spinorial characterizations of surfaces into three-dimensional homogeneous manifolds

Journal of Geometry and Physics 60 (2010) 1045–1061

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Journal of Geometry and Physics

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Spinorial characterizations of surfaces into three-dimensionalhomogeneous manifoldsJulien RothUniversité Paris-Est Marne-la-Vallée, LAMA (UMR 8050), Cité Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France

a r t i c l e i n f o

Article history:Received 4 September 2007Received in revised form 12 March 2010Accepted 17 March 2010Available online 25 March 2010

MSC:53C2753C4053C80

Keywords:Dirac operatorKilling spinorsIsometric immersionsGauss and Codazzi equations

a b s t r a c t

We give spinorial characterizations of isometrically immersed surfaces into three-dimensional homogeneous manifolds with four-dimensional isometry group in terms ofthe existence of a particular spinor field. This generalizes works by Friedrich for R3 andMorel for S3 and H3. The main argument is the interpretation of the energy–momentumtensor of such a spinor field as the second fundamental form up to a tensor depending onthe structure of the ambient space.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Over the past years, the spinorial tool has been used successfully in the study of the geometry and the topology ofsubmanifolds of space forms. The spinorial approach allows one to solve some problems of the geometry of submanifoldsnaturally. For instance, some simple proofs of the Alexandrov theorem in the Euclidean space [1] or in the hyperbolicspace [2] were given and new results about Einstein manifolds or Lagrangian submanifolds of Kähler manifolds [3] wereobtained.Over the same period, several results concerning three-dimensional homogeneous manifolds with four-dimensional

isometry group were obtained, especially concerning minimal and constant mean curvature surfaces in these spaces, byAbresch and Rosenberg (see [4,5], for instance). In this paper, we make use of the spinorial tool in the study of surfacesin these three-dimensional homogeneous manifolds. As in the case of space forms, we can think that a spinorial approachcould lead to solving some open questions in the theory of surfaces in three-dimensional homogeneous manifolds, such asthe existence of an Alexandrov-type theorem in the Heisenberg group Nil3.The first step of such an approach is to understand the spinorial geometry of these surfaces and in particular to give a

spinorial characterization of the surfaces which are isometrically immersed into these homogeneous manifolds. Precisely,we characterize these surfaces by the existence of a special spinor field. Then, we show that the existence of such a specialspinor field is equivalent to the existence of a spinor field solution of a weaker equation (involving the Dirac operator) withan additional condition on the norm of the spinor field (see Theorems 1 and 2).The results that we give in this paper are non-trivial generalizations of [6,7] to these homogeneous manifolds.

E-mail address: [email protected].

0393-0440/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.geomphys.2010.03.007

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http://dx.doi.org/10.1016/j.geomphys.2010.03.007

1046 J. Roth / Journal of Geometry and Physics 60 (2010) 1045–1061

2. Preliminaries

We begin the paper with a short section of preliminaries in order to recall the basic facts about the spin geometry ofhypersurfaces. For further details, the reader can refer to [8–10] for general properties of spin manifolds and [11–14] for thebasics of extrinsic spin geometry.

2.1. Hypersurfaces and induced spin structures

Let (M3, g) be a three-dimensional Riemannian spin manifold andΣM its spinor bundle. We denote by∇M the spinorialLevi-Civita connection onΣM . The Clifford multiplication will be denoted by γM and 〈., .〉 is the natural Hermitian productonΣM compatible with ∇M and γM . Finally, we denote by DM the Dirac operator, locally given by DM =

∑3i=1 γ

M(ei)∇Mei ,where {e1, e2, e3} is a local orthonormal frame of TM .Now let N be an orientable surface of M . Since the normal bundle is trivial, the hypersurface N is also spin. Indeed, the

existence of a normal unit vector field ν globally defined on N induces a spin structure from the spin structure ofM .Then we can consider the intrinsic spinor bundle ΣN of N and the extrinsic spinor bundle on N defined by S := ΣM|N .

Since N is odd-dimensional, there exists a natural identification between these two spinor bundles:

S ≡ ΣN, (1)

and between their connections and Clifford multiplications:

∇S≡ ∇

N and γ S≡ γ N . (2)

The interest of this identification is that we can use restrictions of ambient spinors to study the intrinsic Dirac operator ofN , with the following identities:

∇SX = ∇

MX −

12γM(AX)γM(ν) (3)

γ S(X) = γM(X)γM(ν), (4)

where A is the shape operator of the immersion of N intoM . Eq. (3) is called the spinorial Gauss formula.We finish this section with the following well-known fact. The spinor bundleΣN decomposes in two subbundles under

the action of the complex volume element γ N(ω2) = iγ N(e1)γ N(e2), where {e1, e2} is a local orthonormal frame of TN .Namely, we haveΣN = Σ+N ⊕Σ−N , whereΣ±N is the bundle of eigenspinors for the action of γ N(ω2) associated withthe eigenvalue±1. Hence, any spinor field ϕ decomposes in the following way: ϕ = ϕ++ϕ−, with γ N(ω2)ϕ± = ±ϕ±. Wedenote ϕ = γ N(ω2)ϕ = ϕ+ − ϕ−. Moreover, the Clifford multiplication by a vector exchangesΣ+N andΣ−N . Hence, theDirac operator DN also exchangesΣ+N andΣ−N .

2.2. Three-dimensional homogeneous manifolds with a four-dimensional isometry group

In this section, we give a description of three-dimensional homogeneous manifolds with a four-dimensional isometrygroup. Such a manifold is a Riemannian fibration over a simply connected two-dimensional manifold with constantcurvatureκ and such that the fibers are geodesic.Wedenote by τ the bundle curvature definedbelowby (5), whichmeasuresis the fibration is a Riemannian product or not. When τ vanishes, we get a product manifoldM2(κ)×R, whereM2(κ) is thetwo-dimensional space form of curvature κ .When τ 6= 0, these manifolds are of three types: they have the isometry group of the Berger spheres if κ > 0, of the

Heisenberg group Nil3 if κ = 0, or of ˜PSL2(R) if κ < 0. In what follows, we denote these homogenous manifolds by E(κ, τ ).We begin by giving a short description of E(κ, τ ). For further details, one can refer to [15] or [16].Let E(κ, τ ) be a three-dimensional homogeneous manifold with a four-dimensional isometry group and assume that

τ 6= 0; that is, E(κ, τ ) is not a product manifold M2(κ) × R. As we said, E(κ, τ ) is a Riemannian fibration over a simplyconnected two-dimensionalmanifoldwith constant curvature κ and such that the fibers are geodesic. Now, let ξ be a unitaryvector field tangent to the fibers. We call it the vertical vector field. This vector field is a Killing vector field (correspondingto the translations along the fibers) and satisfying

∇Xξ = τX ∧ ξ, (5)

where ∇ is the Riemannian connection of E(κ, τ ) and ∧ is the vector product in E(κ, τ ); that is, for any X, Y , Z ∈ Γ (T M),

〈X ∧ Y , Z〉 = det {e1,e2,ξ}(X, Y , Z).

Note that, if τ = 0, then ξ = ∂∂t is the unit vector field giving the orientation of R in the productM

2(κ)× R.The manifold E(κ, τ ), with τ 6= 0, admits a local direct orthonormal frame {e1, e2, e3}with

e3 = ξ

J. Roth / Journal of Geometry and Physics 60 (2010) 1045–1061 1047

and such that the Christoffel symbols Γ kij =⟨∇eiej, ek

⟩are

Γ312 = Γ

123 = −Γ

321 = −Γ

213 = τ ,

Γ132 = −Γ

231 = τ −

κ

2τ,

Γiii = Γ

iij = Γ

iji = Γ

jii = 0, ∀ i, j ∈ {1, 2, 3}.

