Spinorial Representation of Submanifolds in a Product of Space Forms

We present a method giving a spinorial characterization of an immersion into a product of spaces of constant curvature. As a first application we obtain a proof using spinors of the fundamental theorem of immersion theory for such target spaces. We also study special cases: we recover previously known results concerning immersions in S2×R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^2\times \mathbb {R}$$\end{document} and we obtain new spinorial characterizations of immersions in S2×R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^2\times \mathbb {R}^2$$\end{document} and in H2×R.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}^2\times \mathbb {R}.$$\end{document} We then study the theory of H=1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=1/2$$\end{document} surfaces in H2×R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}^2\times \mathbb {R}$$\end{document} using this spinorial approach, obtain new proofs of some of its fundamental results and give a direct relation with the theory of H=1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=1/2$$\end{document} surfaces in R1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{1,2}$$\end{document}.


Introduction
Characterizations of immersions in space forms using spinors have been widely studied, as for instance in [8,9,11,14,17,18,19,20] and more recently in [6] (see also the references in these papers).It appears that spin geometry furnishes an elegant formalism for the description of the immersion theory in space forms, especially in low dimension and in relation with the Weierstrass representation formulas.See also the Weierstrass representation obtained in [2] for CMC hypersurfaces in some four-dimensional Einstein manifolds.We are interested here in spinorial characterizations of immersions in a product of space forms.Some special cases have been studied before, as immersions in the products S 2 × R, H 2 × R, S 2 × R 2 and S 3 × R [12,15,16].We propose here a method allowing the treatment of an immersion in an arbitrary product of space forms.The dimension and the co-dimension of the immersion are moreover arbitrary.Let us note that the spinor bundle that we use in the paper is not the usual spinor bundle: in general it is a real bundle, and of larger rank.We used this idea in [6].Let us also mention that even in low dimensions we obtain new results: the theory permits to recover in a unified way the previously known results and to complete them; in particular, we show how to recover the spinorial characterization of an immersion in S 2 × R and we obtain new spinorial characterizations of immersions in S 2 × R 2 and in H 2 × R.
A first application of the general theory is a proof using spinors of the fundamental theorem of immersion theory in a product of space forms.
A second application concerns the theory of CMC surfaces with H = 1/2 in H 2 × R: we show that a component of the spinor field representing the immersion of such a surface is an horizontal lift of the hyperbolic Gauss map, for a connection which depends on the Weierstrass data of the immersion, and we deduce that there exists a two-parameter family of H = 1/2 surfaces in H 2 × R with given hyperbolic 1 Gauss map and Weierstrass data, a result obtained in [7] using different methods.We finally study the spinorial representation of H = 1/2 surfaces in R 1,2 and obtain a direct relation between the two theories.
In order to simplify the exposition we first consider immersions in a product of spheres S m 1 × S n 2 and in a product S m 1 × R n , and we then state without proof the analogous results for immersions in a product of hyperbolic spaces H m 1 × H n 2 and in H m 1 × R n .Using the same ideas it is possible to state analogous results for an arbitrary quantity of factors involving S m 1 , H n 2 and R p , or for space forms with pseudo-riemannian metrics, but these general statements are not included in the paper.
The outline of the paper is as follows.We first study the immersions in a product of spheres S m 1 × S n 2 in Section 1 and the immersions in S m 1 × R n in Section 2. We then state the analogous results for a product of hyperbolic spaces H m 1 × H n 2 and for H m 1 × R n in Section 3. Finally the theory of H = 1/2 surfaces in H 2 × R is studied in Section 4. Some useful auxiliary results are gathered in an appendice at the end of the paper.

Isometric immersions in S m
1 × S n 2 We are interested here in immersions in a product S m 1 × S n 2 of two spheres, of constant curvature c 1 , c 2 > 0. We construct the suitable spinor bundle in Section 1.1, we consider the case of a manifold which is already immersed in S m 1 × S n 2 in Section 1.2, we state and prove the main theorem in Section 1.3 and the fundamental theorem in Section 1.4.
1.1.The suitable spinor bundle.Let M be a p-dimensional riemannian manifold and E → M a vector bundle of rank q, with p + q = m + n, with a bundle metric and a connection compatible with the metric.Let E 2 = M × R 2 → M be the trivial bundle, equipped with its natural metric and its trivial connection.Let us construct a spinor bundle on M equipped with a Clifford action of T M ⊕ E ⊕ E 2 .
