Abstract
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 \(\mathbb {S}^2\times \mathbb {R}\) and we obtain new spinorial characterizations of immersions in \(\mathbb {S}^2\times \mathbb {R}^2\) and in \(\mathbb {H}^2\times \mathbb {R}.\) We then study the theory of \(H=1/2\) surfaces in \(\mathbb {H}^2\times \mathbb {R}\) using this spinorial approach, obtain new proofs of some of its fundamental results and give a direct relation with the theory of \(H=1/2\) surfaces in \(\mathbb {R}^{1,2}\).
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1 Introduction
Characterizations of immersions in space forms using spinors have been widely studied, as for instance in [8, 9, 11, 12, 15, 18,19,20,21] 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 \(\mathbb {S}^2\times \mathbb {R},\) \(\mathbb {H}^2\times \mathbb {R},\) \(\mathbb {S}^2\times \mathbb {R}^2\) and \(\mathbb {S}^3\times \mathbb {R}\) [13, 16, 17]. 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 \(\mathbb {S}^2\times \mathbb {R}\) and we obtain new spinorial characterizations of immersions in \(\mathbb {S}^2\times \mathbb {R}^2\) and in \(\mathbb {H}^2\times \mathbb {R}.\)
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 \(\mathbb {H}^2\times \mathbb {R}\): 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 \(\mathbb {H}^2\times \mathbb {R}\) with given hyperbolic 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 \(\mathbb {R}^{1,2}\) and obtain a direct relation between the two theories.
In order to simplify the exposition we first consider immersions in a product of spheres \(\mathbb {S}_1^m\times \mathbb {S}_2^n\) and in a product \(\mathbb {S}_1^m\times \mathbb {R}^n,\) and we then state without proof the analogous results for immersions in a product of hyperbolic spaces \(\mathbb {H}_1^m\times \mathbb {H}_2^n\) and in \(\mathbb {H}_1^m\times \mathbb {R}^n.\) Using the same ideas it is possible to state analogous results for an arbitrary quantity of factors involving \(\mathbb {S}_1^m,\) \(\mathbb {H}_2^n\) and \(\mathbb {R}^p\), 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 \(\mathbb {S}_1^m\times \mathbb {S}_2^n\) in Sect. 1 and the immersions in \(\mathbb {S}_1^m\times \mathbb {R}^n\) in Sect. 2. We then state the analogous results for a product of hyperbolic spaces \(\mathbb {H}_1^m\times \mathbb {H}_2^n\) and for \(\mathbb {H}_1^m\times \mathbb {R}^n\) in Sect. 3. Finally the theory of \(H=1/2\) surfaces in \(\mathbb {H}^2\times \mathbb {R}\) is studied in Sect. 4. Some useful auxiliary results are gathered in an appendice at the end of the paper.
2 Isometric Immersions in \(\mathbb {S}_1^m\times \mathbb {S}_2^n\)
We are interested here in immersions in a product \(\mathbb {S}_1^m\times \mathbb {S}_2^n\) of two spheres, of constant curvature \(c_1,c_2>0\). We construct the suitable spinor bundle in Sect. 1.1, we consider the case of a manifold which is already immersed in \(\mathbb {S}_1^m\times \mathbb {S}_2^n\) in Sect. 1.2, we state and prove the main theorem in Sect. 1.3 and the fundamental theorem in Sect. 1.4.
2.1 The Suitable Spinor B undle
Let M be a p-dimensional riemannian manifold and \(E\rightarrow M\) a vector bundle of rank q, with \(p+q=m+n,\) with a bundle metric and a connection compatible with the metric. Let \(\mathcal {E}_2=M\times \mathbb {R}^2\rightarrow M\) 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 \(TM\oplus E\oplus \mathcal {E}_2.\) We suppose that M and E are spin, with spin structures \(\widetilde{Q}_M\rightarrow Q_M,\) \(\widetilde{Q}_E\rightarrow Q_E\) and set \(\widetilde{Q}:=\widetilde{Q}_M\times _M\widetilde{Q}_E.\) Let us denote by Spin(N) and Cl(N) the spin group and the Clifford algebra of \(\mathbb {R}^N.\) Associated to the splitting \(\mathbb {R}^{m+n+2}=\mathbb {R}^p\oplus \mathbb {R}^q\oplus \mathbb {R}^2\) we consider
and define the representation
with the bundles
Since the bundle of Clifford algebras constructed on the fibers of \(TM\oplus E\oplus \mathcal {E}_2\) is
with
there is a Clifford action
similar to the usual Clifford action in spin geometry. Let us note that \(\Sigma \) 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 \(\Sigma \) as the bundle of spinors, \(U\Sigma \) as the bundle of unit spinors and \(Cl(TM\oplus E\oplus \mathcal {E}_2)\) as the Clifford bundle acting on the bundle of spinors. There is a natural map
where \(\varphi =[\tilde{s},[\varphi ]]\) and \(\varphi '=[\tilde{s},[\varphi ']]\) in \(\Sigma =\widetilde{Q}\times _{\rho } Cl(m+n+2)\) and \(\tau \) 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 \(\widetilde{s}.\) This map is such that, for all \(\varphi ,\varphi '\in \Sigma \) and \(Z\in TM\oplus E\oplus \mathcal {E}_2,\)
and
Moreover, it is compatible with the connection \(\nabla \) induced on \(\Sigma \) by the Levi-Civita connection on M and the given connection on E :
Lemma 1.1
For all \(X\in TM\) and \(\varphi ,\varphi '\in \Gamma (\Sigma ),\)
where on the left hand side \(\partial \) stands for the usual derivative.
A similar result is proved in [6, Lemma 2.2].
2.2 Spin Geometry of a Submanifold in \(\mathbb {S}_1^m\times \mathbb {S}_2^n\)
We assume in that section that M is a p-dimensional submanifold of \(\mathbb {S}_1^m\times \mathbb {S}_2^n,\) with normal bundle E of rank q and second fundamental form \(B:TM\times TM\rightarrow E,\) denote by \(\nu _1:M\rightarrow \mathbb {R}^{m+1}\) and \(\nu _2:M\rightarrow \mathbb {R}^{n+1}\) the vector fields such that \(\frac{1}{\sqrt{c_1}}\nu _1\) and \(\frac{1}{\sqrt{c_2}}\nu _2\) are the two components of the immersion \(M\rightarrow \mathbb {S}_1^m\times \mathbb {S}_2^n\) and consider the trivial bundle \(\mathcal {E}_2=\mathbb {R}\nu _1\oplus \mathbb {R}\nu _2\rightarrow M.\) We consider spin structures on TM and E, and the bundles \(\Sigma ,\) \(U\Sigma \) and \(Cl(TM\oplus E\oplus \mathcal {E}_2)\) constructed in the previous section. For a convenient choice of the spin structures on TM and E, the bundle \(\Sigma \) identifies canonically with the trivial bundle \(M\times Cl(m+n+2)\), and two connections are defined on \(\Sigma ,\) the connection \(\nabla \) introduced above and the trivial connection \(\partial .\) Since the second fundamental form of M in \(\mathbb {R}^{m+n+2}\) is
where \(X=X_1+X_2\) and \(Y=Y_1+Y_2\) in the decomposition \(TM\subset T\mathbb {S}_1^m\oplus T\mathbb {S}_2^n\) and setting
where \(e_1,\ldots , e_p\) is an orthonormal basis of TM, they satisfy the following Gauss formula:
for all \(\varphi \in \Gamma (\Sigma )\) and all \(X\in TM\); see [3] for the proof of a spinorial Gauss formula in a slightly different context. By formula (5), the constant spinor field \(\varphi ={1_{Cl(m+n+2)}}_{|M}\) satisfies, for all \(X\in TM,\)
2.3 The Main Theorem
We assume that M and \(E\rightarrow M\) are abstract objects as in Sect. 1.1 (i.e. M is not a priori immersed in \(\mathbb {S}_1^m\times \mathbb {S}_2^n\)), and suppose moreover that there is a product structure on \(TM\oplus E,\) i.e. a bundle map \(\mathcal {P}:TM\oplus E\rightarrow TM\oplus E\) such that \(\mathcal {P}^2=id,\) \(\mathcal {P}\ne id.\) Setting \(\mathcal {P}_1:=\text{ Ker } (\mathcal {P}-id)\) and \(\mathcal {P}_2:=\text{ Ker } (\mathcal {P}+id)\) we have \(TM\oplus E=\mathcal {P}_1\oplus \mathcal {P}_2:\) the product structure \(\mathcal {P}\) is equivalent to a splitting of \(TM\oplus E\) into two subbundles \(\mathcal {P}_1\) and \(\mathcal {P}_2,\) that we assume to be respectively of rank m and n.