(6)

Then we have

[e1, e2] = 2τe3, [e2, e3] =κ

2τe1, [e3, e1] =

κ

2τe2.

We will call {e1, e2, ξ} the canonical frame of E(κ, τ ).

2.3. Compatibility equations

Here, we give the basic identities satisfied by surfaces into E(κ, τ ). Let N be an orientable surface of E(κ, τ )with shapeoperatorA associatedwith theunit normal ν.Moreover,wedenote ξ = T+f ν, where the function f is the normal componentof ξ and T is its tangential part. Then, from Eq. (5), we easily deduce the following properties.

Proposition 2.1. For X ∈ X(N), we have

∇XT = f (AX − τ JX) and df (X) = −〈AX − τ JX, T 〉,

where J is the rotation of angle π2 on TN.

Proof. On the one hand, we have

∇Xξ = ∇X (T + f ν)= ∇XT + df (X)ν + f∇Xν= ∇XT + 〈AX, T 〉 ν + df (X)ν − fAX,

and on the other hand

∇Xξ = τX ∧ ξ= τX ∧ (T + f ν)= τ

(〈JX, T 〉 ν − fJX

).

We get the two identities by taking the normal and tangential parts. �

Definition 2.2 (Compatibility Equations). We say that (N, 〈., .〉, A, T , f ) satisfies the compatibility equations for E(κ, τ ) ifand only if, for any X, Y , Z ∈ X(N),

K = det (A)+ τ 2 + (κ − 4τ 2)f 2 (7)

∇XAY −∇YAX − A[X, Y ] = (κ − 4τ 2)f (〈Y , T 〉X − 〈X, T 〉Y ), (8)∇XT = f (AX − τ JX), and (9)df (X) = −〈AX − τ JX, T 〉, (10)

where K is the Gauss curvature of N .

Remark 1. The relations (7) and (8) are the Gauss and Codazzi equations for an isometric immersion into E(κ, τ ) obtainedby a computation of the curvature tensor of E(κ, τ ).

In [17,15], Daniel proves that these compatibility equations are necessary and sufficient for the existence of an isometricimmersion F from N into E(κ, τ )with shape operator dF ◦ A ◦ dF−1 and so that ξ = dF(T )+ f ν.

3. Special spinor fields on surfaces of E(κ, τ)

Now, we describe the special spinor fields which will be involved in our characterization of surfaces into E(κ, τ ).


3.1. Special spinor fields on surfaces ofM2(κ)× R

3.1.1. Spinor bundle ofM2(κ)× RIt is awell-known fact that sinceM2(κ) is spin, thenM2(κ)×R is also spin and the spin structures ofM2(κ) andM2(κ)×R

are in a one-to-one correspondence (see [14] for more details).We will explain explicitly the spinor bundle ofM2(κ)× R. We have seen in Section 2.1 that

Σ(M2(κ)× R)|M2 ≡ ΣM2(κ).

So, if ϕ ∈ Γ (Σ(M2(κ)× R)), then, for any t ∈ R, ϕ(., t) ∈ Γ (ΣM2(κ)). Hence a section ofΣ(M2(κ)× R) is a C∞-map

ϕ:R −→ Γ (Σ(M2(κ)))t 7−→ ϕt .

3.1.2. Restriction to a surfaceIn what follows, we will consider some particular and interesting sections, precisely the sections which do not depend

on t . We have the spinorial Gauss formula,

∇Xϕ = ∇Xϕ +12γ (AX)γ (ν)ϕ,

where ∇ is the spinorial connection onM2(κ) × R, the spinorial connection onM2(κ) is ∇ , the Clifford multiplication onM2(κ)×R is γ , and A is theWeingarten operator of the immersion ofM2(κ) intoM2(κ)×R. SinceM2(κ) is totally geodesicin the productM2(κ)× R, if we take ϕt = ϕ0 a Killing spinor onM2(κ), i.e., ∇Xϕ0 = ηγM2(κ)(X)ϕ0, with κ = 4η2, then wehave

∇Xϕ = ηγM2(κ)(X)ϕ = ηγ (X)γ

(∂

∂t

)ϕ.

On the other hand, the complex volume form ωC3 = −e1 · e2 ·

∂∂t acts as an identity, so we have

γ (e1)γ(∂

∂t

)ϕ = −γ (e2)ϕ, and γ (e2)γ

(∂

∂t

)ϕ = γ (e1)ϕ.

Hence we deduce that∇e1ϕ = −ηγ (e2)ϕ,∇e2ϕ = ηγ (e1)ϕ,∇ ∂

∂tϕ = 0.

(11)

These particular spinor fields will play the same role as Killing spinor fields in the work of Morel [7] for S3 or H3.Now, let (N, 〈·, ·〉) be a surface of M2(κ) × R, oriented by ν. Since (N, 〈·, ·〉) is oriented, it could be equipped with a

spin structure induced from the spin structure ofM2(κ) × R and we have the following identification between the spinorbundles:

Σ(M2(κ)× R)|N ∼= Σ N.

For any X ∈ X(N) and any ψ ∈ Γ (Σ(M2(κ)× R)), we have, from (3),(∇Xψ

)|N = ∇X

(ψ|N

)+12γ N(AX)ψ|N

= ∇X(ψ|N

)+12γ (AX)γ (ν)ψ|N .

If we use this formula for the particular spinor field onM2(κ)× R given by (11), we get

∇Xϕ = ∇Xϕ −12γ N(AX)ϕ

= ηγ (Xt)γ(∂

∂t

)ϕ −

12γ N(AX)ϕ,

where Xt is the part of X tangent toM2(κ); that is,

Xt = X −⟨X,

∂

∂t

⟩∂

∂t= X − 〈X, T 〉 T − f 〈X, T 〉 ν.


So, we deduce that

∇Xϕ = ηγ (X)γ(∂

∂t

)ϕ − η

⟨X,

∂

∂t

⟩γ

(∂

∂t

)γ

(∂

∂t

)ϕ −

12γ N(AX)ϕ

= ηγ (X)γ(∂

∂t

)ϕ + η

⟨X,

∂

∂t

⟩ϕ −

12γ N(AX)ϕ

= ηγ (X)γ (T )ϕ + ηf γ (X)γ (ν)ϕ + η 〈X, T 〉ϕ −12γ N(AX)ϕ.

On the other hand, if we denote by ω = e1 ·Ne2 the real volume element on N , we have the following relations:{

γ (X) = −γ N(X)γ N(ω),γ (ν) = γ N(ω).

By using these two identities, the fact that ω2 = −1, and that ω anti-commutes with vector fields tangent to N , we get

∇Xϕ = ηγN(X)γ N(ω)γ N(T )γ N(ω)ϕ + η 〈X, T 〉ϕ − ηf γ N(X)γ N(ω)γ N(ω)ϕ −

12γ N(AX)ϕ.

Now, we can rewrite this equation in an intrinsic way as follows:

∇Xϕ = ηX · T · ϕ + ηfX · ϕ + η 〈X, T 〉ϕ −12AX · ϕ (12)

where ‘‘·’’ stands for the Clifford multiplication on N . This spinor field satisfies the following property.