We suppose that M and E are spin, with spin structures Let us denote by Spin(N ) and Cl(N ) the spin group and the Clifford algebra of R N .Associated to the splitting R m+n+2 = R p ⊕ R q ⊕ R 2 we consider Spin(p) • Spin(q) ⊂ Spin(p + q) ⊂ Spin(m + n + 2) and define the representation ρ : Spin(p) × Spin(q) → GL(Cl(m + n + 2)) a := (a p , a q ) → ρ(a) : (ξ → a p • a q • ξ) with the bundles Since the bundle of Clifford algebras constructed on the fibers of with there is a Clifford action similar to the usual Clifford action in spin geometry.Let us note that Σ is not the usual spinor bundle, since it is a real vector bundle, associated to a representation which is not irreducible: it is rather a (maybe large) sum of real spinor bundles.We nevertheless interpret the bundle Σ as the bundle of spinors, U Σ as the bundle of unit spinors and Cl(T M ⊕ E ⊕ E 2 ) as the Clifford bundle acting on the bundle of spinors.There is a natural map ., .: where ϕ = [s, [ϕ]] and ) and τ is the involution of Cl(m + n + 2) reversing the order of a product of vectors.Here and in all the paper we use the brackets [.] to denote the component in Cl(m + n + 2) of an element of the spinor or the Clifford bundle in a given spinorial frame s.This map is such that, for all ϕ, ϕ ′ ∈ Σ and Moreover, it is compatible with the connection ∇ induced on Σ by the Levi-Civita connection on M and the given connection on E : Lemma 1.1.For all X ∈ T M and ϕ, ϕ ′ ∈ Γ(Σ), (3) where on the left hand side ∂ stands for the usual derivative.
A similar result is proved in [6, Lemma 2.2].
1.2.Spin geometry of a submanifold in S m 1 × S n 2 .We assume in that section that M is a p-dimensional submanifold of S m 1 × S n 2 , with normal bundle E of rank q and second fundamental form B : T M × T M → E, denote by ν 1 : M → R m+1 and ν 2 : M → R n+1 the vector fields such that 1 √ c1 ν 1 and 1 √ c2 ν 2 are the two components of the immersion M → S m 1 × S n 2 and consider the trivial bundle E 2 = Rν 1 ⊕ Rν 2 → M. We consider spin structures on T M and E, and the bundles Σ, U Σ and Cl(T M ⊕ E ⊕ E 2 ) constructed in the previous section.For a convenient choice of the spin structures on T M and E, the bundle Σ identifies canonically with the trivial bundle M × Cl(m + n + 2), and two connections are defined on Σ, the connection ∇ introduced above and the trivial connection where e 1 , . . ., e p is an orthonormal basis of T M , they satisfy the following Gauss formula: (5) for all ϕ ∈ Γ(Σ) and all X ∈ T M ; see [3] for the proof of a spinorial Gauss formula in a slightly different context.By formula (5), the constant spinor field ϕ = 1 Cl(m+n+2) |M satisfies, for all X ∈ T M, 1.3.The main theorem.We assume that M and E → M are abstract objects as in Section 1.1 (i.e.M is not a priori immersed in S m 1 × S n 2 ), and suppose moreover that there is a product structure on T M ⊕ E, i.e. a bundle map P : T M ⊕ E → T M ⊕E such that P 2 = id, P = id.Setting P 1 := Ker(P −id) and P 2 := Ker(P +id) we have T M ⊕ E = P 1 ⊕ P 2 : the product structure P is equivalent to a splitting of T M ⊕ E into two subbundles P 1 and P 2 , that we assume to be respectively of rank m and n.
Let us fix two unit orthogonal and parallel sections ν 1 , ν 2 of the trivial bundle Theorem 1.The following statements are equivalent: (i) There exist an isometric immersion F : M → S m 1 × S n 2 and a bundle map above F such that Φ(X, 0) = dF (X) for all X ∈ T M, which preserves the bundle metrics, maps the connection on E and the tensor B to the normal connection and the second fundamental form of F , and is compatible with the product structures.
(ii) There exists a section ϕ ∈ Γ(U Σ) solution of for all X ∈ T M, where X = X 1 + X 2 is the decomposition in the product structure P of T M ⊕ E, such that the map commutes with the product structure P and the natural product structure on R m+1 × R n+1 .
Moreover, the bundle map Φ and the immersion F are explicitly given in terms of the spinor field ϕ by the formulas Remark 1. Formulas ( 7) and ( 8) can be regarded as a generalized Weierstrass representation formula.