2.3.1 Statement of the Theorem
Let \(B: TM \times TM \rightarrow E\) be a symmetric tensor. Let us fix two unit orthogonal and parallel sections \(\nu _1,\nu _2\) of the trivial bundle \(\mathcal {E}_2=M\times \mathbb {R}^2\rightarrow M.\)
Theorem 1
The following statements are equivalent:
-
(i)
There exist an isometric immersion \(F:M\rightarrow \mathbb {S}_1^m\times \mathbb {S}_2^n\) and a bundle map \(\Phi : TM \oplus E \rightarrow T(\mathbb {S}_1^m \times \mathbb {S}_2^n)\) above F such that \(\Phi (X,0)= dF(X)\) for all \(X\in TM,\) 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 \(\varphi \in \Gamma (U \Sigma )\) solution of
$$\begin{aligned} \nabla _X \varphi = \frac{1}{2} (\sqrt{c_1}\ X_1 \cdot \nu _1 + \sqrt{c_2}\ X_2 \cdot \nu _2)\cdot \varphi - \frac{1}{2}B(X)\cdot \varphi \end{aligned}$$(7)for all \(X\in TM,\) where \(X= X_1 + X_2\) is the decomposition in the product structure \(\mathcal {P}\) of \(TM\oplus E\), such that the map
$$\begin{aligned} Z\in TM\oplus E\ \mapsto \ \langle \langle Z \cdot \varphi , \varphi \rangle \rangle \in \mathbb {R}^{m+1}\times \mathbb {R}^{n+1} \end{aligned}$$commutes with the product structure \(\mathcal {P}\) and the natural product structure on \(\mathbb {R}^{m+1}\times \mathbb {R}^{n+1}.\)
Moreover, the bundle map \(\Phi \) and the immersion F are explicitly given in terms of the spinor field \(\varphi \) by the formulas
and
Remark 1
Formulas (7) and (8) can be regarded as a generalized Weierstrass representation formula.
2.3.2 Proof of Theorem 1
The proof of “\((i)\Rightarrow (ii)\)” was obtained in Sect. 1.2: if M is immersed in \(\mathbb {S}_1^m\times \mathbb {S}_2^n,\) the spinor field \(\varphi \) is the constant spinor field \(1_{Cl(m+n+2)}\) restricted to M. We prove “\((ii)\Rightarrow (i).\)” We suppose that \(\varphi \in \Gamma ( U \Sigma )\) 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
for all \(X\in TM.\) It preserves the product structures and takes values in \(\mathbb {S}_1^m \times \mathbb {S}_2^n.\)
Proof
Let us consider for \(i=1,2\) the functions \(F_i =\frac{1}{\sqrt{c_i}}\langle \langle \nu _i \cdot \varphi , \varphi \rangle \rangle .\) Recalling the properties (1)–(3) of \(\langle \langle .,.\rangle \rangle \) and since \(\nu _1,\nu _2\) are parallel sections of \(\mathcal {E}_2\) and \(\varphi \) satisfies (7), we have, for \(i=1,2,\)
where we use in the last equality that, for all \(i,j\in \{1,2\},\)
and
(since \(\nu _i,\) \(X_j,\) \(\nu _j\) are three orthogonal vectors if \(i\ne j\)). Since \(F=F_1+F_2\) and \(X=X_1+X_2,\) (9) follows from (10). Let us see now that F takes values in \(\mathbb {S}_1^m \times \mathbb {S}_2^n \subset \mathbb {R}^{m+n+2}\). We assume that \(\nu _1,\nu _2\) are given in an arbitrary frame \(\widetilde{s}\in \widetilde{Q}\) by \(\nu _1=[\widetilde{s},e_{1}^o]\) and \(\nu _2=[\widetilde{s},e_{2}^o]\) where \(e_1^o\) and \(e_2^o\) are the last two vectors of the canonical basis of \(\mathbb {R}^{m+n+2}=\mathbb {R}^{p+q}\oplus \mathbb {R}^2.\) Since \([\varphi ]\) belongs to \(Spin(p+q+2)\), we have
which implies that \(F_1\) and \(F_2\) take values in the spheres of \(\mathbb {R}^{m+n+2}\) of radius \(1/\sqrt{c_1}\) and \(1/\sqrt{c_2}\) respectively. We then have to check that \(F_1\) and \(F_2\) take respectively values in \(\mathbb {R}^{m+1}\) and \(\mathbb {R}^{n+1}:\) since dF preserves the product structures, we have for \(X=X_1+X_2 \in TM\) that
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 \(\mathbb {R}^{m+1}\) and \(\mathbb {R}^{n+1}\) 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 \(\mathbb {S}_1^m \times \mathbb {S}_2^n\), 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 \(\Phi \) takes values in \(T(\mathbb {S}_1^m \times \mathbb {S}_2^n).\) By Lemma 1.2, if \(X\in TM,\) \(\Phi (X)\) belongs to \(T(\mathbb {S}_1^m \times \mathbb {S}_2^n).\) For \(Z_1, Z_2 \in TM \oplus E \oplus \mathcal {E}_2,\) we have in \(Cl(m+n+2)\)
since \(\tau [\varphi ][\varphi ]=1\) and \(\langle Z_1, Z_2 \rangle = \langle [Z_1], [Z_2] \rangle \) belongs to \(\mathbb {R}.\) Let us note that \(\widetilde{\Phi }(\nu _1)\) and \(\widetilde{\Phi }(\nu _2)\) are normal to \(T(\mathbb {S}_1^m \times \mathbb {S}_2^n)\): for \(i=1,2,\) we have at \(p\in M\)
which is the unit normal to \(\mathbb {S}_i\) at p. Thus, since \(\widetilde{\Phi }\) preserves the metric, for \(Y\in E,\) \(\Phi (Y)\) belongs to \(T(\mathbb {S}_1^m \times \mathbb {S}_2^n).\) Let us now compute, for \(Z\in \Gamma (TM\oplus E),\)
where \(p_{\mathbb {S}_1^m \times \mathbb {S}_2^n}\) is the projection \(\mathbb {R}^{m+1}\times \mathbb {R}^{n+1}\rightarrow T(\mathbb {S}_1^m \times \mathbb {S}_2^n\)). We have
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
Since
where \(B^*:TM\times E\rightarrow TM\) is so that \(\langle B(X,Y),Z\rangle =\langle Y,B^*(X,Z)\rangle \) for all \(X,Y\in TM\) and \(Z\in E\) (Lemma A.2 in Appendix A), we deduce that
and
This formula implies the following expressions for the second fundamental form \(B^F\) and the normal connection \(\nabla '^F\) of the immersion F : if \(Z\in \Gamma (TM)\) is such that \(\nabla Z=0\) at the point where we do the computations, then \(B^F(X,Z)=\Phi (B(X,Z));\) if \(Z\in \Gamma (E)\), then \(\nabla '^F_X(\Phi (Z))=\Phi (\nabla '_XZ).\) This finishes the proof of the lemma (and of Theorem 1).
2.4 The Fundamental Theorem in \(\mathbb {S}_1^m\times \mathbb {S}_2^n\)
We give here a proof using spinors of the fundamental theorem of the immersion theory in \(\mathbb {S}_1^m\times \mathbb {S}_2^n\). This result has been proved independently by Kowalczyk [10] and Lira–Tojeiro–Vitório [14].
2.4.1 Statement of the Theorem
Let \(\mathcal {P}\) and \(\mathcal {P}'\) be the product structures of \(TM \oplus E\) and \(\mathbb {R}^{m+n+2}=\mathbb {R}^{m+1}\times \mathbb {R}^{n+1}\). We define \(f:TM \rightarrow TM, h:TM \rightarrow E,s:E \rightarrow TM\) and \(t:E \rightarrow E\) such that
We set, for \(U,V,W\in TM\oplus E,\)
We first write the compatibility equations necessary for the existence of a non-trivial spinor field solution of (7):
Proposition 1.4
Let \(\varphi \in \Gamma (U\Sigma )\) be a solution of (7) such that
commutes with the product structures, i.e. satisfies
for all \(Z \in TM \oplus E.\) 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 \(\nabla '\) on E, the following fundamental equations hold: for all \(X,Y,Z\in TM\) and \(N\in E,\)
where \(\widetilde{\nabla }\) stands for the natural connection on \(T^*M\otimes T^*M\otimes E.\) Moreover, if we use the same symbol \(\widetilde{\nabla }\) to denote the natural connections on \(T^*M\otimes TM,\) \(T^*M\otimes E,\) \(E^*\otimes TM\) and \(E^*\otimes E,\) we have, for \(X,Y\in TM\) and \(Z\in E,\)
Equations (13), (14) and (15) are respectively the equations of Gauss, Ricci and Codazzi. Equations (16)–(19) are additional equations traducing that \(\Phi \) commutes with the product structures \(\mathcal {P}\) and \(\mathcal {P}',\) with \(\mathcal {P}'\) parallel in \(\mathbb {R}^{m+n+2}.\) All these equations are necessary for the existence of an immersion \(M\rightarrow \mathbb {S}_1^m\times \mathbb {S}_2^n\) with second fundamental form B and normal connection \(\nabla '.\) It appears that they are also sufficient:
Theorem 2
Let us assume that \(B:TM \times TM \rightarrow E\) 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 \(\dim \text{ Ker }(\mathcal {P}-id)=m\) and \(\dim \text{ Ker }(\mathcal {P}+id)=n.\) Then there exists \(\varphi \in \Gamma (U \Sigma )\) solution of (7) such that the map
commutes with the product structures \(\mathcal {P}\) and \(\mathcal {P}'\). The spinor field \(\varphi \) is moreover unique up to the natural right action of \(Spin(m+1)\cdot Spin(n+1)\) on \(U\Sigma .\) In particular, there is an isometric immersion \(F:M\rightarrow \mathbb {S}_1^m\times \mathbb {S}_2^n\) and a bundle isomorphism \(\Phi :TM\oplus E\rightarrow T(\mathbb {S}_1^m\times \mathbb {S}_2^n)\) above F identifying E, B and \(\nabla '\) to the normal bundle, the second fundamental form and the normal connection of F in \(\mathbb {S}_1^m\times \mathbb {S}_2^n.\) The immersion is moreover unique up to the natural action of \(SO(m+1)\times SO(n+1)\) on \(\mathbb {S}_1^m\times \mathbb {S}_2^n.\)
Section 1.4.2 is devoted to the proof of Proposition 1.4, and Sect. 1.4.3 to the proof of Theorem 2.