Proposition 3.1. Let ϕ be a solution of Eq. (12).

(i) If η = 12 , then the norm of ϕ is constant.

(ii) If η = i2 , then the norm of ϕ satisfies, for any X ∈ X(N),

X |ϕ|2 = <e 〈iX · T · ϕ + ifX · ϕ, ϕ〉 .

Proof. We need to compute X |ϕ|2 for X ∈ X(N). We have

X |ϕ|2 = 2<e 〈∇Xϕ, ϕ〉 .

We replace∇Xϕ by the expression given by (12), and we use the compatibility of the Clifford multiplication with the scalarproduct to get the result. �

3.2. Special spinor fields on surfaces of E(κ, τ ), τ 6= 0

3.2.1. Special spinor fields on E(κ, τ ), τ 6= 0We will give here the expression of special spinor fields on E(κ, τ )which will replace parallel or Killing spinors that we

have in space forms. For this, let us endow E(κ, τ ) with its trivial spin structure. We can consider a constant section ϕ ofthe spinor bundle. From the local expression of the spinorial connection and of the Christoffel symbols Γ kij given by (6), wecan compute ∇Xϕ. We recall that the complex volume element ωC

3 = −e1 · e2 · ξ acts as an identity, which implies that

e1 · e2 · ϕ = ξ · ϕ, e2 · ξ · ϕ = e1 · ϕ, and ξ · e1 · ϕ = e2 · ϕ.

The expression of the spinorial connection is, for any X ∈ X(E(κ, τ )),

∇Xϕ = X(ϕ)+14

3∑i,j=1

⟨∇Xei, ej

⟩γ (ei)γ (ej)ϕ.

So, we deduce from (6) that, for a constant section ϕ,∇e1ϕ =

12τe1 · ϕ,

∇e2ϕ =12τe2 · ϕ,

∇ξϕ =12

( κ2τ− τ

)ξ · ϕ.

(13)


3.2.2. Restriction to a surfaceIn this section, we give the restriction of these spinors to a surface of E(κ, τ ). By a computation similar to the one of

Section 3.1.2, we obtain, after restriction to N , a spinor field satisfying

∇Xϕ = −τ

2X · ω · ϕ +

α

2〈X, T 〉 T · ω · ϕ −

α

2f 〈X, T 〉ω · ϕ −

12AX · ϕ, (14)

where we have set α = 2τ − κ2τ . From this equation, we get the following consequence for the norm of a solution.

Proposition 3.2. The norm of a spinor field solution of (14) is constant.

4. Isometric immersions into M2(κ)× R

4.1. Statement of the result

Here, we state the main result of the present paper, which gives a spinorial characterization of surfaces isometricallyimmersed into the product spaces S2 × R and H2 × R. Before stating the theorem, we want to recall that the energy–momentum tensor Qϕ associated with a spinor field ϕ is the symmetric 2-tensor defined by

Qϕ(X, Y ) =12<e⟨X · ∇Yϕ + Y · ∇Xϕ,

ϕ

|ϕ|2

⟩, (15)

on the complementary set of zeros of ϕ.

Theorem 1. Let η ∈ R or iR, η 6= 0 and κ = 4η2. Let (N, 〈., .〉) be a connected, oriented and simply connected Riemanniansurface. Let T be a vector field, and f and H two real functions on N satisfying f 2 + ‖T‖2 = 1. Let A be a field of endomorphismson TN with tr (A) = 2H. The following three statements are equivalent.(i) There exists an isometric immersion F fromN into S2(κ)×R (resp.H2(κ)×R) ofmean curvature H such that theWeingartenoperator related to the normal ν is given by

dF ◦ A ◦ dF−1

and such that∂

∂t= dF(T )+ f ν.

(ii) There exists a non-trivial spinor field ϕ satisfying

∇Xϕ = ηX · T · ϕ + ηfX · ϕ + η 〈X, T 〉ϕ −12AX · ϕ,

where A satisfies

∇XT = fAX, and df (X) = −〈AX, T 〉.

(iii) There exists a nowhere vanishing spinor field ϕ satisfying

Dϕ = Hϕ − ηT · ϕ − 2ηf ϕ,

of constant norm (resp. satisfying X |ϕ|2 = <e(iX · T · ϕ + ifX · ϕ, ϕ

)), and such that

∇XT = 2fQϕ(X)− fB(X) and df = −2Qϕ(T )+ B(T ),

where Qϕ is the energy–momentum tensor associated with the spinor field ϕ and B is the symmetric 2-tensor defined by

B(X, Y ) = −2<e⟨η 〈X, Y 〉 T · ϕ,

ϕ

|ϕ|2

⟩− 2<e

⟨ηf 〈X, Y 〉ϕ,

ϕ

|ϕ|2

⟩−<e

⟨η (〈X, T 〉 Y + 〈Y , T 〉 X) · ϕ,

ϕ

|ϕ|2

⟩.

Note that

• if η = 12 , then B = f Id ;

• if η = i2 , then B satisfiesB11 = −<e 〈iT · ϕ, ϕ〉 − <e

⟨iT1e1 · ϕ,

ϕ

|ϕ|2

⟩B12 = B21 =

12<e⟨i(T1e2 + T2e1) · ϕ,

ϕ

|ϕ|2

⟩B22 = −<e 〈iT · ϕ, ϕ〉 − <e

⟨iT2e2 · ϕ,

ϕ

|ϕ|2

⟩.


4.2. Special spinor fields and compatibility equations

For the first time, we show that the existence of a solution of Eq. (12) implies the Gauss and Codazzi equations. We willsee that the integrability conditions for this equation are precisely the Gauss and Codazzi equations.

Proposition 4.1. Let (N, 〈·, ·〉) be an oriented surface with a non-trivial solution of Eq. (12) and such that Eqs. (9) and (10) aresatisfied. Then, the Gauss and Codazzi equations for M2(κ)× R are also satisfied.

Proof. The proof of this proposition is based on the computation of the spinorial curvature applied to the spinor field ϕsolution of Eq. (12); i.e.

R(X, Y )ϕ = ∇X∇Yϕ −∇Y∇Xϕ −∇[X,Y ]ϕ,

for X, Y ∈ X(N). Using the expression given by (12), Eqs. (9) and (10), the fact that ω2 = −1, and that ω anti-commuteswith the vector fields tangent to N , we get

∇X∇Yϕ = ηfY · AX · ϕ︸︷︷︸α1(X,Y )

+ η2Y · T · X · T · ϕ︸︷︷︸α2(X,Y )

+ η2fY · T · X · ϕ︸︷︷︸α3(X,Y )

−η

2Y · T · AX · ϕ︸︷︷︸−α4(X,Y )

− η 〈AX, T 〉 Y · ϕ︸︷︷︸−α5(X,Y )

+ η2fY · X · T · ϕ︸︷︷︸α6(X,Y )

+ η2 〈X, T 〉 Y · T · ϕ︸︷︷︸α7(X,Y )

+ η2f 2Y · X · ϕ︸︷︷︸α8(X,Y )

+ η2f 〈X, T 〉 Y · ϕ︸︷︷︸α9(X,Y )

−η

2fY · AX · ϕ︸︷︷︸−α10(X,Y )

+ ηf 〈Y , AX〉ϕ︸︷︷︸α11(X,Y )

+ η2 〈Y , T 〉 X · T · ϕ︸︷︷︸α12(X,Y )

+ η2f 〈Y , T 〉 X · ϕ︸︷︷︸α13(X,Y )

+ η2 〈X, T 〉〈Y , T 〉ϕ︸︷︷︸α14(X,Y )

−η

2〈Y , T 〉 AX · ϕ︸︷︷︸−α15(X,Y )

−12∇X (AY ) · ϕ︸︷︷︸−α16(X,Y )

−η

2AY · X · T · ϕ︸︷︷︸−α17(X,Y )

−η

2fAY · X · ϕ︸︷︷︸−α18(X,Y )

−η

2〈X, T 〉 AY · ϕ︸︷︷︸−α19(X,Y )

+14AY · AX · ϕ︸︷︷︸α20(X,Y )

+ η∇XY · T · ϕ︸︷︷︸α21(X,Y )

+ ηf∇XY · ϕ︸︷︷︸α22(X,Y )

+ η 〈∇XY , T 〉ϕ︸︷︷︸α23(X,Y )

.