1.3.2.Proof of Theorem 1.The proof of "(i) ⇒ (ii)" was obtained in Section 1.2: if M is immersed in S m 1 × S n 2 , the spinor field ϕ is the constant spinor field 1 Cl(m+n+2) restricted to M. We prove "(ii) ⇒ (i)".We suppose that ϕ ∈ Γ(U Σ) is a solution of (7) and obtain (i) as a direct consequence of the following two lemmas: Lemma 1.2.The map F defined by ( 8) satisfies (9) dF (X) = X • ϕ, ϕ , for all X ∈ T M. It preserves the product structures and takes values in S m 1 × S n 2 .Proof.Let us consider for i = 1, 2 the functions Recalling the properties (1)-(3) of ., .and since ν 1 , ν 2 are parallel sections of E 2 and ϕ satisfies (7), we have, for i = 1, 2, where we use in the last equality that, for all i, j ∈ {1, 2}, (since ν i , X j , ν j are three orthogonal vectors if i = j).Since F = F 1 + F 2 and X = X 1 + X 2 , (9) follows from (10) ) for i = 1, 2 which implies that F 1 and F 2 take values in the spheres of R m+n+2 of radius 1/ √ c 1 and 1/ √ c 2 respectively.We then have to check that F 1 and F 2 take respectively values in R m+1 and R n+1 : since dF preserves the product structures, we have for since by (10) dF 1 (X) = dF (X 1 ) and dF 2 (X) = dF (X 2 ) we conclude that dF 1 and dF 2 take values in R m+1 and R n+1 respectively, and so do F 1 and F 2 .
Lemma 1.3.The map is a bundle map which preserves the metrics, identifies E with the normal bundle of the immersion F in S m 1 × S n 2 , and sends the connection on E and the tensor B to the normal connection and the second fundamental form of the immersion F .

Proof. Let us first see that Φ takes values in
which is the unit normal to S i at p. Thus, since Φ preserves the metric, for We focus on the second term.From the Killing type equation ( 7), we have For i = 1, 2, we have and thus Moreover, we have where This formula implies the following expressions for the second fundamental form B F and the normal connection ∇ ′F of the immersion F : if Z ∈ Γ(T M ) is such that ∇Z = 0 at the point where we do the computations, then This finishes the proof of the lemma (and of Theorem 1).
We give here a proof using spinors of the fundamental theorem of the immersion theory in S m 1 × S n 2 .This result has been proved independently by Kowalczyk [10] and Lira-Tojeiro-Vitório [13].
1.4.1.Statement of the theorem.Let P and P ′ be the product structures of T M ⊕E and We set, for U, V, W ∈ T M ⊕ E, We first write the compatibility equations necessary for the existence of a non-trivial spinor field solution of ( 7): Proposition 1.4.Let ϕ ∈ Γ(U Σ) be a solution of ( 7) such that for all Z ∈ T M ⊕ E. If R T stands for the curvature tensor of the Levi-Civita connection on M and R N for the curvature tensor of the connection ∇ ′ on E, the following fundamental equations hold: for all X, Y, Z ∈ T M and where ∇ stands for the natural connection on T * M ⊗ T * M ⊗ E.Moreover, if we use the same symbol ∇ to denote the natural connections on Equations ( 13), ( 14) and ( 15) are respectively the equations of Gauss, Ricci and Codazzi.Equations ( 16)-( 19) are additional equations traducing that Φ commutes with the product structures P and P ′ , with P ′ parallel in R m+n+2 .All these equations are necessary for the existence of an immersion M → S m 1 × S n 2 with second fundamental form B and normal connection ∇ ′ .It appears that they are also sufficient: Theorem 2. Let us assume that B : T M × T M → E is symmetric and such that the Gauss, Ricci and Codazzi Equations ( 13), ( 14) and ( 15) hold together with ( 16)- (19).Let us moreover suppose that dim Ker(P − id) = m and dim Ker(P + id) = n.Then there exists ϕ ∈ Γ(U Σ) solution of (7) such that the map commutes with the product structures P and P ′ .The spinor field ϕ is moreover unique up to the natural right action of Spin(m + 1) • Spin(n + 1) on U Σ. In particular, there is an isometric immersion F : M → S m 1 ×S n 2 and a bundle isomorphism 2 ) above F identifying E, B and ∇ ′ to the normal bundle, the second fundamental form and the normal connection of The immersion is moreover unique up to the natural action of SO(m + 1) × SO(n + 1) on S m 1 × S n 2 .Section 1.4.2 is devoted to the proof of Proposition 1.4, and Section 1.4.3 to the proof of Theorem 2.