2.4.2 Proof of Proposition 1.4
We assume that \(X,Y\in \Gamma (TM)\) are such that \(\nabla X=\nabla Y=0\) at the point where we do the computations. A direct computation using (7) twice yields
with
and
Lemma 1.5
In local orthonormal frames \(\{e_j\}_{1\le j \le p}\) of TM and \(\{n_r\}_{1 \le r \le q}\) of E we have
These expressions respectively mean that \(\mathcal {A}\in TM\otimes E\) represents the transformation
\(\mathcal {B}\in \Lambda ^2TM\oplus \Lambda ^2E\) represents the transformation
together with
and \(\mathcal {C}\in \ \Lambda ^2TM\ \oplus \ \Lambda ^2E\ \oplus \ TM\otimes E\) represents
and
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), \(\mathcal {A}\) represents the transformation (24) and \(\mathcal {B}\) 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
is
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 \(TM\oplus E\) is given by
Comparing Eqs. (20) and (37) and since \(\varphi \) is represented in a frame \(\widetilde{s}\in \widetilde{Q}\) by an element of \(Spin(m+n+2),\) invertible in \(Cl(m+n+2),\) we deduce that
and
Now the right hand side of (38) represents the transformations \(Z\in TM\mapsto R^T(X,Y)Z\in TM\) and \(N\in E\mapsto R^N(X,Y)N\in E.\) The Eqs. (13)–(15) of Gauss, Ricci and Codazzi follow from this and Lemma 1.5. Let us now prove that Eqs. (16)–(19) are consequences of the fact that \(\Phi \) commutes with the product structures \(\mathcal {P}\) and \(\mathcal {P}',\) Eq. (12), where the product structure \(\mathcal {P}'\) is parallel. We have by (12)
Assuming that \(\nabla X=0\) 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 \(X\in TM\)
and for \(X \in E\)
Using that \(\Phi \) is injective on the fibers and decomposing \(\nabla _Y \mathcal {P}(X)\), \(\mathcal {P}(B(Y,X))\) and \(\mathcal {P}(B^*(Y,X))\) in their tangent and normal parts, we get
if \(X \in TM,\) and
if \(X \in E.\) Finally, taking the tangent and the normal parts of each one of the last two equations we get (16)–(19).
Remark 2
Equation (39) is in fact equivalent to the antisymmetric part of (16)–(19).
2.4.3 Proof of Theorem 2
Let us set, for \(X\in TM\) and \(\varphi \in \Gamma (U\Sigma ),\)
We consider \(U\Sigma \rightarrow M\) as a principal bundle of group \(Spin(p+q+2),\) where the action is the multiplication on the right
for all \(a\in Spin(p+q+2).\) The connection \(\nabla '\) may be considered as given by a connection 1-form on this principal bundle, since so is \(\nabla \) and the term
defines a vertical and invariant vector field on \(U\Sigma .\) 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 \(U\Sigma \rightarrow M\) has a global parallel section: this yields \(\varphi \in \Gamma (U\Sigma )\) such that \(\nabla '\varphi =0,\) i.e. a non-trivial solution of (7). Let us verify that Eqs. (16)–(19) imply that the map
is compatible with the product structures, i.e. verifies \(\Phi (\mathcal {P}(X)) = \mathcal {P}' (\Phi (X))\) for all \(X \in TM \oplus E.\) The sum of (16) and (17) gives, for \(X,Y\in TM,\)
Similarly, for \(X\in E\) and \(Y\in TM,\) (18) and (19) imply that
As in the proof of Theorem 1, \(\Phi \) is a bundle map above the immersion
The product structure \(\mathcal {P}\) on \(TM \oplus E\) extends to a product structure \(\widetilde{\mathcal {P}}\) on \(TM \oplus E \oplus \mathcal {E}_2\) by setting \(\widetilde{\mathcal {P}}(\nu _1)=\nu _1\) and \(\widetilde{\mathcal {P}}(\nu _2)=-\nu _2.\) Let us consider the trivial connection \(\partial \) induced on \(TM\oplus E\oplus \mathcal {E}_2\) by the bundle isomorphism
Lemma 1.6
We have for \(Y\in \Gamma (TM\oplus E)\) and \(X\in TM\)
and \((\partial -\nabla )_X\nu _i=X_i,\) \(i=1,2.\)
Proof
Assuming that \(\nabla _X Y = 0\) at the point where we do the computations, we have by definition \((\partial - \nabla )_XY=\widetilde{\Phi }^{-1} (\partial _X \Phi (Y) ),\) and the formula is a consequence of (11). Finally, \((\partial -\nabla )_X\nu _i=\partial _X\nu _i=\widetilde{\Phi }^{-1} (\partial _X N_i)=X_i.\)
Lemma 1.7
The product structure \(\widetilde{\mathcal {P}}\) is parallel with respect to \(\partial \).
Proof
Using (45) twice, for X, Y tangent to M we have
since \(\langle Y_1, \widetilde{\mathcal {P}}(X) \rangle = \langle \widetilde{\mathcal {P}}(Y_1), \widetilde{\mathcal {P}}(X) \rangle = \langle Y_1, X \rangle \) and \(\langle Y_2, \widetilde{\mathcal {P}} (X) \rangle = - \langle \widetilde{\mathcal {P}} (Y_2), \widetilde{\mathcal {P}}(X) \rangle = - \langle Y_2, X \rangle ,\) and we conclude with (43) that \((\partial _X \widetilde{\mathcal {P}})(Y) = 0.\) The computation for \(Y\in \Gamma (E)\) is analogous. For \(Y=\nu _1\) we have
which implies that \((\partial _X \widetilde{\mathcal {P}})(\nu _1) = 0\) since
The computations for \(X=\nu _2\) are analogous.
Since \(\widetilde{\mathcal {P}}\) and \(\mathcal {P}'_{\vert M}\) are parallel sections of endomorphisms of
and since \(id +\widetilde{\mathcal {P}}\) and \(id- \widetilde{\mathcal {P}}\) have rank \(m+1\) and \(n+1,\) there exists \(A \in O(m+n+2)\) such that
on \(F^*T\mathbb {R}^{n+m+2}\). We consider \(a \in Spin(m+n+2)\) such that \(Ad(a)= A^{-1}\) and the spinor field \(\varphi ':= \varphi \cdot a \in U\Sigma :\) it is still a solution of (7) and \(\widetilde{\Phi }'(X):=\langle \langle X \cdot \varphi ', \varphi ' \rangle \rangle \) is such that
The map \(\widetilde{\Phi }'\) thus satisfies \(\widetilde{\Phi }' \circ \widetilde{\mathcal {P}} \circ \widetilde{\Phi }'^{-1}= \mathcal {P}'\) which implies that \(\Phi ':TM\oplus E\rightarrow F^*T(\mathbb {S}_1^m\times \mathbb {S}_2^n)\) is compatible with the product structures \(\mathcal {P}\) and \(\mathcal {P}'.\) Finally, it is clear from the proof that if a solution \(\varphi \) of (7) is such that \(\Phi :X\mapsto \langle \langle X\cdot \varphi ,\varphi \rangle \rangle \) commutes with the product structures, then the other solutions of (7) satisfying this property are of the form \(\varphi \cdot a\) with \(a\in Spin(m+n+2)\) such that Ad(a) belongs to \(SO(m+1)\times SO(n+1),\) i.e. with \(a\in Spin(m+1)\cdot Spin(n+1).\)
3 Isometric Immersions in \(\mathbb {S}_1^m\times \mathbb {R}^n\)
We now consider immersions in \(\mathbb {S}_1^m\times \mathbb {R}^n\) where \(\mathbb {S}_1^m\) is a m-dimensional sphere of curvature \(c_1>0\). After the statement of the main theorem in Sect. 2.1, we study the special cases \(\mathbb {S}^2\times \mathbb {R}\) and \(\mathbb {S}^2\times \mathbb {R}^2\) in Sects. 2.2 and 2.3.
In that section M still denotes a p-dimensional riemannian manifold and \(E\rightarrow M\) 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 \(\mathcal {E}_1:=M\times \mathbb {R}\rightarrow M,\) with its natural metric and the trivial connection, and fix a unit parallel section \(\nu _1\) of \(\mathcal {E}_1.\) We finally consider the representation associated to the splitting \(\mathbb {R}^{m+n+1}=\mathbb {R}^p\oplus \mathbb {R}^q\oplus \mathbb {R}\)
(the last map is given by the left multiplication) and the bundles (associated to a spin structure \(\widetilde{Q}:=\widetilde{Q}_M\times _M\widetilde{Q}_E\) of TM and E)
and
We finally suppose that a product structure \(\mathcal {P}\) is given on \(TM\oplus E\) as in Sect. 1.3.