That is,

∇X∇Yϕ =

23∑i=1

αi(X, Y ).

Obviously, by symmetry, we have

∇Y∇Xϕ =

23∑i=1

αi(Y , X).

On the other hand, we have

∇[X,Y ]ϕ = η[X, Y ] · T · ϕ︸︷︷︸β1([X,Y ])

+ ηf [X, Y ] · ϕ︸︷︷︸β2([X,Y ])

+ η 〈[X, Y ], T 〉ϕ︸︷︷︸β3([X,Y ])

−12A[X, Y ] · ϕ︸︷︷︸−β4([X,Y ])

.

Since the connection ∇ is torsion-free, we get ∇XY −∇YX − [X, Y ] = 0, which implies that

α21(X, Y )− α21(Y , X)− β1([X, Y ]) = 0,α22(X, Y )− α22(Y , X)− β2([X, Y ]) = 0,α23(X, Y )− α23(Y , X)− β3([X, Y ]) = 0.

Moreover, by symmetry, we have

α11(X, Y )− α11(Y , X) = 0 and α14(X, Y )− α14(Y , X) = 0.

Other terms vanish by symmetry. Namely,

α1(X, Y )+ α10(X, Y )+ α18(X, Y )− α1(Y , X)− α10(Y , X)− α18(Y , X) = 0,α3(X, Y )+ α6(X, Y )− α3(Y , X)− α6(Y , X) = 0,

and

α4(X, Y )+ α5(X, Y )+ α15(X, Y )+ α17(X, Y )+ α19(X, Y )−α4(Y , X)− α5(Y , X)− α15(Y , X)− α17(X, Y )− α19(X, Y ) = 0.


The terms α2, α7, α8 et α12 can be combined. Indeed, if we set

α = α2 + α7 + α8 + α12,

then

α(X, Y )− α(Y , X) = η2[f 2 (Y · X − X · Y )+ Y · T · X · T − X · T · Y · T

]· ϕ

= η2[f 2 (Y · X − X · Y )+ ‖T‖2 (Y · X − X · Y )

]· ϕ − 2η2 (〈X, T 〉 Y · T − 〈Y , T 〉 X · T ) · ϕ,

by taking X and Y as a direct orthonormal frame {e1, e2}, we have

α(e1, e2)− α(e2, e1) = 2η2[−T1e2 · (T1e1 + T2e2)+ T2e1 · (T1e1 + T2e2)

]· ϕ − 2η2e1 · e2 · ϕ

= −2η2ω · ϕ + 2η2(T 21ω · +T1T2 − T1T2 + T

22ω·

)ϕ

= −2η2f 2ω · ϕ.

Always for X = e1 and Y = e2,

α9(e1, e2)+ α13(e1, e2)− α9(e2, e1)− α13(e2, e1) = 2T · ω · ϕ = −2J(T ) · ϕ,

where J is the rotation of positive angle π2 on TN . Finally, we get

R(e1, e2)ϕ =14(Ae2 · Ae1 − Ae1 · Ae2) · ϕ − 2η2f 2ω · ϕ −

12d∇A(e1, e2) · ϕ − 2J(T ) · ϕ.

Using the Ricci identity

R(e1, e2)ϕ = −12R1212e1 · e2 · ϕ,

and since κ = 4η2, we haveR1212 − det (A)− κ f 2︸︷︷︸G

ω · ϕ =d∇A(e1, e2)+ κ fJ(T )︸︷︷︸

C

· ϕ;that is,

Gω · ϕ = C · ϕ,

where G is a function and C a vector field. Since, ω · ϕ = −iϕ, we obtain

C · ϕ± = ±iGϕ∓.

Thus,

‖C‖2ϕ± = −G2ϕ±.

Since ϕ is a non-trivial solution of (12), by Proposition 3.1, ϕ+ and ϕ− cannot vanish at the same point. Then C = 0 andG = 0. But G = 0 is the Gauss equation and C = 0 is the Codazzi equation, which achieves the proof. �

Hence, by Daniel’s result, there exists an isometric immersion from N into E(κ, τ ).

4.3. Special spinor fields and Dirac equation

If a spinor field ϕ is a solution Eq. (12), then, we deduce immediately that ϕ is also a solution of the corresponding Diracequation

Dϕ = Hϕ − ηT · ϕ − 2ηf ϕ. (16)

In this section, we will see that this Dirac equation, which is weaker than Eq. (12), is in fact equivalent for spinors with anorm satisfying the condition of Proposition 3.1. We write ϕ = ϕ+ + ϕ− with ϕ± ∈ Σ±, and then we have

Dϕ± = Hϕ∓ − ηT · ϕ± − 2ηf ϕ∓. (17)

Now, we define the following endomorphisms:

Q±ϕ (X, Y ) = <e⟨∇Xϕ

±, Y · ϕ∓⟩,


and

B±(X, Y ) = −<e⟨ηX · T · ϕ±, Y · ϕ∓

⟩−<e

⟨η 〈X, T 〉ϕ±, Y · ϕ∓

⟩−<e

⟨ηfX · ϕ∓, Y · ϕ∓

⟩.

On the other hand, we set

A± = Q±ϕ + B±, (18)

and

W =A+

|ϕ−|2−A−

|ϕ+|2.

From now, we will divide our study in two parts, η = 12 and η =

i2 . We give all the details for the case η =

12 and only

specify the small differences for the case η = i2 .

4.3.1. The case η = 12

In this section, we assume that the norm of ϕ is constant and we will show thatW is identically zero. For that, we willshow thatW is symmetric, and trace-free, with rank less or equal to 1. First, we give this elementary lemma.

Lemma 4.2. The endomorphisms Q±ϕ and B± satisfy

1. tr (Q±ϕ ) = −H|ϕ∓|2+12 <e

⟨T · ϕ±, ϕ∓

⟩+ f |ϕ∓|2,

2. tr (B±) = − 12 <e⟨T · ϕ±, ϕ∓

⟩− f |ϕ∓|2,

3. Q±ϕ (e1, e2) = Q±ϕ (e2, e1)+

12 <e

⟨ω · T · ϕ±, ϕ∓

⟩,

4. B±(e1, e2) = B±(e2, e1)− 12 <e

⟨ω · T · ϕ±, ϕ∓

⟩.