1.4.2.Proof of Proposition 1.4.We assume that X, Y ∈ Γ(T M ) are such that ∇X = ∇Y = 0 at the point where we do the computations.A direct computation using (7) twice yields Lemma 1.5.In local orthonormal frames {e j } 1≤j≤p of T M and {n r } 1≤r≤q of E we have These expressions respectively mean that A ∈ T M ⊗E represents the transformation Proof.The expression (21) directly follows from the definition (4) of B(X).For (22) we refer to [6, Lemma 5.2] where a similar computation is carried out.By Lemma A.1, formula (95), A represents the transformation (24) and B the transformations (25) and (26).We now prove (23).Using and the analogous expressions for Y, straightforward computations yield By Lemma A.3 in the Appendix, the right hand terms (32) and (33) represent the transformations ( 27) and (28); the term (34) represents the transformation (29) since the commutator in the Clifford bundle similarly, the terms (35) and (36) represent the transformations (30) and (31).Formula (23) then follows from Lemmas A.1 and A.2 in the Appendix.
The curvature tensor of the spinorial connection on T M ⊕ E is given by Comparing Equations ( 20) and (37) and since ϕ is represented in a frame s ∈ Q by an element of Spin(m + n + 2), invertible in Cl(m + n + 2), we deduce that Now the right hand side of (38) represents the transformations The equations ( 13)-( 15) of Gauss, Ricci and Codazzi follow from this and Lemma 1.5.Let us now prove that Equations ( 16)-( 19) are consequences of the fact that Φ commutes with the product structures P and P ′ , Equation (12), where the product structure P ′ is parallel.We have by (12) (40) Assuming that ∇X = 0 at the point where we do the computations and recallling (11) we have and the left hand side of ( 40) is given by by (11) again, the right hand side of ( 40) is given by and since for i = 1, 2 we deduce that for X ∈ T M and for Using that Φ is injective on the fibers and decomposing ∇ Y P(X), P(B(Y, X)) and P(B * (Y, X)) in their tangent and normal parts, we get Finally, taking the tangent and the normal parts of each one of the last two equations we get ( 16)-( 19).
We consider U Σ → M as a principal bundle of group Spin(p + q + 2), where the action is the multiplication on the right for all a ∈ Spin(p + q + 2).The connection ∇ ′ may be considered as given by a connection 1-form on this principal bundle, since so is ∇ and the term defines a vertical and invariant vector field on U Σ. The compatibility equations ( 13)- (19) imply that this connection is flat (the computations are similar to the computations in the previous section).Since it is flat and assuming moreover that M is simply connected, the principal bundle U Σ → M has a global parallel section: this yields ϕ ∈ Γ(U Σ) such that ∇ ′ ϕ = 0, i.e. a non-trivial solution of (7).Let us verify that equations ( 16)- (19) imply that the map is compatible with the product structures, i.e. verifies Φ(P(X)) = P ′ (Φ(X)) for all X ∈ T M ⊕ E. The sum of ( 16) and ( 17) gives, for X, Y ∈ T M, Similarly, for X ∈ E and Y ∈ T M, (18) and (19) imply that (44) As in the proof of Theorem 1, Φ is a bundle map above the immersion The product structure P on T M ⊕E extends to a product structure P on T M ⊕E ⊕ E 2 by setting P(ν 1 ) = ν 1 and P(ν 2 ) = −ν 2 .Let us consider the trivial connection Proof.Assuming that ∇ X Y = 0 at the point where we do the computations, we have by definition , and the formula is a consequence of (11).Finally, (∂ − ∇) Lemma 1.7.The product structure P is parallel with respect to ∂.
Proof.Using (45) twice, for X, Y tangent to M we have The computations for X = ν 2 are analogous.
Since P and P ′ |M are parallel sections of endomorphisms of and since id+ P and id− P have rank m+1 and n+1, there exists A ∈ O(m+n+2) such that We consider a ∈ Spin(m + n + 2) such that Ad(a) = A −1 and the spinor field 2 ) is compatible with the product structures P and P ′ .Finally, it is clear from the proof that if a solution ϕ of ( 7) is such that Φ : X → X • ϕ, ϕ commutes with the product structures, then the other solutions of ( 7) satisfying this property are of the form ϕ•a with a ∈ Spin(m+ n+ 2) such that Ad(a) belongs to SO(m + 1) × SO(n + 1), i.e. with a ∈ Spin(m + 1) • Spin(n + 1).