3.1 Statement of the Theorem
Theorem 3
We suppose that M is simply connected. Let \(B: TM \times TM \rightarrow E\) be a symmetric tensor. The following statements are equivalent:
-
(i)
There exist an isometric immersion \(F:M\rightarrow \mathbb {S}_1^m\times \mathbb {R}^n\) and a bundle map \(\Phi : TM \oplus E \rightarrow T\mathbb {S}_1^m \times \mathbb {R}^n\) above F such that \(\Phi (X,0)= dF(X)\) for all \(X\in TM,\) 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 \(\varphi \in \Gamma (U \Sigma )\) solution of
$$\begin{aligned} \nabla _X \varphi = \frac{1}{2} \sqrt{c_1}\ X_1 \cdot \nu _1\cdot \varphi - \frac{1}{2}B(X)\cdot \varphi \end{aligned}$$(46)for all \(X\in TM,\) where \(X= X_1 + X_2\) is the decomposition in the product structure \(\mathcal {P}\) of \(TM\oplus E\), such that the map
$$\begin{aligned} Z\in TM\oplus E\ \mapsto \ \langle \langle Z \cdot \varphi , \varphi \rangle \rangle \in \mathbb {R}^{m+1}\times \mathbb {R}^{n} \end{aligned}$$commutes with the product structures \(\mathcal {P}\) and \(\mathcal {P}'.\)
Moreover, the bundle map \(\Phi \) and the immersion F are explicitly given in terms of the spinor field \(\varphi \) by the formulas
and \(F=(F_1,F_2)\in \mathbb {S}_1^m\times \mathbb {R}^n\) with
Brief indications of the proof: setting
and using (16)–(19) it is not difficult to see that \(\Phi _2\) 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 \(\varphi ,\) 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 \(\mathbb {S}_1^m\times \mathbb {R}^n.\)
3.2 Surfaces in \(\mathbb {S}^2\times \mathbb {R}\)
The aim is to recover the spinorial characterization of an immersion in \(\mathbb {S}^2\times \mathbb {R}\) given in [16]. Let us consider \(\Sigma _0=\widetilde{Q}\times _{\rho }Cl^0(4).\) If \(e_0^o,e_1^o,e_2^o,e_3^o\) is an orthonormal basis of \(\mathbb {R}^4,\) where \(e_0^o\) belongs to the second factor of \(\mathbb {S}^2\times \mathbb {R},\) we set \(\omega :=-e_0^o\cdot e_1^o\cdot e_2^o\cdot e_3^o,\) consider the two ideals \(\mathcal {I}_1:=Cl^0(4)\cdot \frac{1}{2}\left( 1-\omega \right) \) and \(\mathcal {I}_2:=Cl^0(4)\cdot \frac{1}{2}\left( 1+\omega \right) \) of \(Cl^0(4)\) and the splitting \(Cl^0(4)=\mathcal {I}_1\oplus \mathcal {I}_2.\) It induces a decomposition
with \(\Sigma _1=\widetilde{Q}\times _{\rho }\mathcal {I}_1\) and \(\Sigma _2=\widetilde{Q}\times _{\rho }\mathcal {I}_2.\) Let us consider the map
to identify \(\Sigma _2\) with \(\Sigma _1,\) and an identification
such that \((X\cdot \psi )^*=X\cdot \nu _1\cdot (\psi )^*\) for all \(X\in TM\oplus E\) and \(\psi \in \Sigma M\otimes \Sigma E.\) We set \(\psi _1,\psi _2\in \Sigma M\otimes \Sigma E\) such that \(\psi _1^*=\varphi _1\) and \(\psi _2^*=u(\varphi _2).\) Since \(\varphi _1\) and \(\varphi _2\) are both normalized solutions of
where N is a unit normal and \(S:TM\rightarrow TM\) is the corresponding shape operator of M in \(\mathbb {S}^2\times \mathbb {R},\) \(\psi _1\) and \(\psi _2\in \Sigma M\otimes \Sigma E\) are so that
and
with \(|\psi _1|=|\psi _2|=1.\) Now the condition expressing that \(\Phi \) commutes with the product structures gives the following:
Lemma 2.1
For a convenient choice of the unit section \(V\in \Gamma (TM\oplus E)\) generating the distinguished line \(\mathcal {P}_2\) of the product structure \(\mathcal {P}\) of \(TM\oplus E\), we have
Proof
Choosing \(V\in \mathcal {P}_2\) so that \(\Phi (V)=e_0^o,\) we have
that is \([V][\varphi ]=[\varphi ]e_0^o.\) Writing \([\varphi ]=[\varphi _1]+[\varphi _2]\in \mathcal {I}_1\oplus \mathcal {I}_2\) and since the right-multiplication by \(e_0^o\) exchanges the ideals \(\mathcal {I}_1\) and \(\mathcal {I}_2\) (since \(\omega \cdot e_0^o=-e_0^o\cdot \omega \)), we deduce that \([V][\varphi _1]=[\varphi _2]e_0^o\) and \([V][\varphi _2]=[\varphi _1]e_0^o.\) We thus have \([V][\nu _1][\varphi _1]=-[\nu _1][\varphi _2]e_0^o\) that is \(V\cdot \nu _1\cdot \varphi _1=-\nu _1\cdot \varphi _2\cdot e_0^o,\) which readily implies (52).
Equations (50) and (51) and the lemma imply that \(\psi _1\) and \(\psi _2\) satisfy
and
The spinor field \(\psi :=\psi _1-\psi _2\in \Sigma M\otimes \Sigma E\) is a solution of
By (52), we have \(\langle \psi _1,\psi _2\rangle = \langle \psi _1,V\cdot \psi _1\rangle =- \langle V\cdot \psi _1,\psi _1\rangle =- \langle \psi _2,\psi _1\rangle =0,\) that is \(\psi _1\) and \(\psi _2\) are orthogonal in \(\Sigma M\otimes \Sigma E,\) which implies that \(|\psi |=\sqrt{2}.\) Finally, since \(E=\mathbb {R}N\) there is an identification
such that \((X\cdot \psi )^*=X\cdot N\cdot (\psi )^*.\) Using that \(X_1=X-\langle X,V\rangle V\) and \(V=T+fN,\) we readily get from (53) that
This is the spinorial characterization of an immersion in \(\mathbb {S}^2\times \mathbb {R}\) obtained in [16].
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 \(\mathbb {S}^3\times \mathbb {R},\) or of surfaces in \(\mathbb {S}^2\times \mathbb {R}^2,\) obtained in [13, 17]. We rather focus below on the new case of hypersurfaces in \(\mathbb {S}^2\times \mathbb {R}^2\).