Proof. We begin by the computation of the trace of B± and Q±ϕ . We have

tr (Q±ϕ ) = Q±

ϕ (e1, e1)+ Q±

ϕ (e2, e2)

= <e⟨∇e1ϕ

±, e1 · ϕ∓⟩+⟨∇e2ϕ

±, e2 · ϕ∓⟩

= −<e⟨Dϕ±, ϕ∓

⟩= −H <e

⟨ϕ∓, ϕ∓

⟩+12<e⟨T · ϕ±, ϕ∓

⟩+<e

⟨f ϕ∓, ϕ∓

⟩= −H|ϕ∓|2 +

12<e⟨T · ϕ±, ϕ∓

⟩+ f |ϕ∓|2

and

tr (B±) = B±(e1, e1)+ B±(e2, e2)

= −<e⟨T · ϕ±, ϕ∓

⟩+12<e⟨T · ϕ±, ϕ∓

⟩−<e

⟨f ϕ∓, ϕ∓

⟩= −

12<e⟨T · ϕ±, ϕ∓

⟩− f |ϕ∓|2.

Then, we compute Q±ϕ (e1, e2).

Q±ϕ (e1, e2) = <e⟨∇e1ϕ

±, e2 · ϕ∓⟩

= <e⟨e1 · ∇e1ϕ

±, e1 · e2 · ϕ∓⟩

= <e⟨Dϕ±, ω · ϕ∓

⟩−<e

⟨e2 · ∇e2ϕ

±, e1 · e2 · ϕ∓⟩

= <e⟨Dϕ±, ω · ϕ∓

⟩+ Q±ϕ (e2, e1).

Using (17), we obtain

<e⟨Dϕ±, ω · ϕ∓

⟩= <e

⟨Hϕ∓, ω · ϕ∓

⟩−12<e⟨T · ϕ±, ω · ϕ∓

⟩− f <e

⟨ϕ∓, ω · ϕ∓

⟩= −

12<e⟨T · ϕ±, ω · ϕ∓

⟩=12<e⟨ω · T · ϕ±, ϕ∓

⟩.


Finally, we compute B±(e1, e2)− B±(e2, e1).

B±(e1, e2)− B±(e2, e1) = −12<e⟨e1 · T · ϕ±, e2 · ϕ∓

⟩−12<e⟨T1ϕ±, e2 · ϕ∓

⟩−12<e⟨fe1 · ϕ∓, e2 · ϕ∓

⟩+12<e⟨e2 · T · ϕ±, e1 · ϕ∓

⟩+12<e⟨T2ϕ±, e1 · ϕ∓

⟩+12<e⟨fe2 · ϕ∓, e1 · ϕ∓

⟩=12<e⟨e2 · e1 · T · ϕ±, ϕ∓

⟩−12<e⟨e1 · e2 · T · ϕ±, ϕ∓

⟩+12<e⟨(T1e2 − T2e1) · ϕ±, ϕ∓

⟩= −<e

⟨ω · T · ϕ±, ϕ∓

⟩+12<e⟨ω · T · ϕ±, ϕ∓

⟩= −

12<e⟨ω · T · ϕ±, ϕ∓

⟩. �

This lemma implies immediately thatW is symmetric and trace-free. Now, we give this last lemma which shows that therank ofW is at most 1.

Lemma 4.3. The endomorphism field W satisfies

<e⟨W (X) · ϕ−, ϕ+

⟩= 0

for any tangent vector field X.

Proof. First, since ϕ has constant norm, for any tangent vector field X , we have

0 = X |ϕ|2

= X(|ϕ+|2 + |ϕ−|2

)= 2<e

(∇Xϕ

+, ϕ+)+ 2<e

(∇Xϕ

−, ϕ−)

= 2<e(U(X) · ϕ−, ϕ+

), (19)

where U(X) := Q+ϕ (X)|ϕ−|2−Q−ϕ (X)|ϕ+|2

. In order to simplify the notation, we set

U+(X) :=Q+ϕ (X)

|ϕ−|2and U−(X) :=

Q−ϕ (X)

|ϕ+|2.

On the other hand, we denote

V+(X) :=B+(X)|ϕ−|2

, V−(X) :=B−(X)|ϕ+|2

and V = V+ − V−.

Let us compute<e(V (X) · ϕ−, ϕ+

).

<e(V (X) · ϕ−, ϕ+

)= <e

(V+(X) · ϕ−, ϕ+

)−<e

(V−(X) · ϕ−, ϕ+

)= <e

(V+(X) · ϕ−, ϕ+

)+<e

(V−(X) · ϕ+, ϕ−

).

But,

<e(V+(X) · ϕ−, ϕ+

)= <e

(V+(X, e1)e1 · ϕ−, ϕ+

)+<e

(V+(X, e2)e2 · ϕ−, ϕ+

).

Now, we compute the term V+(X, ej) for j = 1, 2.

V+(X, ej) = −12<e⟨X · T · ϕ+, ej ·

ϕ−

|ϕ−|2

⟩−12<e 〈X, T 〉

⟨ϕ+, ej ·

ϕ−

|ϕ∓|2

⟩−12<e⟨fX · ϕ−, ej ·

ϕ−

|ϕ−|2

⟩.

Since{e1 ·

ϕ−

|ϕ−|, e2 ·

ϕ−

|ϕ−|

}is a local orthonormal frame of Γ (Σ+N) for the scalar product<e 〈., .〉, we deduce the following:

V+(X) · ϕ− = −12X · T · ϕ+ −

12〈X, T 〉ϕ+ −

12fX · ϕ−.

Then, we conclude that

<e⟨V+(X) · ϕ−, ϕ+

⟩= −

12<e⟨X · T · ϕ+, ϕ+

⟩−12<e⟨〈X, T 〉ϕ+, ϕ+

⟩−12<e⟨fX · ϕ+, ϕ−

⟩.


We can easily see that, for any vector field X ,

<e⟨X · T · ϕ+, ϕ+

⟩+<e

⟨〈X, T 〉ϕ+, ϕ+

⟩= 0,

which yields

<e⟨V+(X) · ϕ−, ϕ+

⟩= −

12<e⟨fX · ϕ+, ϕ−

⟩.

In the same way, we show that

<e⟨V−(X) · ϕ+, ϕ−

⟩= −

12<e⟨fX · ϕ−, ϕ+

⟩.

Finally, we conclude that

<e⟨V (X) · ϕ−, ϕ+

⟩= 0. (20)

SinceW = U + V , we deduce from (19) and (20) that

<e⟨W (X) · ϕ−, ϕ+

⟩= 0.

This achieves the proof of the lemma. �

The fact that

<e⟨W (X) · ϕ−, ϕ+

⟩= 0

implies thatW has rank at most 1. Since, by Lemma 4.2,W is also symmetric and trace-free, we deduce the following.

Proposition 4.4. The endomorphism field W is identically zero.

Now, we can state the following proposition.

Proposition 4.5. If ϕ is a non-trivial solution of the Dirac equation (16)with constant norm, then ϕ is also a solution of Eq. (12).

Proof. We set F := A+ + A−. The fact thatW = 0 implies that

F|ϕ|2=A+

|ϕ−|2=A−

|ϕ+|2. (21)

On the other hand, we have seen that, for any vector field X ,

∇Xϕ = U+(X) · ϕ− + U−(X) · ϕ+

=A+

|ϕ−|2· ϕ− +

A−

|ϕ+|2· ϕ+ − V+(X) · ϕ− − V−(X) · ϕ+,

which gives, with (21),

∇Xϕ =F(X)|ϕ|2· ϕ − V+(X) · ϕ− − V−(X) · ϕ+. (22)

The tensor F is symmetric because A+ and A− are symmetric. Then, by setting A := −2F , Eq. (22) gives

∇Xϕ =12X · T · ϕ +

12fX · ϕ +

12〈X, T 〉ϕ −

12AX · ϕ.