Isometric immersions in S m
1 × R n We now consider immersions in S m 1 × R n where S m 1 is a m-dimensional sphere of curvature c 1 > 0. After the statement of the main theorem in Section 2.1, we study the special cases S 2 × R and S 2 × R 2 in Sections 2.2 and 2.3.
In that section M still denotes a p-dimensional riemannian manifold and E → M a metric bundle of rank q with p+q = m+n, equipped with a connection compatible with the metric.We consider here the trivial bundle E 1 := M × R → M, with its natural metric and the trivial connection, and fix a unit parallel section ν 1 of E 1 .We finally consider the representation associated to the splitting R m+n+1 = R p ⊕R q ⊕R ρ : Spin(p)×Spin(q) → Spin(p)•Spin(q) ⊂ Spin(m+n+1) → Aut(Cl(m+n+1)) (the last map is given by the left multiplication) and the bundles (associated to a spin structure and We finally suppose that a product structure P is given on T M ⊕ E as in Section 1.3.

2.1.
Statement of the theorem.
Theorem 3. We suppose that M is simply connected.Let B : T M × T M → E be a symmetric tensor.The following statements are equivalent: (i) There exist an isometric immersion F : M → S m 1 × R n and a bundle map Φ : T M ⊕ E → T S m 1 × R n above F such that Φ(X, 0) = dF (X) for all X ∈ T M, which preserves the bundle metrics, maps the connection on E and the tensor B to the normal connection and the second fundamental form of F , and is compatible with the product structures.
(ii) There exists a section ϕ ∈ Γ(U Σ) solution of (46) for all X ∈ T M, where X = X 1 + X 2 is the decomposition in the product structure P of T M ⊕ E, such that the map commutes with the product structures P and P ′ .
Moreover, the bundle map Φ and the immersion F are explicitly given in terms of the spinor field ϕ by the formulas Brief indications of the proof: setting and using ( 16)-( 19) it is not difficult to see that Φ 2 is a closed 1-form and F 2 is well defined if M is simply connected.Formulas (47) and (48) thus give an explicit expression for F = (F 1 , F 2 ) in terms of the spinor field ϕ, and the theorem may then be proved by direct computations as in the previous sections.
Here again, as in the case of a product of two spheres, we can obtain a spinorial proof of the fundamental theorem of immersions theory in S m 1 × R n .

Surfaces in
to identify Σ 2 with Σ 1 , and an identification . Since ϕ 1 and ϕ 2 are both normalized solutions of where N is a unit normal and S : T M → T M is the corresponding shape operator of M in S 2 × R, ψ 1 and ψ 2 ∈ ΣM ⊗ ΣE are so that (50) . Now the condition expressing that Φ commutes with the product structures gives the following: Lemma 2.1.For a convenient choice of the unit section V ∈ Γ(T M ⊕E) generating the distinguished line P 2 of the product structure P of T M ⊕ E, we have , which readily implies (52).
Equations ( 50) and (51) and the lemma imply that ψ 1 and ψ 2 satisfy and The spinor field ψ := By (52), we have This is the spinorial characterization of an immersion in S 2 × R obtained in [15].
Remark 3. Similarly, it is possible to obtain as a consequence of Theorem 3 the characterizations in terms of usual spinor fields of immersions of surfaces or hypersurfaces in S 3 × R, or of surfaces in S 2 × R 2 , obtained in [12,16].We rather focus below on the new case of hypersurfaces in S 2 × R 2 .
As in the previous section we consider the map u : to identify Σ 2 with Σ 1 , and for We traduce in the following lemma the condition expressing that Φ commutes with the product structures: it shows that ψ 2 (and thus ϕ and therefore the immersion) is essentially determined by ψ 1 : Proof.The condition expressing that Φ commutes with the product structures reads Φ(P 2 ) = {0} × R 2 , and V 1 , V 2 are well-defined.The proof is then identical to the proof of Lemma 2.1 above.Equation (56) implies that (58) where N and S respectively denote a unit normal vector and the corresponding shape operator of M in Under the Clifford action of the volume element for all X ∈ T M and Ψ ∈ ΣM, where the complex structure on Σ + 1 is given by the right-action of e o 0 • e o 1 .We write together with Conversely, the existence of two spinor fields Ψ 1 , Ψ 2 ∈ Γ(ΣM ) solutions of ( 59)-( 63) implies the existence of an isometric immersion of M into S 2 × R 2 : we may indeed construct ϕ ∈ Γ(U Σ) solution of (46) from Ψ 1 and Ψ 2 , just doing step by step the converse constructions; it is such that the map Φ : X → X • ϕ, ϕ commutes with the product structures.