3.3 Hypersurfaces in \(\mathbb {S}^2\times \mathbb {R}^2\)
Let us assume that M is a 3-dimensional manifold. The aim is to obtain the characterization of an immersion of M in \(\mathbb {S}^2\times \mathbb {R}^2\) in terms of usual spinor fields. Suppose that \(\varphi \in \Gamma (U\Sigma )\) represents the immersion of M in \(\mathbb {S}^2\times \mathbb {R}^2,\) as in Theorem 3 (with \(m=2\), \(n=2\)). Let us set \(\Sigma _0=\widetilde{Q}\times _{\rho }Cl^0(5).\) If \(e_0^o,e_1^o,e_2^o,e_3^o,e_4^o\) is an orthonormal basis of \(\mathbb {R}^5,\) where \(e_0^o,e_1^o\) is a basis of the second factor of \(\mathbb {S}^2\times \mathbb {R}^2\), we set \(\omega :=-e_0^o\cdot e_1^o\cdot e_2^o\cdot e_3^o,\) consider the two ideals \(\mathcal {I}_1:=Cl^0(5)\cdot \frac{1}{2}\left( 1-\omega \right) \) and \(\mathcal {I}_2:=Cl^0(5)\cdot \frac{1}{2}\left( 1+\omega \right) \) of \(Cl^0(5)\) and the splitting \(Cl^0(5)=\mathcal {I}_1\oplus \mathcal {I}_2.\) It induces a decomposition
with \(\Sigma _1=\widetilde{Q}\times _{\rho }\mathcal {I}_1\) and \(\Sigma _2=\widetilde{Q}\times _{\rho }\mathcal {I}_2.\) As in the previous section we consider the map
to identify \(\Sigma _2\) with \(\Sigma _1,\) and for
an identification
such that \((X\cdot \psi )^*=X\cdot \nu _1\cdot \psi ^*\) for all \(X\in TM\oplus E\) and \(\psi \in \Sigma _1'.\) Let us set \(\psi _1,\psi _2\in \Gamma (\Sigma '_1)\) such that
They satisfy
and
We traduce in the following lemma the condition expressing that \(\Phi \) commutes with the product structures: it shows that \(\psi _2\) (and thus \(\varphi \) and therefore the immersion) is essentially determined by \(\psi _1:\)
Lemma 2.2
For \(V_1,V_2\in \Gamma (TM\oplus E)\) such that \(\Phi (V_1)=e_0^o\) and \(\Phi (V_2)=e_1^o\) we have \(V_1\cdot \psi _1=\psi _2\) and \(V_2\cdot \psi _1=-\psi _2\cdot e_0^o\cdot e_1^o,\) and therefore \(V_1\cdot V_2\cdot \psi _1=\psi _1\cdot e_0^o\cdot e_1^o.\)
Proof
The condition expressing that \(\Phi \) commutes with the product structures reads \(\Phi (\mathcal {P}_2)=\{0\}\times \mathbb {R}^{2}\), 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
where N and S respectively denote a unit normal vector and the corresponding shape operator of M in \(\mathbb {S}^2\times \mathbb {R}^2.\) Let us write \(V_1=T_1+f_1N,\) \(V_2=T_2+f_2N\) and \(X_1=X-\langle X,T_1\rangle V_1-\langle X,T_2\rangle V_2.\) Under the Clifford action of the volume element \(-e_0\cdot e_1\cdot e_2\cdot e_3\in Cl(TM\oplus E)\) the bundle \(\Sigma '_1\) splits into \(\Sigma '_1=\Sigma _1^+\oplus \Sigma _1^-\). There is a \(\mathbb {C}\)-linear isomorphism
so that \((X\cdot \Psi )^*=X\cdot N\cdot \Psi ^*\) for all \(X\in TM\) and \(\Psi \in \Sigma M,\) where the complex structure on \(\Sigma _1^+\) is given by the right-action of \(e_0^o\cdot e_1^o.\) We write \(\psi _1=\psi _1^++\psi _1^-\) in \(\Sigma '_1=\Sigma _1^+\oplus \Sigma _1^-\) and consider \(\Psi _1,\Psi _2\in \Gamma (\Sigma M)\) such that \(\Psi _1^*=\psi _1^+\) and \(\Psi _2^*=N\cdot \psi _1^-.\) From (58) we have
and
together with
Moreover,
and
Conversely, the existence of two spinor fields \(\Psi _1,\Psi _2\in \Gamma (\Sigma M)\) solutions of (59)–(63) implies the existence of an isometric immersion of M into \(\mathbb {S}^2\times \mathbb {R}^2:\) we may indeed construct \(\varphi \in \Gamma (U\Sigma )\) solution of (46) from \(\Psi _1\) and \(\Psi _2\), just doing step by step the converse constructions; it is such that the map \(\Phi :X\mapsto \langle \langle X\cdot \varphi ,\varphi \rangle \rangle \) commutes with the product structures.
Remark 4
Two non-trivial spinor fields \(\Psi _1,\Psi _2\in \Gamma (\Sigma M)\) solutions of (59)–(60) are in fact such that \(|\Psi _1|^2+|\Psi _2|^2\) is a constant, and may thus be supposed so that (61) holds.
4 Isometric Immersions in \(\mathbb {H}_1^m\times \mathbb {H}_2^n\) and \(\mathbb {H}_1^m\times \mathbb {R}^n\)
We state here the analogous results for immersions in \(\mathbb {H}_1^m\times \mathbb {H}_2^n\) and \(\mathbb {H}_1^m\times \mathbb {R}^n,\) where \(\mathbb {H}_1^m\) and \(\mathbb {H}_2^n\) are spaces of constant curvature \(c_1,c_2<0.\) We do not include the proofs since they only differ by obvious sign adjustments from the proofs of the previous sections. Here M still denotes a p-dimensional riemannian manifold and \(E\rightarrow M\) 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 \(\mathcal {P}\) is given on \(TM\oplus E\) as in Sect. 1.3. We denote by \(\mathbb {R}^{r,s}\) the space \(\mathbb {R}^{r+s}\) with the metric with signature
by Cl(r, s) its Clifford algebra and by Spin(r, s) its spin group. For immersions in \(\mathbb {H}_1^m\times \mathbb {H}_2^n\) we consider the trivial bundle \(\mathcal {E}_2:=M\times \mathbb {R}^{2,0}\rightarrow M,\) with the natural negative metric and the trivial connection, and two orthonormal and parallel sections \(\nu _1,\nu _2\) of that bundle. We also consider the representation associated to the splitting \(\mathbb {R}^{2,m+n}=\mathbb {R}^{2,0}\oplus \mathbb {R}^{0,p}\oplus \mathbb {R}^{0,q} \)
and the bundles (associated to a spin structure \(\widetilde{Q}:=\widetilde{Q}_M\times _M\widetilde{Q}_E\) of TM and E)
and
Theorem 4
Let \(B: TM \times TM \rightarrow E\) be a symmetric tensor. The following statements are equivalent:
-
(i)
There exist an isometric immersion \(F:M\rightarrow \mathbb {H}_1^m\times \mathbb {H}_2^n\) and a bundle map \(\Phi : TM \oplus E \rightarrow T(\mathbb {H}_1^m \times \mathbb {H}_2^n)\) above F such that \(\Phi (X,0)= dF(X)\) for all \(X\in TM,\) 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 \(\varphi \in \Gamma (U \Sigma )\) solution of
$$\begin{aligned} \nabla _X \varphi = -\frac{1}{2} (\sqrt{|c_1|}\ X_1 \cdot \nu _1 + \sqrt{|c_2|}\ X_2 \cdot \nu _2)\cdot \varphi - \frac{1}{2}B(X)\cdot \varphi \end{aligned}$$(64)for all \(X\in TM,\) where \(X= X_1 + X_2\) is the decomposition in the product structure \(\mathcal {P}\) of \(TM\oplus E\), such that the map
$$\begin{aligned} Z\in TM\oplus E\ \mapsto \ \langle \langle Z \cdot \varphi , \varphi \rangle \rangle \in \mathbb {R}^{1,m}\times \mathbb {R}^{1,n} \end{aligned}$$commutes with the product structures \(\mathcal {P}\) and \(\mathcal {P}'.\)
Moreover, the bundle map \(\Phi \) and the immersion F are explicitly given in terms of the spinor field \(\varphi \) by the formulas
and
For immersions in \(\mathbb {H}_1^m\times \mathbb {R}^n\) we consider \(\mathcal {E}_1:=M\times \mathbb {R}^{1,0}\rightarrow M,\) a parallel section \(\nu _1\) of \(\mathcal {E}_1\) such that \(\langle \nu _1,\nu _1\rangle =-1\) and the bundles
and
Theorem 5
Let \(B: TM \times TM \rightarrow E\) be a symmetric tensor. The following statements are equivalent:
-
(i)
There exist an isometric immersion \(F:M\rightarrow \mathbb {H}_1^m\times \mathbb {R}^n\) and a bundle map \(\Phi : TM \oplus E \rightarrow T\mathbb {H}_1^m \times \mathbb {R}^n\) above F such that \(\Phi (X,0)= dF(X)\) for all \(X\in TM,\) 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 \(\varphi \in \Gamma (U \Sigma )\) solution of
$$\begin{aligned} \nabla _X \varphi = -\frac{1}{2}\sqrt{|c_1|}\ X_1 \cdot \nu _1\cdot \varphi - \frac{1}{2}B(X)\cdot \varphi \end{aligned}$$(66)for all \(X\in TM,\) where \(X= X_1 + X_2\) is the decomposition in the product structure \(\mathcal {P}\) of \(TM\oplus E\), such that the map
$$\begin{aligned} Z\in TM\oplus E\ \mapsto \ \langle \langle Z \cdot \varphi , \varphi \rangle \rangle \in \mathbb {R}^{1,m}\times \mathbb {R}^{n} \end{aligned}$$commutes with the product structures \(\mathcal {P}\) and \(\mathcal {P}'.\)
Moreover, the bundle map \(\Phi \) and the immersion F are explicitly given in terms of the spinor field \(\varphi \) by the formulas
and \(F=(F_1,F_2)\in \mathbb {H}_1^m\times \mathbb {R}^n\) with
As in the positive curvature case, it is possible to deduce a spinorial proof of the fundamental theorem of immersions theory in \(\mathbb {H}_1^m\times \mathbb {H}_2^n\) or \(\mathbb {H}_1^m\times \mathbb {R}^n\).
5 CMC Surfaces with \(H=1/2\) in \(\mathbb {H}^2\times \mathbb {R}\)
We consider the immersion of a surface with \(H=1/2\) in \(\mathbb {H}^2\times \mathbb {R}\subset \mathbb {R}^{1,2}\times \mathbb {R}\) represented by a spinor field \(\varphi \) 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 \(\mathbb {H}^2\times \mathbb {R},\) it is of the form \((N',\nu )\) in \(\mathbb {R}^{1,2}\times \mathbb {R},\) and by \(\nu _1\) the unit vector normal to \(\mathbb {H}^2\times \mathbb {R}\) so that the immersion reads \(F=(\nu _1,h)\in \mathbb {H}^2\times \mathbb {R}.\) The function \(\nu \) is the angle function of the immersion, and we assume that it is always positive (the surface has regular vertical projection), and the function \(h:M\rightarrow \mathbb {R}\) is the height function of the immersion. We fix a conformal parameter \(z=x+iy\) of the surface, in which the metric reads \(\mu ^2(dx^2+dy^2).\) The matrix of the shape operator in the basis \(\partial _x/\mu ,\partial _y/\mu \) reads
and we set the following two important quantities
Following [7] \((Q_0,\tau _0)\) 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 \(\mathbb {C}\)), 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 \(\mathbb {H}^2\times \mathbb {R}\) and in \(\mathbb {R}^{1,2}\).
