =−<[Sν1, Sν2](e1), e2>+h21h12−h11h22+ 2j12t12, (20) (∇e1B)(e2, ek)−(∇e2B)(e1, ek) =j2khe1−j1khe2+ 2j12hek (21) It is clear that equations (10) to (21) are necessary condition for surfaces in CP2(4c).

Conversely, given(M2, 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 forCP2(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 ofCP2, 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 CP2(4c)then, there exists an isometric immersion Φ : M −→ CP2(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(ΦX) = Φ(jX) +Φ(hX),e J(eΦξ) = Φ(sX) +Φ(tξ),e

whereJ is the canonical complex structure ofCP2(4c)and this isometric immersion is unique up to an isometry ofCP2(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(M2, g)of CP2(4c)is saidcomplexif the tangent bundle of M is stable by the complex structure ofCP2(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 ofCP2(4c)becomes







h= 0, s= 0, j2=−idT M, t2=−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)(e2, ek)−(∇e2B)(e1, ek) = 0


A surface(M2, g)ofCP2(4c)is said totally real ifJ(T M)is transversal toT M. In the particular case whenJ(T M) = N M, we say that(M2, g)isLagrangian. In that case, we havej = 0andt = 0. Hence, the compatibility equations for Lagranigan surfaces of CP2(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 =−<[Sν1, Sν2](e1), e2>+h21h12−h11h22

(∇e1B)(e2, ek)−(∇e2B)(e1, ek) = 0



We consider a specialSpinc structure onCP2carrying 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 = ϕ01 ∈ Γ(Σ0CP2⊕Σ1CP2)satisfies the following:



∇eXϕ1=−X+·ϕ0, Thus, we have


2X·ϕ+ i

2J X·ϕ,

whereϕ =ϕ0−ϕ1is the conjugate ofϕfor the action of the complex volume element ω4C = −e1·e2·e3·e4. Indeed,Σ0CP2 ⊂ Σ+CP2andΣ1CP2 = ΣCP2. 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 Spincstructure.

Now, letMbe a surface ofCP2with normal bundle denoted byN M. By the identification of the Clifford multiplications and theSpincGauss formula, we have



2X·ϕ+ i

2J X·ϕ.

In intrinsic terms, it can be written as

Xϕ = −1


2X·ϕ+ i

2jX·ϕ+ i

2hX·ϕ, (24)


whereηis given by

η(X) =




ej·B(ej, X). (25)

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



2X·ϕ− i

2jX·ϕ−i 2hX·ϕ.

Now, let us go back to an instrinsic setting by considering(M2, 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{e1, e2} be a normal local orthonormal frame ofT Mand{ν1, ν2}a local orthonormal frame ofE. We have

e1e2ϕ = −1





4η(e2)·j(e1)·ϕ− i



2∇e1e2·ϕ +1

4e2·e1·ϕ− i

4e2·j(e1)·ϕ− i

4e2·h(e1)·ϕ +i

2∇e1(j(e2))·ϕ− i


4j(e2)·e1·ϕ +1


4j(e2)·h(e1)·ϕ+ i





4h(e2)·j(e1)·ϕ +1


= −1







4e2·η(e1)·ϕ +i

2∇e1(j(e2))·ϕ+ i



4e2·j(e1)·ϕ+ i



4e2·h(e1)·ϕ+ i

4h(e2)·e1·ϕ +1

4j(e2)·je1·ϕ +1

4h(e2)·he1·ϕ +1

4 j(e2)·h(e1)·ϕ+h(e2)·je1·ϕ


4 η(e2)·h(e1)·ϕ+h(e2)·η(e1)·ϕ


4 η(e2)·j(e1)·ϕ+j(e2)·η(e1)·ϕ




I = −1




2∇e1e2·ϕ II = +i

2∇e1(j(e2))·ϕ+ i


III = −i

4e2·j(e1)·ϕ+ i


IV = −i

4e2·h(e1)·ϕ+ i


V = +1


V I = +1


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


4e2·η(e1)·ϕ We point out that


2η([e1, e2])·ϕ−1

2[e1, e2]·ϕ

| {z }

I([ee 1,e2])


2j([e1, e2])·ϕ+ i

2h([e1, e2])·ϕ

| {z }



Some terms are vanishing as shown in the following lemma.