Remark 4. Two non-trivial spinor fields Ψ 1 , Ψ 2 ∈ Γ(ΣM ) solutions of ( 59)-( 60) are in fact such that |Ψ 1 | 2 + |Ψ 2 | 2 is a constant, and may thus be supposed so that (61) holds.Here M still denotes a p-dimensional riemannian manifold and E → M a metric bundle of rank q with a connection compatible with the metric and such that p+q = m+n.We suppose that a product structure P is given on T M ⊕ E as in Section 1.3.We denote by R r,s the space R r+s with the metric with signature

Isometric immersions in
by Cl(r, s) its Clifford algebra and by Spin(r, s) its spin group.For immersions in H m 1 × H n 2 we consider the trivial bundle E 2 := M × R 2,0 → M, with the natural negative metric and the trivial connection, and two orthonormal and parallel sections ν 1 , ν 2 of that bundle.We also consider the representation associated to the splitting and the bundles (associated to a spin structure Theorem 4. Let B : T M × T M → E be a symmetric tensor.The following statements are equivalent: (i) There exist an isometric immersion above F such that Φ(X, 0) = dF (X) for all X ∈ T M, which preserves the bundle metrics, maps the connection on E and the tensor B to the normal connection and the second fundamental form of F , and is compatible with the product structures.
(ii) There exists a section ϕ ∈ Γ(U Σ) solution of (64) for all X ∈ T M, where X = X 1 + X 2 is the decomposition in the product structure P of T M ⊕ E, such that the map commutes with the product structures P and P ′ .Moreover, the bundle map Φ and the immersion F are explicitly given in terms of the spinor field ϕ by the formulas For immersions in Theorem 5. Let B : T M × T M → E be a symmetric tensor.The following statements are equivalent: (i) There exist an isometric immersion for all X ∈ T M, which preserves the bundle metrics, maps the connection on E and the tensor B to the normal connection and the second fundamental form of F , and is compatible with the product structures.
(ii) There exists a section ϕ ∈ Γ(U Σ) solution of (66) for all X ∈ T M, where X = X 1 + X 2 is the decomposition in the product structure P of T M ⊕ E, such that the map commutes with the product structures P and P ′ .
Moreover, the bundle map Φ and the immersion F are explicitly given in terms of the spinor field ϕ by the formulas As in the positive curvature case, it is possible to deduce a spinorial proof of the fundamental theorem of immersions theory in We consider the immersion of a surface with R represented by a spinor field ϕ as in Theorem 5 (with m = 2 and n = 1).Let us first introduce some notation.We denote by N the unit vector normal to the surface and tangent to H 2 × R, it is of the form (N ′ , ν) in R 1,2 × R, and by ν 1 the unit vector normal to H 2 × R so that the immersion reads F = (ν 1 , h) ∈ H 2 × R. The function ν is the angle function of the immersion, and we assume that it is always positive (the surface has regular vertical projection), and the function h : M → R is the height function of the immersion.We fix a conformal parameter z = x + iy of the surface, in which the metric reads µ 2 (dx 2 + dy 2 ).The matrix of the shape operator in the basis and we set the following two important quantities Following [7] (Q 0 , τ 0 ) are called the Weierstrass data of the immersion, and we will see below that they appear naturally in the equations satisfied by the spinor field representing the immersion in adapted coordinates.We will then compute the hyperbolic Gauss map in terms of these data (we will recall the definition below) and we will interpret geometrically the relation between the spinor field and the hyperbolic Gauss map.Using these observations we will show that conversely the hyperbolic Gauss map and its Weierstrass data determine a family of spinor fields (parameterized by C), and thus a family of immersions, a result obtained in [7] by other methods.We will finally use this spinorial approach to describe directly the correspondence between the theories of H = 1/2 surfaces in H 2 × R and in R 1,2 .