5.1 The Clifford Algebra and the Spin Group of \(\mathbb {R}^{1,3}\)
Let us consider the algebra of complex quaternions \(\mathbb {H}^{\mathbb {C}}:=\mathbb {H}\otimes \mathbb {C}\). An element a of \(\mathbb {H}^{\mathbb {C}}\) is of the form
its complex norm is
and its complex conjugate is the complex quaternion
where \(\overline{a_i}\) denotes the usual complex conjugate of \(a_i.\) Let us associate to
the complex quaternion
where \(JI=-IJ=-K.\) Using the Clifford map
we easily obtain that
and
i.e. the identification
For the sake of simplicity, we will frequently use below the natural identifications of \(Cl^0(1,3)\) and \(Cl^1(1,3)\) with \(\mathbb {H}^{\mathbb {C}}\). We will moreover use the models
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 \(\nu \) is the angle function.
5.2 The Killing Type Equation in Adapted Coordinates
In a fixed spinorial frame \(\widetilde{s}\) above the orthonormal frame \(s=(\partial _x/\mu ,\partial _y/\mu ,N,\)\(\nu _1),\) the spinor field is represented by \([\varphi ]=g\in S^3_{\mathbb {C}},\) and we consider the components
so that \(g=g_1+g_2.\) Let us note that
and
It will be convenient to consider the following norm on \(\frac{1}{2}\left( 1+iI\right) \mathbb {H}^{\mathbb {C}}\): writing an element \(g_1'\) belonging to \(\frac{1}{2}\left( 1+iI\right) \mathbb {H}^{\mathbb {C}}\) in the form
for some (unique) \(a,b\in \mathbb {C}\), we define its Hermitian norm \(|g_1'|^2:=|a|^2-|b|^2.\)
Proposition 4.1
The component \(g_1':=\sqrt{\mu }g_1\) satisfies
Moreover the compatibility conditions of the equation read
and \(g_1'\) is such that \(|g_1'|^2=\sqrt{\tau _0}.\)
The equation of \(g'_1\) only depends on the Weierstrass data \((Q_0,\tau _0).\) In the statement and in the rest of the section we use the following notation: \(z=x+Iy,\) \(dz=dx+Idy,\) \(\partial _z=1/2(\partial _x-I\partial _y),\) \(\overline{z}=x-Iy,\) \(d\overline{z}=dx-Idy\) and \(\partial _{\overline{z}}=1/2(\partial _x+I\partial _y),\) i.e. the complex parameter z is with respect to the complex structure I.
Remark 5
The Abresch–Rosenberg differential is the quadratic differential \(-Q_0dz^2\). 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 \(\widetilde{s}\) the product structure \(T+\nu N\) reads
and \(h_z,\) \(\mu \) and \(\nu \) satisfy the relations
and
Moreover, the two components \(g_1\) and \(g_2\) of the spinor field are linked by
Proof of Lemma 4.2 In \(\widetilde{s},\) we represent the vectors \(\partial _x/\mu ,\) \(\partial _y/\mu \) and N of the basis s by, respectively, J, JI and \(I\in \mathbb {H}^{\mathbb {C}}.\) Since h is the second component of the immersion we have \(dh(X)=\langle X,T\rangle \) and \(T=1/\mu ^2(\partial _xh\ \partial _x+\partial _yh\ \partial _y)\), which gives
and (72). Writing that \(T+\nu N\) is parallel in \(\mathbb {H}^2\times \mathbb {R}\) we obtain the two equations \(\nabla T-\nu S=0\) and \(\langle S,T\rangle +d\nu =0.\) The first equation yields (73): the first term gives \([\nabla T]=d[T]-aI[T]\) with \([T]=\frac{2}{\mu } h_z J\) and where a is the Levi-Civita connection form
and the second term gives
with \(\sqrt{\tau _0}=\mu \nu .\) The second equation yields (74) by a computation of \(\langle S,T\rangle =-\frac{1}{2}([S] [T]+[T] [S])\) in \(\mathbb {H}^{\mathbb {C}}\) using (76) and (78). Finally, since the spinor field preserves the product structure we have \(\langle \langle (T+\nu N)\cdot \varphi ,\varphi \rangle \rangle =I,\) which reads \(([T]+\nu [N])\widehat{g}=gI\) with \(g=g_1+g_2,\) \([T]=2/\mu \ h_zJ\) and \([N]=I;\) the component in \(1/2(1+iI)\mathbb {H}^{\mathbb {C}}\) of that expression yields \(2h_z/\mu Jg_1+\nu I g_2=\widehat{g_1}I\) and (75).
Proof of Proposition 4.1 Since \(X_1=X-X_2=X-\langle X,T\rangle (T+\nu N)\) with \([X]=\mu d\overline{z}J,\) \(\langle X,T\rangle =dh(X)\) and \([T]+\nu [N]\) given by (72), we have
and since
we obtain
Using that \([\nabla \varphi ]=dg-\frac{1}{2}aIg\), (78) and (80), the Killing type equation (66) with \([\nu _1]=i1\) and \([B(X)]=[S(X)]\cdot [N]\) reads
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
and \(g_1'=\sqrt{\mu }g_1\) satisfies (70). A computation using (70) twice then shows that
which proves (71). For the last claim, \(g=a_01+a_1I+a_2J+a_3K\) with \(a_j\in \mathbb {C}\) is such that \(gI\widehat{\overline{g}}=[T]+\nu [N]\) belongs to \(\mathbb {R}I\oplus \mathbb {R}J\oplus \mathbb {R}K\) (recall the last step of the proof of Lemma 4.2), i.e. the component of \(gI\widehat{\overline{g}}\) on i1 is zero, which yields
moreover, \(\nu \) is defined as the component along the direction I of the vector \(\langle \langle N\cdot \varphi ,\varphi \rangle \rangle =\overline{g}I\widehat{g}\) normal to the immersion, which yields by a direct computation
Since \(g_1'=\frac{1}{2}(1+iI)(\sqrt{\mu }(a_0-ia_1)+\sqrt{\mu }(a_2-ia_3)J),\) we get using (81)
5.3 The Product Structure and the Second Component of the Spinor Field
It appears that the product structure (\(\nu \) and \(h_z/\mu \) in (72)) may be determined independently of \(g_1'\) and \(g_2'\). This relies on the following key lemma of [7]:
Lemma 4.3
[7, Lemma 10] The function \(h_z\) satisfies the system
If we fix \(z_0\) and \(\theta _0\in \mathbb {C},\) the system admits a unique globally defined solution \(h_z\) satisfying the initial condition \(h_z(z_0)=\theta _0.\)
The system is a consequence of the compatibility equations (73) and (74). Since the equations only depend on the Weierstrass data \((Q_0,\tau _0),\) \(h_z\) only depends on these data and on the choice of the initial condition \(h_z(z_0)\in \mathbb {C}\). Moreover, since
(by (76) and since \(|T|^2=1-\nu ^2\) with \(\tau _0=\mu ^2\nu ^2\)) \(\mu \) and \(\nu \) are also determined by \((Q_0,\tau _0)\) and \(h_z(z_0).\) We finally observe that the other component \(g_2':=\sqrt{\mu }g_2\) of the spinor field is given by
(Eq. (75)) and is thus determined by \(g_1'\) if \(h_z\) is known: so \(g_1'\) determines a family of spinor fields parametrized by \(\mathbb {C};\) the parameter corresponds to the choice of an initial condition for the determination of the product structure.
5.4 The Hyperbolic Gauss Map
If N is normal to M and tangent to \(\mathbb {H}^2\times \mathbb {R},\) \(\nu _1\) is normal to \(\mathbb {H}^2\times \mathbb {R}\) in \(\mathbb {R}^{1,3}\) and \(\nu \) is the angle function of M as above, it is defined in [7] as the map
It belongs to the light-cone \(\{X\in \mathbb {R}^{1,3}:\ |X|^2=0\}\), and since \(\nu =\langle N,I\rangle \) it is of the form \(G=G'+I\) where \(G':\mathbb {C}\rightarrow \mathbb {H}^2\) takes values in the model (69) of \(\mathbb {H}^2\). We will frequently identify G and \(G'\) below. In terms of the spinor field representing the immersion, since \([N]=I,\) \([\nu _1]=i1\) and \([\varphi ]=g\) in \(\widetilde{s}\) it is written
and we see that it only depends on the component \(g_1'\) of the spinor field. A direct computation using (70) shows that
where \((u_1,u_2)\) is the positively oriented orthonormal basis of \(T_G\mathbb {H}^2\)
(setting \(g_1'=1/2(1+iI)(a+bJ),\) direct computations show that this is indeed a positively oriented orthonormal basis of \(T_G\mathbb {H}^2\)). Since \(H(u_1,u_1)=H(u_2,u_2)=1\) and \(H(u_1,u_2)=0,\) we also readily get that
As observed in [7], since \(Q_0\) is a holomorphic function, \(G:M\rightarrow \mathbb {H}^2\) is harmonic, and following that paper we will say that \(Q_0:\mathbb {C}\rightarrow \mathbb {C}\) and \(\tau _0:\mathbb {C}\rightarrow (0,+\infty )\) such that (85) and (71) hold form the Weierstrass data of the map \(G:\mathbb {C}\rightarrow \mathbb {H}^2.\) At a regular point of G (i.e. where \(Q_0\ne 0\)), computations show that conversely (85) implies the existence of a unique positively oriented orthonormal basis \((u_1,u_2)\) of \(T_G\mathbb {H}^2\) such that (83) holds. We will assume in the rest of the paper that the set of singular points of G has empty interior.