Lemma 4.1. We have

X(e1, e2)−X(e2, e1) = 0 (26) IV(e1, e2)−IV(e2, e1) = 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−12B(ej, X) =ej·η(X)−η(X)·ej. Hence X(e1, e2)−X(e2, e1) =−1

8B(e2, e1) +1

8B(e1, e2) = 0.



IV(e1, e2)−IV(e2, e1) =


4(e2·h(e1)−h(e2)·e1)·ϕ+ i


= 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(e1, e2)−II(e2, e1)−IIf([e1, e2])

= i

2∇e1(j(e2))·ϕ+ i 2∇e

1(h(e2))·ϕ− i

2∇e2(j(e1))·ϕ− i 2∇e



2j([e1, e2])·ϕ− i

2h([e1, e2])·ϕ

= i 2

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

= i 2

s(B(e1, e2))·ϕ+Sh(e2)e1·ϕ+t(B(e1, e2))·ϕ−B(e1, j(e2))·ϕ

−s(B(e1, e2))·ϕ−Sh(e1)e2·ϕ−t(B(e1, e2))·ϕ+B(e2, je1)·ϕ

= i 2

Sh(e2)e1·ϕ−Sh(e1)e2·ϕ−B(e1, j(e2))·ϕ+B(e2, je1)·ϕ

(30) Moreover, we calculate

−B(e1, j(e2))·ϕ+B(e2, je1)·ϕ

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

= 2g(j(e1), e2)H·ϕ (31)



= −< Sh(e1)e2, e1> e1·ϕ−< Sh(e1)e2, e2> e2·ϕ +< Sh(e2)e1, e1> e1·ϕ+< Sh(e2)e1, e2> e2·ϕ

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

+< B(e1, e1), h(e2)> e1·ϕ+< B(e1, e2), h(e2)> e2·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)


= 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)



IX(e1, e2)−IX(e2, e1)

= −i

4η(e2)·j(e1)·ϕ− i

4j(e2)·η(e1)·ϕ+ i

4η(e1)·j(e2)·ϕ+ i


= i 4

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

−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


= i 4

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

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


= i

4 −2g(j(e1), e2)B(e2, e2) + 2g(j(e2, e1))B(e1, e1)


= −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(e1, e2)−V(e2, e1) =−1

2hj(e2), e1)i2e1·e2, (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, e2)−V II(e2, e1) = 1

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

·ϕ(36) Proof.

(1) We denote byjkl =g(jek, el). Sincejis antisymmetric, we havejkl =−jlkand so

V(e1, e2)−V(e2, e1)

= 1

4 j(e2)·j(e1)−j(e1)·j(e2)


= 1


= 1


= −1

2g(j(e1), e2)2e1·e2·ϕ (37)



V I(e1, e2)−V I(e2, e1)

= 1


= 1

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



= 1

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

III(e1, e2)−III(e2, e1) = −i

4 e2·j(e1)−j(e2)·e1−e1·j(e2) +j(e1)·e2


= −i

4 −j12+j21+j21−j12


= ig(e2, j(e1))ϕ (38)

(4) We have

V II(e1, e2)−V II(e2, e1)

= 1

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


= 1

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



= 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




ej· (∇0e1B)(e2, ej))−(∇0e2B)(e1, ej)


2g([Sν1, Sν2])(e1), e21·ν2·ϕ


2e1·e2·ϕ, (40)

where∇0is the natural connection onTM⊗TM⊗EandBkl=B(ek, el)


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

Re1,e2ϕ = 1


2KEν1·ν2·ϕ+ i

M+E(e1, e2)ϕ, (41) withαM+E(e1, e2) = 0becausej= 0. From the other hand, we have


Re1,e2ϕ (42)