This concludes the proof. �

4.3.2. The case η = i2

The scheme of the proof is exactly the same as for η = 12 . The twominor differences are that the expression of the spinor

field solution of the Dirac equation is different and the norm of this spinor is not constant, but satisfies

X |ϕ|2 = <e 〈iX · T · ϕ + ifX · ϕ, ϕ〉 .

We do not give the details but all the lemmas of last section have an analogue in the case η = i2 , andwe obtain the following

proposition.

Proposition 4.6. If ϕ is a non-trivial solution of the Dirac equation (16) with its norm satisfying

X |ϕ|2 = <e 〈iX · T · ϕ + ifX · ϕ, ϕ〉 ,

then ϕ is also a solution of Eq. (12).


Indeed, as in the case η = 12 , we have this elementary lemma.

Lemma 4.7. The endomorphisms Q±ϕ and B± satisfy

1. tr (Q±ϕ ) = −H|ϕ∓|2+12 <e

⟨iT · ϕ±, ϕ∓

⟩.

2. tr (B±) = − 12 <e⟨iT · ϕ±, ϕ∓

⟩.

3. Q±ϕ (e1, e2) = Q±ϕ (e2, e1)∓

12 <e

⟨ω · T · ϕ±, ϕ∓

⟩± f |ϕ∓|2,

4. B±(e1, e2) = B±(e2, e1)± 12 <e

⟨ω · T · ϕ±, ϕ∓

⟩∓ f |ϕ∓|2.

The proof of this lemma is straightforward.Now, we have all ingredients to prove Theorem 1.

4.4. Proof of Theorem 1

The proof is the same for the two cases.The equivalence between (i) and (ii) have been proved in Section 4.2. On the other hand, it is clear that (ii) implies (iii).

Finally, we have to show that (iii) implies (ii). We have seen in Section 4.3 that a spinor field solution of the Dirac equation(16) with the norm assumption is also a solution of Eq. (12). We just have to show that

∇XT = fAX and df (X) = −〈AX, T 〉 .

These two identities just come from the assumptions

∇XT = 2fQϕ(X)− fB(X) and df = −2Qϕ(T )+ B(T ),

together with the expression of A = −2F = −2 A++A−

|ϕ|2, with A± given by (18). This achieves the proof. �

5. Isometric immersions into E(κ, τ), τ 6= 0

In this section, we give a spinorial characterization for surfaces into E(κ, τ )with τ 6= 0.

5.1. Statement of the result

First, we state the main result about spinorial characterization of surfaces isometrically immersed into E(κ, τ ).

Theorem 2. Let κ, τ ∈ R, τ 6= 0 such that κ 6= 4τ 2. We set α = 2τ − κ2τ . Let (N, 〈., .〉) be a connected, simply connected and

oriented Riemannian surface. Let T be a vector field, and f and H two functions on N satisfying f 2 +‖T‖2 = 1. Let A be a field ofsymmetric endomorphisms on TN with tr (A) = 2H. The following three statements are equivalent.

(i) There exists an isometric immersion F from N into E(κ, τ )with mean curvature H such that theWeingarten operator relatedto the normal ν is given by

dF ◦ A ◦ dF−1

and such that

ξ = dF(T )+ f ν.

(ii) There exists a non-trivial spinor field ϕ solution of the equation

∇Xϕ = −τ

2X · ω · ϕ +

α

2〈X, T 〉T · ω · ϕ −

α

2f 〈X, T 〉ω · ϕ −

12AX · ϕ,

where A satisfies

∇XT = f (AX − τ JX), and df (X) = −〈AX − τ JX, T 〉.

(iii) There exists a non-trivial spinor field ϕ solution of the Dirac equation

Dϕ = Hϕ + τω · ϕ −α

2‖T‖2ω · ϕ −

α

2fT · ωϕ,

with constant norm and such that

∇XT = 2fQϕ(X)+ fB(T )+ f τ J(T )

and

df = −2Qϕ(T )− B(T )− τ J(T ),


where Qϕ is the energy–momentum tensor associated with ϕ and B is the tensor defined in an orthonormal frame {e1, e2} bythe following matrix: αT1T2

α

2

(T 21 − T

22

)α

2

(T 21 − T

22

)−αT1T2

,with Ti = 〈T , ei〉.

The proof of Theorem 2 uses the same arguments as in the proof of Theorem 1 and Daniel’s theorem [15] for isometricimmersions into E(κ, τ ).

5.2. Special Spinor field and compatibility equations

Proposition 5.1. If (N, 〈., .〉) admits a spinor field solution of Eq. (14) and if Eqs. (9) and (10) are fulfilled, then the Gauss andCodazzi equations for E(κ, τ ) are satisfied.

Proof. We compute the spinorial curvature of the spinor field ϕ solution of (14).

∇X∇Yϕ = −τ

2(∇XY ) · ω · ϕ︸︷︷︸

α1(X,Y )

+τ 2

4Y · X · ϕ︸︷︷︸α2(X,Y )

−ατ

4〈X, T 〉 Y · T · ϕ︸︷︷︸α3(X,Y )

−ατ

4f 〈X, T 〉 Y · ϕ︸︷︷︸α4(X,Y )

−τ

4Y · AX · ω · ϕ︸︷︷︸

α5(X,Y )

+α

2〈∇XY , T 〉 T · ω · ϕ︸︷︷︸

α6(X,Y )

+α

2f 〈Y , AX〉 T · ω · ϕ︸︷︷︸

α7(X,Y )

−ατ

2f 〈Y , JX〉 T · ω · ϕ︸︷︷︸

α8(X,Y )

+α

2f 〈Y , T 〉 AX · ω · ϕ︸︷︷︸

α9(X,Y )

−ατ

2f 〈Y , T 〉 JX · ω · ϕ︸︷︷︸

α10(X,Y )

−ατ

2〈Y , T 〉 T · X · ϕ︸︷︷︸α11(X,Y )

−α2

4〈X, T 〉〈Y , T 〉 ‖T‖2ϕ︸︷︷︸

α12(X,Y )

+α2

4f 〈X, T 〉〈Y , T 〉 T · ϕ︸︷︷︸

α13(X,Y )

+α

4〈Y , T 〉 T · AX · ω · ϕ︸︷︷︸

α14(X,Y )

+α

2〈Y , T 〉〈AX, T 〉ω · ϕ︸︷︷︸

α15(X,Y )

−ατ

2〈Y , T 〉〈JX, T 〉ω · ϕ︸︷︷︸

α16(X,Y )

−α

2〈∇XY , T 〉ω · ϕ︸︷︷︸

α17(X,Y )

−α

2f 2 〈Y , AX〉ω · ϕ︸︷︷︸

α18(X,Y )

+ατ

2f 2 〈Y , JX〉ω · ϕ︸︷︷︸α19(X,Y )

+ατ

4f 〈Y , T 〉 X · ϕ︸︷︷︸α20(X,Y )

+α2

4f 〈X, T 〉〈Y , T 〉 T · ϕ︸︷︷︸

α21(X,Y )

−α2

4f 2 〈X, T 〉〈Y , T 〉ϕ︸︷︷︸

α22(X,Y )

−α

4f 〈Y , T 〉 AX · ω · ϕ︸︷︷︸

α23(X,Y )

−τ

2

(∇XAY

)· ϕ︸︷︷︸

α24(X,Y )

+τ

4AY · X · ω · ϕ︸︷︷︸α25(X,Y )

−α

4〈X, T 〉 AY · T · ω · ϕ︸︷︷︸

α26(X,Y )

+α

4〈X, T 〉 AY · ω · ϕ︸︷︷︸

α27(X,Y )

+14AY · AX · ϕ︸︷︷︸α28(X,Y )

.