5.5 Interpretation in Terms of a Principal Bundle
If \(g_1'=\frac{1}{2}\left( 1+iI\right) (a+bJ)\) and \(g_1''=\frac{1}{2}\left( 1+iI\right) (a'+b'J)\) belong to \(\frac{1}{2}\left( 1+iI\right) \mathbb {H}^{\mathbb {C}},\) we consider the hermitian product \(\langle g_1',g_1''\rangle =a\overline{a'}-b\overline{b'}\) and the norm \(|g_1'|^2=|a|^2-|b|^2.\) We consider the set
and the map
This is the projection of a principal bundle, with group of structure \(\mathbb {C}^*\) acting by multiplication. This bundle is moreover equipped with a natural invariant connection form \(\omega _0\) given by
for all \(g_1'\in \mathcal {V}\) and v tangent to \(\mathcal {V}\) at \(g_1'\): the horizontal distribution at \(g_1'\) is the complex line orthogonal to the line \(\mathbb {C}.g_1'\) (the fiber of \(\pi \)) with respect to the hermitian product \(\langle .,.\rangle \) introduced above. We consider the bundle induced from the bundle \(\pi :\mathcal {V}\rightarrow \mathbb {H}^2\) by the Gauss map \(G:\mathbb {C}\rightarrow \mathbb {H}^2\)
and the hypersurface
with the projection \(p_1:\mathcal {H}\rightarrow \mathbb {C},\) \((z,g_1')\mapsto z;\) this is a \(S^1\) principal bundle. If \(p_2:\mathcal {H}\rightarrow \mathcal {V},\) \((z,g_1')\mapsto g_1'\) is the other projection we consider the 1-form
It is a connection form on the \(S^1\) principal bundle \(\mathcal {H}\rightarrow \mathbb {C}:\)
-
it takes values in \(\underline{S^1}=i\mathbb {R}:\) \((U,V)\in T_{(z,g_1')}\mathcal {H}\) satisfies \(\Re e\langle V,g_1'\rangle =\frac{1}{2}d(\sqrt{\tau _0})(U)\) and
$$\begin{aligned} \omega _{(z,g_1')}(U,V)= & {} -(\log \sqrt{\tau _0})_zU+\frac{1}{\sqrt{\tau _0}}\langle V,g_1'\rangle \\= & {} -\frac{1}{2}\left( (\log \sqrt{\tau _0})_zU-(\log \sqrt{\tau _0})_{\overline{z}}\overline{U}\right) +\frac{i}{\sqrt{\tau _0}}\Im m\langle V,g_1'\rangle \ \in i\mathbb {R}; \end{aligned}$$ -
it is \(S^1\)-invariant: \(S^1\) only acts on the component \(g_1'\) and \(\omega _0\) is \(S^1\)-invariant;
-
it is normalized on vertical vectors:
$$\begin{aligned} \omega \left( \frac{d}{d\theta }_{|\theta =0}(z,e^{i\theta }g_1')\right) ={\omega _0}_{g_1'}(ig_1')=i. \end{aligned}$$
Proposition 4.4
The component \(g_1':\mathbb {C}\rightarrow \mathcal {V}\) of the spinor field which represents the immersion naturally identifies to a section of \(\mathcal {H}\rightarrow \mathbb {C},\) which is parallel with respect to the connection \(\omega .\)
Proof
\(g_1'\) defines a section \(\sigma :z\mapsto (z,g_1'(z))\) of \(\mathcal {H}\rightarrow \mathbb {C}\) since \(\pi (g_1')=G\) and \(|g_1'|^2=\sqrt{\tau _0}.\) It satisfies
since \(p_1\circ \sigma =id,\) \(p_2\circ \sigma =g_1'\) and by (70); indeed, the right-hand term of (70) is horizontal: if \(g_1'=\frac{1}{2}(1+iI)(a+bJ),\) then \(J\widehat{g_1'}I=\frac{1}{2}(1+iI)(\overline{b}+\overline{a}J)\) is orthogonal to \(g_1'\) with respect to the hermitian product \(\langle .,.\rangle .\)
Remark 6
The connection \(\omega \) is flat since it admits a parallel section. In fact a computation shows that \(\omega \) defined by (86) is flat if and only if \((Q_0,\tau _0)\) satisfy the conditions (71).
Corollary 1
\(g_1'\) is determined by the Gauss map G and its Weierstrass data \((Q_0,\tau _0)\) up to a sign.
Proof
A parallel section of \(\mathcal {H}\rightarrow \mathbb {C}\) is unique up to the multiplication by a complex number \(e^{i\theta }\in S^1,\) and since \(g_1'\) is a solution of (70) the section \(g_1'':=e^{i\theta } g_1'\) satisfies
it is a solution of (70) if and only if \(e^{i\theta }=\pm 1.\)
5.6 Surfaces with Prescribed Hyperbolic Gauss Map
We assume here that \(G:\mathbb {C}\rightarrow \mathbb {H}^2\) is a given map with Weierstrass data \((Q_0,\tau _0)\) 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 \(\mathbb {H}^2\times \mathbb {R}\) with hyperbolic Gauss map G and Weierstrass data \((Q_0,\tau _0).\) The family is parameterized by \(\mathbb {C}\times \mathbb {R}.\)
Proof
Since the connection \(\omega \) is flat (Remark 6), the principal bundle \(\mathcal {H}\rightarrow \mathbb {C}\) admits a globally defined parallel section that we may consider as a map \(\sigma :\mathbb {C}\rightarrow \mathcal {V}\) such that \(\pi \circ \sigma =G.\) At a regular point of G, since \(d\sigma -(\log \sqrt{\tau _0})_zdz\sigma \) is horizontal and projects onto \(d\pi (d\sigma )=d(\pi \circ \sigma )=dG\) it satisfies
for some \(\theta \in \mathbb {R}\) (indeed, the term \(1/\sqrt{\tau _0}\left( Q_0dz+\tau _0/4d\overline{z}\right) J\widehat{\sigma }I\) is horizontal at \(\sigma \) and a computation shows that its projection by \(d\pi \) is of the form (83), with an orthonormal basis \((u_1',u_2')\) of \(T_G\mathbb {H}^2\) which is perhaps different to the basis \((u_1,u_2)\); taking \(\theta \) 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 \(\theta \) is a constant, and in view of (87) the section \(g_1':=e^{-i\theta }\sigma \) 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 \(\mathbb {C}\)), the solution \(g_2'\) given by (82) and set \(g:=1/\sqrt{\mu }(g_1'+g_2'):\) it belongs to \(S^3_{\mathbb {C}}\) (since \(H(g,g)=1\) by a computation), and we consider the spinor field \(\varphi \) whose component is g in \(\widetilde{s},\) and the corresponding immersions into \(\mathbb {H}^2\times \mathbb {R}\); they depend on \(\mathbb {C}\times \mathbb {R}\) 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 \(F=(i\overline{g}\widehat{g},h)\) (formula (67)) lead to the expression of the immersion in terms of \(G_z,\) \(Q_0,\) \(\tau _0\) and h given in [7, Theorem 11].