= −1 2




ej·((∇e1B)(e2, ej)−(∇e2B)(e1, ej))·ϕ


2(|B12|2−< B11, B22>)e1·e2·ϕ +1

2 <[Sν1, Sν2](e1), e2> ν1·ν2·ϕ



2(h21h12−h11h221·ν2·ϕ (43) We get finally thatT·ϕ= 0, whereT ∈(Λ2M⊗1⊕T M⊗E⊕1⊗Λ2E). Thus, since ϕ+andϕnever vanish, we haveT = 0by [4][Lemma 3.4] and it gives

KM =< B11, B22>−|B12|2+ 1,

KE=−<[Sν1, Sν2](e1), e2>−(h21h12−h22h11), (∇e1B)(e2, ej)−(∇e2B)(e1, ej) = 0,

which are Gauss, Ricci and Codazzi equations for a Lagrangian surface inCP2and so the conditions (23) are fulfuilled. Hence, by Proposition 3.2, we conclude that there exists a Lagrangian isomertic immersion from(M, g)intoCP2withEas 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.


Assume thats= 0;h= 0soαM+E(e1, e2) =−2. We takeje1 =e2andtν12, i.e, g(je1, e2) =g(tν1, ν2) = 1.

We calculate and we get Re1,e2ϕ = −1


2KNν1·ν2·ϕ+ i

M+N(e1, e2

= −1



= T·ϕ−iϕ, (44)

whereT is a2-form. From the other hand, we have Re1,e2ϕ = −e1·e2·ϕ+iϕ

−1 2




ej·((∇e1B)(e2, ej)−(∇e2B)(e1, ej))·ϕ


2(|B12|2−< B11, B22>)e1·e2·ϕ +1

2 <[Sν1, Sν2](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




2(|B12|2−< B11, B22>)e1·e2·ϕ


2 <[Sν1, Sν2](e1), e2> ν1·ν2·ϕ +1





ej·((∇e1B)(e2, ej)−(∇e2B)(e1, ej))·ϕ (46)

We give now the following Lemma

Lemma 6.1. LetT be a2form, i.e,T ∈Λ2M ⊗1⊕Λ1M ⊗Λ1E⊕1⊗Λ2E. Assume that

T ·ϕ−iϕ−iϕ= 0,

and writeT =Tte1·e2+Tnν1·ν2+Tm,whereTm∈Λ1M⊗Λ1E. Then, Tt=−1, Tn = 0, and Tm= 0.


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


2X·ϕ−+− i



2X·ϕ−−+ i



2X·ϕ+++ i



2X·ϕ+−− i


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

T ·ϕ−iϕ−iϕ= 0, becomes

Tte1·e2·(ϕ++−+)+(Tn+1)ν1·ν2·(ϕ++−+)+Tm·(ϕ++−+) =iϕ=i(ϕ++−ϕ−+) Taking the scalar product withϕ++then withϕ−+, we get

Tt+Tn+ 1 =−1,

−Tt+Tn+ 1 = 1,

which gives Tn = −1,Tt = −1 andTm = 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 CP2 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.



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






hijei·ξj, wherehi,j=hhei, ξji.

As in [4] and [32], we will show that this equation with an appropiate condition on the norm of bothϕ+andϕis 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+|2and|ϕ|2. Indeed, after decomposition ontoΣ+andΣ, (24) becomes



2X·ϕ∓ i

2jXϕ∓ i 2hXϕ. From this we deduce that

X(|ϕ±|2) = Re


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


X· ∇Yϕ+−1

2(X+ijX+ihX)·Y ·ϕ, ξ·ϕ+

(49) + 1


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


2X·ϕ∓ i

2jX·ϕ∓ i

2hX·ϕ, ϕ±

thenϕis solution of Equation(24)



2X·ϕ+ i

2jX·ϕ+ i 2hX·ϕ whereηis defined byη(X) =−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(M2, 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


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