As in the previous case, lots of terms vanish by symmetry, and for X = e1 and Y = e2, we have

R(e1, e2) =14(Ae2 · Ae1 − Ae1 · Ae2) · ϕ −

τ 2

2ω · ϕ −

12d∇(e1, e2)+ 2ατ fT · ω · ϕ + ατ f 2ω · ϕ.

Using the fact that ατ = 2τ 2 − κ2 and the Ricci identity, we get(

R1212 − det (A)− τ 2 − (κ − 4τ 2)f 2︸︷︷︸G

)ω · ϕ =

(d∇(e1, e2)+ (κ − 4τ 2)fJT︸︷︷︸

C

)· ϕ;

that is,

Gω · ϕ = C · ϕ,


where G is a real function and C is a vector field. Moreover, since ω · ϕ = −iϕ, we deduce that

C · ϕ± = ±iGϕ∓,

and so

‖C‖2ϕ± = −G2ϕ±.

Since ϕ is a solution of (14), its norm is constant. Therefore, ϕ+ and ϕ− cannot vanish at the same point, which yields C = 0and G = 0. But G = 0 is the Gauss equation and C = 0 is the Codazzi equation. This achieves the proof. �

This proposition proves together with Daniel’s result that (ii) implies (i) in Theorem 2. Note that (i) implies (ii) is obviousfrom the section of preliminaries. Now, we will show that (iii) implies (ii).

5.3. Special spinor fields and Dirac equation

If a spinor field ϕ is a solution of Eq. (14), thenwe deduce immediately that ϕ is also a solution of the corresponding Diracequation

Dϕ = Hϕ + τω · ϕ −α

2‖T‖2ω · ϕ − 2ηfTω · ϕ. (23)

In this section,wewill see that this Dirac equation is equivalent to Eq. (14) for spinors of constant norm.We setϕ = ϕ++ϕ−,with ϕ± ∈ Σ±N . Then, we have

Dϕ± = Hϕ∓ + τω · ϕ∓ −α

2‖T‖2ω∓ · ϕ − 2ηfTω · ϕ±. (24)

Now, we define the following endomorphisms:

Q±ϕ (X, Y ) = <e⟨∇Xϕ

±, Y · ϕ∓⟩,

and

B±(X, Y ) = ∓<e(iα

2〈X, T 〉T · ϕ∓, Y · ϕ∓

)∓ <e

(iα

2f 〈X, T 〉 · ϕ±, Y · ϕ∓

).

On the other hand, we set

A± = Q±ϕ + B±, (25)

and

W =A+

|ϕ−|2−A−

|ϕ+|2.

From Eq. (24), we deduce easily that

Lemma 5.2. 1. tr (Q±ϕ ) = −H|ϕ±|2±

α2 f <e

(iT · ϕ±, ϕ∓

),

2. Q±ϕ (e1, e2) = Q±ϕ (e2, e1)+

(τ − α

2 ‖T‖2)|ϕ∓|2 + α

2 f <e(T · ϕ±, ϕ∓

),

3. tr (B±) = ∓ α2 f <e

(iT · ϕ±, ϕ∓

),

4. B±(e1, e2) = B±(e2, e1)+ α2 ‖T‖

2|ϕ∓|2 − α

2 f <e(T · ϕ±, ϕ∓

).

Proof. The proof is similar to the proof of Lemma 4.2, but here we use Eq. (24). �

Thus, we deduce immediately the following:

Lemma 5.3. The endomorphisms A± satisfy

1. tr (A±) = −H|ϕ∓|2,2. A±(e1, e2) = A±(e2, e1)+ τ |ϕ∓|2.

Now, we give the next lemma showing thatW is of rank at most 1.

Lemma 5.4. The endomorphism field W satisfies

<e⟨W (X) · ϕ+, ϕ−

⟩= 0.


Proof. First of all, since ϕ is of constant norm, for any tangent vector field X , we have

0 = X |ϕ|2

= X(|ϕ+|2 + |ϕ−|2

)= 2<e

(∇Xϕ

+, ϕ+)+ 2<e

(∇Xϕ

−, ϕ−)

= 2<e(U(X) · ϕ−, ϕ+

), (26)

where U(X) := Q+ϕ (X)|ϕ−|2−Q−ϕ (X)|ϕ+|2

. So we set

U+(X) :=Q+ϕ (X)

|ϕ−|2and U−(X) :=

Q−ϕ (X)

|ϕ+|2,

and

V+(X) :=B+(X)|ϕ−|2

, V−(X) :=B−(X)|ϕ+|2

et V = V+ − V−.

Now, we compute<e(V (X) · ϕ−, ϕ+

).

<e(V (X) · ϕ−, ϕ+

)= <e

(V+(X) · ϕ−, ϕ+

)−<e

(V−(X) · ϕ−, ϕ+

)= <e

(V+(X) · ϕ−, ϕ+

)+<e

(V−(X) · ϕ+, ϕ−

).

But

<e(V±(X) · ϕ∓, ϕ±

)= <e

(V±(X, e1)e1 · ϕ∓, ϕ±

)+<e

(V±(X, e2)e2 · ϕ∓, ϕ±

).

Moreover, we have, for j = 1, 2,

V±(X, ej) = ∓α

2<e(i〈X, T 〉T · ϕ∓, ej ·

ϕ∓

|ϕ∓|2

)−α

2<e(if 〈X, T 〉 · ϕ±, ej

ϕ∓

|ϕ∓|2

).

Since{e1 ·

ϕ∓

|ϕ∓|, e2 ·

ϕ∓

|ϕ∓|

}is a local orthonormal frame of Γ (Σ±N) for the inner product<e 〈., .〉, we deduce that

V±(X) · ϕ∓ = ∓iα

2〈X, T 〉 T · ϕ∓ ∓ i

α

2f 〈X, T 〉ϕ±.

Thus, we deduce easily that

<e⟨V±(X) · ϕ∓, ϕ±

⟩= ∓

α

2<e⟨〈X, T 〉 T · ϕ∓, ϕ±

⟩.

So, we conclude that

<e(V (X) · ϕ−, ϕ+

)= 0. (27)

SinceW = U + V , we deduce from (26) and (27) that

<e(W (X) · ϕ−, ϕ+

)= 0.

This achieves the proof. �

Now, sinceW is a symmetric 2-tensor with rank at most 1 and trace-free, thenW vanishes identically.We conclude this section with this last proposition.

Proposition 5.5. If ϕ is a non-trivial solution of the Dirac equation (23) with constant norm, then ϕ is a solution of Eq. (14).

Proof. We set F := A+ + A−. The fact thatW = 0 implies

F|ϕ|2=A+

|ϕ−|2=A−

|ϕ+|2. (28)

Moreover, we have already seen that, for any vector field X ,

∇Xϕ = U+(X) · ϕ− + U−(X) · ϕ+

=A+

|ϕ−|2· ϕ− +

A−

|ϕ+|2· ϕ+ − V+(X) · ϕ− − V−(X) · ϕ+,


which from (28) yields

∇Xϕ =F(X)|ϕ|2· ϕ − V+(X) · ϕ− − V−(X) · ϕ+. (29)

We note that the tensor F is not symmetric, so we consider the following symmetric (2, 0)-tensor S:

S(X, Y ) := −12|ϕ|2

(F(X, Y )+ F(Y , X)) .