5.7 Link with \(H=1/2\) Surfaces in \(\mathbb {R}^{1,2}\)
We first describe with spinors the immersions of \(H=1/2\) surfaces in \(\mathbb {R}^{1,2}\) 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 \(\mathbb {C}\)) of spinor fields representing \(H=1/2\) surfaces in \(\mathbb {H}^2\times \mathbb {R}\): this gives a simple interpretation of the natural correspondence between \(H=1/2\) surfaces in \(\mathbb {R}^{1,2}\) and in \(\mathbb {H}^2\times \mathbb {R}\). Since the proofs are very similar to proofs of the preceding sections, we will omit many details. We consider the model
which is the subgroup of \(Spin(1,3)=S^3_{\mathbb {C}}\) fixing \(I\in \mathbb {R}^{1,3}\) (under the double covering \(Spin(1,3)\rightarrow SO(1,3)\)) and thus also leaving
globally invariant. By [4], if \(\widetilde{Q}\) is a spin structure of M and \(\rho :Spin(2)\rightarrow Spin(1,2)\) a natural representation, a spacelike immersion of a surface in \(\mathbb {R}^{1,2}\) is represented by a spinor field \(\psi \in \widetilde{Q}\times _{\rho }Spin(1,2)\) solution of the Killing type equation
for all \(X\in TM,\) where \(\nu _1\) is the upward vector normal to M in \(\mathbb {R}^{1,2}\) so that \(|\nu _1|^2=-1\) and \(S=\nabla \nu _1:TM\rightarrow TM\) is the shape operator. Explicitly, the immersion is a primitive of the 1-form \(\xi (X)=\langle \langle X\cdot \psi ,\psi \rangle \rangle .\) Fixing a conformal parameter \(z=x+iy\) of the surface in which the metric reads \(\mu ^2(dx^2+dy^2)\) and choosing a spinorial frame \(\widetilde{s}\) above \((\partial _x/\mu ,\partial _y/\mu ),\) the spinor field reads as a map \(v:\mathbb {C}\rightarrow Spin(1,2)\). Assuming that the matrix of S reads as (68), we set here
Proposition 4.5
The function \(v':=\sqrt{\mu }v\) satisfies
The compatibility conditions of the equation are (71). Moreover \(|v'|^2:=H(v',v')=\sqrt{\tau _0}.\)
Proof
As in the proof of Proposition 4.1, in the spinorial frame \(\widetilde{s}\) the Killing type equation (89) reads
which, for \(H=1/2\) and the definitions of \(Q_0\) and \(\tau _0\), gives
Using (77), we obtain (90). Finally, \(|v'|^2=H(v',v')=\mu H(v,v)=\mu =\sqrt{\tau _0}.\)
We now consider the usual Gauss map \(G:\mathbb {C}\rightarrow \mathbb {H}^2\) of the surface, still with the model (69). Since \(G=\langle \langle \nu _1.\psi ,\psi \rangle \rangle \) with
it reads
The projection
is a principal bundle of group of structure \(H=\{\cos \theta +\sin \theta I,\theta \in \mathbb {R}\}\) (acting on the left) that we equip with a natural connection: we consider the decomposition of the Lie algebra \(\underline{Spin(1,2)}=\mathfrak {h}\oplus \mathfrak {m}\) with \(\mathfrak {h}=\mathbb {R}I\) and \(\mathfrak {m}=iJ(\mathbb {R}1\oplus \mathbb {R}I),\) the projection \(p_1\) onto the first factor \(\mathfrak {h}\) and the connection form
where \(\omega _{MC}=d\sigma \ \sigma ^{-1}\in \Omega ^1(Spin(1,2),\underline{Spin(1,2)})\) is the Maurer–Cartan form. The bundle \(\pi ':Spin(1,2)\rightarrow \mathbb {H}^2\) and the Gauss map \(G:\mathbb {C}\rightarrow \mathbb {H}^2\) induce a bundle
with the projection \(p_1:G^*Spin(1,2)\rightarrow \mathbb {C},\) \((z,v)\mapsto z.\) If \(p_2:G^*Spin(1,2)\rightarrow Spin(1,2),\) \((z,v)\mapsto v\) is the second projection, we consider the connection form
The following results are similar to results obtained in the previous section:
Proposition 4.6
The component \(v:\mathbb {C}\rightarrow Spin(1,2)\) of the spinor field representing the immersion is naturally a section of \(G^*Spin(1,2)\rightarrow \mathbb {C}.\) It is horizontal for the connection \(\omega .\)
Proof
This is a translation of (91), similar to Proposition 4.4.
Corollary 3
v is determined by the Gauss map G and its Weierstrass data \((Q_0,\tau _0)\) up to a sign.
Proof
It is analogous to the proof of Corollary 1.
We now suppose that a map \(G:\mathbb {C}\rightarrow \mathbb {H}^2\) is given with Weierstrass data \((Q_0,\tau _0).\) We moreover suppose that the set of singular points of G has empty interior.
Corollary 4
[1] There exists a \(H=1/2\) surface in \(\mathbb {R}^{1,2}\) with Gauss map G and Weierstrass data \((Q_0,\tau _0).\) It is unique up to a translation in \(\mathbb {R}^{1,2}.\)
Proof
As in the proof of Corollary 2, a horizontal section v of \(G^*Spin(1,2)\rightarrow \mathbb {C}\) is a map such that
for some function \(\theta ':\mathbb {C}\rightarrow \mathbb {R}\) which has to be constant, and \(\widetilde{v}=e^{-I\theta '}v\) is a section solution of (91) such that \(\pi '(\widetilde{v})=G.\) Since such a solution is unique up to a sign, the spinor field \(\psi \) is determined up to a sign and the immersion is unique up to a translation (since the immersion in \(\mathbb {R}^{1,2}\) is finally obtained by the integration of the 1-form \(\xi (X)=\langle \langle X\cdot \psi ,\psi \rangle \rangle \)).
Proposition 4.7
The correspondence
transforms a solution \(v'\) of (90) such that \(\pi '(v'/|v'|)=G\) to a solution \(g_1'\) of (70) such that \(\pi (g_1')=G+I,\) and vice-versa.
Proof
Assuming that (91) holds we compute
For the last term, since \(Iv'=\widehat{v'}I\) (v belongs to Spin(1, 2)) we observe that
which shows that \(g_1'\) satisfies (70). We finally note that \(|g_1'|^2=\mu \) and therefore
The first term \(i\overline{v}\widehat{v}\) is the Gauss map G. Since \(I\widehat{v}=vI\) and \(\overline{v}v=1,\) the second term \(\overline{v}I\widehat{v}\) is the constant I. So \(\pi (g_1')=G+I\) and \(g_1'\) lies above the same (hyperbolic) Gauss map G. Conversely, if now \(g_1'\in 1/2(1+iI)\mathbb {H}^C\) is a solution of (70) we consider the unique \(v'=\alpha _01+\alpha _1I+i\alpha _2J+i\alpha _3K\) such that (93) holds, and similar computations show that \(v'\) is a solution of (90) above G, which proves the proposition.
Corollary 5
Immersions of CMC surfaces with \(H=1/2\) in \(\mathbb {R}^{1,2}\) (up to translations) correspond to 2-parameter families of immersions of CMC surfaces with \(H=1/2\) in \(\mathbb {H}^2\times \mathbb {R}\) and regular vertical projection (up to vertical translations).
Proof
In the correspondence (93), \(g_1'\) determines a family of spinor fields parametrized by \(\mathbb {C}\), and therefore a 2-parameter family of immersions in \(\mathbb {H}^2\times \mathbb {R}\) up to a vertical translation, whereas \(v'\) determines a spinor field and therefore an immersion in \(\mathbb {R}^{1,2}\) up to a translation.
Note that the correspondence preserves the (hyperbolic) Gauss maps and their Weierstrass data.
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Acknowledgements
Pierre Bayard wishes to thank Julien Roth and the Laboratoire d’Analyse et de Mathématiques Appliquées at Eiffel University for their hospitality from august 2021 to july 2022; he also wishes to thank the program PASPA-DGAPA of the National Autonomous University of Mexico for support. The authors wish to thank an anonymous referee for many comments which improved the writing of the paper.
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Appendix A: Skew-Symmetric Operators and Clifford Algebra
Appendix A: Skew-Symmetric Operators and Clifford Algebra
We consider \(\mathbb {R}^N\) with its standard scalar product \(\langle .,.\rangle .\) If \(\eta \) and \(\eta '\) belong to the Clifford algebra Cl(N), we set
where the dot \(\cdot \) is the Clifford product. We denote by \((e_1,\ldots ,e_N)\) the canonical basis of \(\mathbb {R}^N.\) The next two lemmas were proved in the Appendix A of [5]. We include the statements here for the convenience of the readers.
Lemma A.1
Let \(u:\mathbb {R}^N\rightarrow \mathbb {R}^N\) be a skew-symmetric operator. Then the bivector
represents u in the sense that, for all \(\xi \in \mathbb {R}^N,\) \([\underline{u},\xi ]=u(\xi ).\) We also have the formula
We now assume that \(\mathbb {R}^N=\mathbb {R}^p\oplus \mathbb {R}^q,\) \(p+q=N.\)
Lemma A.2
Let us consider a linear map \(u:\mathbb {R}^p\rightarrow \mathbb {R}^q\) and its adjoint \(u^*:\mathbb {R}^q\rightarrow \mathbb {R}^p.\) Then the bivector
represents
in the sense that, for all \(\xi =\xi _p+\xi _q\in \mathbb {R}^N,\) \([\underline{u},\xi ]=u(\xi _p)-u^*(\xi _q).\) Moreover we have
We moreover have the following
Lemma A.3
Consider, for \(U,V\in \mathbb {R}^N,\) the skew-symmetric operator
It is represented by
i.e., for all \(W\in \mathbb {R}^N,\)
Proof
Let us write
Using \(\langle U,W\rangle =-\frac{1}{2}\left( U\cdot W+W\cdot U\right) \) and \(\langle V,W\rangle =-\frac{1}{2}\left( V\cdot W+W\cdot V\right) \) we get
This last expression is
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Basilio, A., Bayard, P., Lawn, MA. et al. Spinorial Representation of Submanifolds in a Product of Space Forms. Adv. Appl. Clifford Algebras 33, 55 (2023). https://doi.org/10.1007/s00006-023-01302-x
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DOI: https://doi.org/10.1007/s00006-023-01302-x