4.1.The Clifford algebra and the Spin group of R 1,3 .Let us consider the algebra of complex quaternions H C := H ⊗ C.An element a of H C is of the form a = a 0 1 + a 1 I + a 2 J + a 3 K, a 0 , a 1 , a 2 , a 3 ∈ C, its complex norm is H(a, a) = a 2 0 + a 2 1 + a 2 2 + a 2 3 ∈ C and its complex conjugate is the complex quaternion where a i denotes the usual complex conjugate of a i .Let us associate to where JI = −IJ = −K.Using the Clifford map we easily obtain that For the sake of simplicity, we will frequently use below the natural identifications of Cl 0 (1, 3) and Cl 1 (1, 3) with H C .We will moreover use the models (69) ∈ R} and we will decompose the special direction I of the product structure in the form where T is tangent and N is normal to the immersion, and ν is the angle function.C , and we consider the components It will be convenient to consider the following norm on 1 2 (1 + iI) H C : writing an element g ′ 1 belonging to 1 2 (1 + iI) H C in the form for some (unique) a, b ∈ C, we define its hermitian norm Moreover the compatibility conditions of the equation read The equation of g ′ 1 only depends on the Weierstrass data (Q 0 , τ 0 ).In the statement and in the rest of the section we use the following notation: z = x + Iy, dz = dx+Idy, ∂ z = 1/2(∂ x −I∂ y ), z = x−Iy, dz = dx−Idy and ∂ z = 1/2(∂ x +I∂ y ), i.e. the complex parameter z is with respect to the complex structure I.
Remark 5.The Abresch-Rosenberg differential is the quadratic differential −Q 0 dz 2 .It rather appears here as a 1-form.
We will need for the computations the following form of the compatibility equations for the product structure: Lemma 4.2.In the frame s the product structure T + νN reads and h z , µ and ν satisfy the relations Moreover, the two components g 1 and g 2 of the spinor field are linked by Proof of Lemma 4. Proof of Proposition 4.1: Since X 1 = X − X 2 = X − X, T (T + νN ) with [X] = µdzJ, X, T = dh(X) and [T ] + ν[N ] given by (72), we have Using that [∇ϕ] = dg − 1 2 aIg, (78) and (80), the Killing type equation (66) with We take H = 1/2 and we multiply both sides of the equation by 1/2(1 + iI) on the left: since the first right-hand term vanishes and we get Using (75) we obtain and using finally that (this is a consequence of (74) and ( 79)) together with (77) we get √ µg 1 satisfies (70).A computation using (70) twice then shows that which proves (71).For the last claim, g = a 0 1 + a 1 I + a 2 J + a 3 K with a j ∈ C is such that gI g = [T ] + ν[N ] belongs to RI ⊕ RJ ⊕ RK (recall the last step of the proof of Lemma 4.2), i.e. the component of gI g on i1 is zero, which yields (81) ℑm(a 0 a 1 ) = ℑm(a 2 a 3 ); moreover, ν is defined as the component along the direction I of the vector N • ϕ, ϕ = gI g normal to the immersion, which yields by a direct computation 4.3.The product structure and the second component of the spinor field.
It appears that the product structure (ν and h z /µ in (72)) may be determined independently of g ′ 1 and g ′ 2 .This relies on the following key lemma of [7]: Lemma 10] The function h z satisfies the system If we fix z 0 and θ 0 ∈ C, the system admits a unique globally defined solution h z satisfying the initial condition h z (z 0 ) = θ 0 .
The system is a consequence of the compatibility equations ( 73) and (74).Since the equations only depend on the Weierstrass data (Q 0 , τ 0 ), h z only depends on these data and on the choice of the initial condition h z (z 0 ) ∈ C.Moreover, since (by (76) and since |T | 2 = 1 − ν 2 with τ 0 = µ 2 ν 2 ) µ and ν are also determined by (Q 0 , τ 0 ) and h z (z 0 ).We finally observe that the other component g ′ 2 := √ µg 2 of the spinor field is given by (82) 75)) and is thus determined by g ′ 1 if h z is known: so g ′ 1 determines a family of spinor fields parametrized by C; the parameter corresponds to the choice of an initial condition for the determination of the product structure.4.6.Surfaces with prescribed hyperbolic Gauss map.We assume here that G : C → H 2 is a given map with Weierstrass data (Q 0 , τ 0 ) and that the set of singular points of G has empty interior.The following result was obtained in [7].
Corollary 2. There exists a family of CMC surfaces with H = 1/2 in H 2 × R with hyperbolic Gauss map G and Weierstrass data (Q 0 , τ 0 ).The family is parameterized by C × R.