Moreover, from Lemma 5.3, we getS(e1, e1) = −F(e1, e1)/|ϕ|2,S(e2, e2) = −F(e2, e2)/|ϕ|2,S(e1, e2) = −F(e1, e2)/|ϕ|2 +

τ

2,

S(e2, e1) = −F(e2, e1)/|ϕ|2 −τ

2.

(30)

From these relations and (29), we deduce that

∇Xϕ = −τ

2X · ω · ϕ +

α

2〈X, T 〉 T · ω · ϕ −

α

2f 〈X, T 〉ω · ϕ − SX · ϕ;

that is, ϕ is solution of (14) if we set A = 2S. �

5.4. Proof of Theorem 2

Now, we have all ingredients to finish the proof of Theorem 2. Equivalence between points (i) and (ii) has been provedin Section 5.2, and clearly (ii) implies (iii) by taking the trace. The last thing to prove is that (iii) implies (ii). We have provedin Section 5.3 that a spinor field solution of the Dirac equation (23) and of constant norm is a solution of Eq. (14). The lastpoints to check is that

∇XT = f (AX − τ JX) and df (X) = −〈AX − τ JX, T 〉 .Here again, from the expression of A = 2S given by (30) and (25) and the assumptions

∇XT = 2fQϕ(X)+ fB(T )+ f τ J(T ) and df = −2Qϕ(T )− B(T )− τ J(T ),we get the required identities. �

6. Two final remarks

6.1. Other special spinor fields on surfaces of E(κ, τ ), τ 6= 0

We want to point out that the choice of our special spinor fields in E(κ, τ ) with τ 6= 0 is not the only possible choice.Indeed, we could have proceeded as in the case of products, that is τ = 0, by considering a Killing spinor on the base of thefibration and lifting it to the ambient space. One can refer to [18] for a study of spinors on Riemannian submersions.This choice gives us a unitary approach for any τ . Nevertheless, with this approach, we obtain special spinor fields on

surfaces of E(κ, τ ), τ 6=with more complicated derivatives. Indeed, we have the following spinor field on E(κ, τ ):∇e1ϕ = −ηe2 · ϕ,∇e2ϕ = ηe1 · ϕ,∇ξϕ =

τ

2ξ · ϕ.

(31)

Note that, for τ = 0, we have the spinor field onM2(κ)× R given in (11). Thus, after restriction to a surface of E(κ, τ ), weobtain the following spinor field:

∇Xϕ = −12A(X) · ϕ + ηX · T · ϕ + ηfX · ϕ + ηf 〈X, T 〉ϕ −

τ

2〈X, T 〉T · ω · ϕ +

τ

2f 〈X, T 〉ω · ϕ, (32)

and therefore

Dϕ = Hϕ − 2ηT · ϕ − 2ηf ϕ + ηfT · ϕ +τ

2‖T‖2ω · ϕ +

τ

2fT · ω · ϕ, (33)

with 4η2 = κ . Note that, if τ = 0, we have exactly Eqs. (12) and (16) for surfaces ofM2(κ)× R.In the case where τ 6= 0, the computation of the spinorial curvature of a spinor field solution of (32) is very long, and the

equivalence between Eqs. (32) and (33) with a norm assumption is very technical.Thus, although we are convinced that both cases τ = 0 and τ 6= 0 can be treated in a unified way, we have decided, for

the sake of clarity, to work with spinors obtained by constant sections to make the proof more readable in the case τ 6= 0.


6.2. The Lawson correspondence in three-dimensional homogeneous manifolds

In [15], Daniel gives a Lawson correspondence for constant mean curvature (cmc) surfaces into three-dimensionalhomogeneous manifolds. The Lawson correspondence is a local isometric correspondence between cmc surfaces intotwo different three-dimensional homogeneous manifolds. Precisely, let E(κ1, τ1) and E(κ2, τ2) be two three-dimensionalhomogeneous manifolds with a four-dimensional isometry group with κ1 − 4τ 21 = κ2 − 4τ 22 . Then, a surface of constantmean curvature H1 in E(κ1, τ1) has a sister surface of constant mean curvature H2 in E(κ2, τ2) with τ 21 + H

21 = τ 22 + H

22 .

In particular, this gives a local isometric correspondence between minimal surfaces in the Heisenberg group and cmc 12surfaces into H2 × R.This result is a generalization of the Lawson correspondence for cmc surfaces in space forms. It would be very interesting

to understand these two correspondences in terms of spinors. Note that the classical Lawson correspondence (for cmcsurfaces in space forms) is still not well understood via spinors.

References

[1] O. Hijazi, S. Montiel, X. Zhang, Dirac operator on embedded hypersurfaces, Math. Res. Lett. 8 (2001) 195–208.[2] O. Hijazi, S. Montiel, A. Roldán, Dirac operator on hypersurfaces in negatively curved manifolds, Ann. Global Anal. Geom. 23 (2003) 247–264.[3] O. Hijazi, S. Montiel, F. Urbano, Spinc geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds, Math. Z. 253 (4)(2006).

[4] U. Abresch, H. Rosenberg, A Hopf differential for constant mean curvature surfaces in Sn × R and Hn × R, Acta Math. 193 (2) (2004) 141–174.[5] H. Rosenberg, Minimal surfaces inMn

× R, Illinois J. Math. 46 (4) (2002) 1177–1195.[6] T. Friedrich, On the spinor representation of surfaces in euclidian 3-space, J. Geom. Phys. 28 (1998) 143–157.[7] B. Morel, Surfaces in S3 and H3 via spinors, Actes du Séminaire de Théorie Spectrale et Géométrie de l’Institut Fourier de Grenoble 23 (2005) 9–22.[8] J.P. Bourguignon, O. Hijazi, J.-L. Milhorat, A. Moroianu, A spinorial approach to Riemannian and conformal geometry, Monograph (in preparation).[9] T. Friedrich, Dirac Operators in Riemannian Geometry, in: A.M.S. Graduate Studies in Math., vol. 25, 2000.[10] B. Lawson, M.-L. Michelson, Spin Geometry, Princeton University Press, 1989.[11] C. Bär, Extrinsic bounds for eigenvalues of the Dirac operator, Ann. Global Anal. Geom. (1998).[12] O. Hijazi, S. Montiel, Extrinsic Killing spinors, Math. Z. 244 (2003) 337–347.[13] B. Morel, Eigenvalue estimates for the Dirac–Schrödinger operators, J. Geom. Phys. 38 (2001) 1–18.[14] C. Bär, P. Gauduchon, A. Moroianu, Generalized cylinders in semi-Riemannian and spin geometry, Math. Z. 249 (2005) 545–580.[15] B. Daniel, Isometric immersions into 3-dimensional homogenous manifolds, Comment. Math. Helv. 82 (1) (2007).[16] P. Scott, The geometries of 3-manifolds, Bull. Lond. Math. Soc. 15 (5) (1983) 401–487.[17] B. Daniel, Isometric immersions into Sn × R and Hn × R and application to minimal surfaces, Trans. Amer. Math. Soc. 361 (12) (2009) 6255–6282.[18] A. Moroianu, Opérateur de Dirac et submersions riemanniennes, École Polytechnique, 1996.

spinorial characterizations of surfaces into three-dimensional homogeneous manifolds

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