Proof.Since the connection ω is flat (Remark 6), the principal bundle H → C admits a globally defined parallel section that we may consider as a map σ : for some θ ∈ R (indeed, the term 1/ √ τ 0 (Q 0 dz + τ 0 /4dz) J σI is horizontal at σ and a computation shows that its projection by dπ is of the form (83), with an orthonormal basis (u ′ 1 , u ′ 2 ) of T G H 2 which is perhaps different to the basis (u 1 , u 2 ); taking θ such that (u ′ 1 , u ′ 2 ) matches with (u 1 , u 2 ), the right hand term of (88) is the horizontal lift of dG).The compatibility conditions of (88) imply that θ is a constant, and in view of (87) the section g ′ 1 := e −iθ σ is a solution of (70).g ′ 1 is uniquely determined up to a sign, as in Corollary 1.Note that this equation extends by continuity to the singular points of G.We then consider a product structure h z given by Lemma 4.3 (there is a family of solutions, depending on a parameter belonging to C), the solution g ′ 2 given by (82) and set g := 1/ √ µ(g ′ 1 + g ′ 2 ) : it belongs to S 3 C (since H(g, g) = 1 by a computation), and we consider the spinor field ϕ whose component is g in s, and the corresponding immersions into H 2 × R; they depend on C × R since a last integration is required to obtain h from h z .
Remark 7. The spinorial representation formula permits to recover the explicit representation formula of the immersion in terms of all the data: calculations from the representation formula F = (ig g, h) (formula (67)) lead to the expression of the immersion in terms of G z , Q 0 , τ 0 and h given in [7,Theorem 11].4.7.Link with H = 1/2 surfaces in R 1,2 .We first describe with spinors the immersions of H = 1/2 surfaces in R 1,2 and deduce that they are entirely determined by their Gauss map and its Weierstrass data.This is a proof using spinors of a result obtained in [1] with other methods.We then obtain a simple algebraic relation between the spinor fields representing these immersions and families (parametrized by C) of spinor fields representing H = 1/2 surfaces in H 2 × R: this gives a simple interpretation of the natural correspondence between H = 1/2 surfaces in R 1,2 and in H 2 × R. Since the proofs are very similar to proofs of the preceding sections, we will omit many details.We consider the model Spin(1, 2) = {α 0 1 + α 1 I + iα 2 J + iα 3 JI, α j ∈ R, α 2 0 + α 2 1 − α 2 2 − α 2 3 = 1} which is the subgroup of Spin(1, 3) = S 3 C fixing I ∈ R 1,3 (under the double covering Spin(1, 3) → SO(1, 3)) and thus also leaving R 1,2 := {ix 0 1 + x 2 J + x 3 JI, x 0 , x 2 , x 3 ∈ R} ⊂ R 1,3 globally invariant.By [4], if Q is a spin structure of M and ρ : Spin Proof.This is a traduction of (91), similar to Proposition 4.4.
Corollary 3. v is determined by the Gauss map G and its Weierstrass data (Q 0 , τ 0 ) up to a sign.
Proof.It is analogous to the proof of Corollary 1.
We now suppose that a map G : C → H 2 is given with Weierstrass data (Q 0 , τ 0 ).We moreover suppose that the set of singular points of G has empty interior.
Proof.As in the proof of Corollary 2, a horizontal section v of G * Spin(1, 2) → C is a map such that for some function θ ′ : C → R which has to be constant, and v = e −Iθ ′ v is a section solution of (91) such that π ′ ( v) = G.Since such a solution is unique up to a sign, the spinor field ψ is determined up to a sign and the immersion is unique up to a translation (since the immersion in R 1,2 is finally obtained by the integration of the 1-form ξ(X) = X • ψ, ψ ).
Proposition 4.7.The correspondence transforms a solution v ′ of (90) such that π ′ (v ′ /|v ′ |) = G to a solution g ′ 1 of (70) such that π(g ′ 1 ) = G + I, and vice-versa.Proof.Assuming that (91) holds we compute For the last term, since Iv ′ = v ′ I (v belongs to Spin(1, 2)) we observe that which shows that g ′ 1 satisfies (70).We finally note that |g ′ 1 | 2 = µ and therefore The first term iv v is the Gauss map G. Since I v = vI and vv = 1, the second term vI v is the constant I.So π(g ′ 1 ) = G + I and g ′ 1 lies above the same (hyperbolic) X , and we conclude with (43) that (∂ X P)(Y ) = 0.The computation for Y ∈ Γ(E) is analogous.For Y = ν 1 we have