A FUNDAMENTAL THEOREM FOR SUBMANIFOLDS OF MULTIPRODUCTS OF REAL SPACE FORMS

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A FUNDAMENTAL THEOREM FOR

SUBMANIFOLDS OF MULTIPRODUCTS OF REAL SPACE FORMS

Marie-Amélie Lawn, Julien Roth

To cite this version:

Marie-Amélie Lawn, Julien Roth. A FUNDAMENTAL THEOREM FOR SUBMANIFOLDS OF

MULTIPRODUCTS OF REAL SPACE FORMS. Advances in Geometry, De Gruyter, 2017, 17 (3),

pp.323-338. �hal-01140769�

MULTIPRODUCTS OF REAL SPACE FORMS

MARIE-AM ´ELIE LAWN AND JULIEN ROTH

Abstract. We prove a Bonnet theorem for isometric immersions of submanifolds into the products of an arbitrary number of simply connected real space forms. Then, we prove the existence of associated families of minimal surfaces in such products. Finally, in the case of S²×S², we give acomplexversion of the main theorem in terms of the two canonical complex structures ofS²×S².

1. Introduction

It is a classical problem of submanifold theory to determine when a Riemannian manifold (Mⁿ, g) can be immersed into a fixed Riemannian manifold ( ¯M^n+p,¯g). The well-known Gauss, Ricci and Codazzi equations relate the intrinsic and extrinsic curvatures, and any submanifold of any Rie- mannian manifold must satisfy them. Conversely, the classical Bonnet theorem [1] states that on a surface, given first and second fundamental forms satisfying the Gauss and Codazzi equations, this surface is locally embeddable into the Euclidean 3-spaceR³. This result can be generalized to higher codimension [10], and the classical Fundamental Theorem of Submanifolds states that, in fact, the Gauss, Codazzi and Ricci equations are necessary and sufficient conditions for a Rie- manniann-dimensional manifold to admit a (local) immersion into a space of constant sectional curvature of dimensionn+d.

If the ambient space is not of constant sectional curvature, proving fundamental theorems is tech- nically difficult and there are few results known. Moreover, the Gauss, Codazzi and Ricci equa- tions are in general not sufficient anymore and other conditions are required in order to produce the immersion. In [3], Daniel gave such a characterization for surfaces in the three-dimensional Thurston geometries with four-dimensional isometry groups, by computing the Christoffel sym- bols explicitly and using the technique of Cartan moving frames. In higher dimensions, he also stated in [2] necessary and sufficient conditions for an n-dimensional Riemannian manifold to be isometrically immersible into the productsSⁿ×RandHⁿ×R, also using the moving frame technique. This allowed him to study the existence of associated families in the case of minimal surfaces. This result was later generalized by the second author [9] in the case where the ambient space is a Lorentzian product. Very recently, Ortega and the first author [7] proved fundamental theorems characterizing immersions of hypersurfaces into (quasi-)Einstein manifolds, specifically Robertson-Walker warped products. These spaces play an important role in standard models of cosmology, arising as solutions of the non-vacuum Einstein equations, and have therefore a great importance in Lorentzian geometry. As an application, conditions were obtained for 3-dimensional hypersurfaces in Robertson-Walker spacetimes to be foliated by surfaces whose mean curvature vector is either lightlike or zero (including maximal surfaces, marginally outer

2010Mathematics Subject Classification. 53A42, 53C20, 53C21.

Key words and phrases. Almost-umbilical hypersurfaces, space forms, constant scalar curvature.

1

trapped surfaces (MOTS), and mixed cases), hence providing an helpful tool for the study of horizons on Robertson-Walker spacetimes with spacelike or timelike causal character, including marginally outer trapped tubes.

Extending the result of [2], Kowalczyk and Lira-Tojeiro-Vit´orio proved independently in [6] and [8] the existence and uniqueness of isometric immersions in a product of two spaces forms of constant sectional curvature. In this paper we generalize their result to immersions into multi- products ˜P =M₁× · · · ×M_m of real space form of arbitrary dimension and arbitrary sectional curvature. The key idea is to use the projections π_i, i∈1, . . . , minto each of the factors of the product. Each projection induces then two operators on the tangent bundle and two operators on the normal bundle of the submanifold satisfying some properties and some compatibility equations which can be deduced from the Gauss and Weingarten formulas of the immersion. We prove that, conversely, these conditions together with the Gauss, Codazzi and Ricci equations are necessary and sufficient conditions to immerse a Riemannian manifold isometrically into such an ambient space.

As an application we then prove the existence of a one-parameter associated family of isometric immersions for minimal surfaces in multiproducts. Finally we consider the special case where the ambient space is S²×S² and give a complex version of our fundamental theorem in terms of the induced complex structures.

The authors want to thank F. Torralbo for helpful discussions.

2. Multiproducts of space forms and their submanifolds

We consider the product space (Pe =M1× · · · ×Mm,ge=g1⊕ · · · ⊕gm) where (Mi, gi) is the simply connected real space form of dimensionniand constant sectional curvatureci. Moreover, without loss of generality, we assume thatci6= 0 fori∈ {1,· · ·, m−1}and thatcmmay possibly be zero. We denote by πi the projection of any tangent vectorX onT Mk. These projections satisfy the following relations



















eg(πiX, Y) =eg(X, πiY) for anyX, Y ∈Γ(T P), πi◦πi=πi,

πi◦πj = 0 ifi6=j,

∇πe i= 0, rank(π_i) =n_i, Pm

i=1π_i= Id_T

Pe. Moreover, the curvature tensor of Pe is given by

R(X, Ye )Z=

m

X

i=1

c_i

hπ_iY, π_iZiπ_iX− hπ_iX, π_iZiπ_iY . (1)

Now, we consider a Riemannian manifold (Mⁿ, g) isometrically immersed intoPe. We denote by N M the normal bundle and by∇^⊥ the normal connection and byB:T M ×T M −→N M the second fundamental form. For any ξ∈N M,Aξ is the Weingarten operator associated toξand defined by eg(A_ξX, Y) =eg(B(X, Y), ξ), withX, Y vectors tangent toM.

For any i∈ {1,· · · , m}, the projectionπ_i induces the existence of the following four operators f_i:T M →T M,h_i:T M →N M, s_i:N M →T M andt_i :N M →N M, such that

π_iX=f_iX+h_iX and π_iξ=s_iξ+t_iξ.

(2)

From the symmetry of the πi, we obtain that, for anyi∈ {1,· · · , m},fi and ti are symmetric and for any X∈Γ(T M) andξ∈Γ(N M)

(3) eg(hiX, ξ) =eg(X, siξ).

In addition, from the fact that

m

X

i=1

πi=Id_T

Pe, we get the following identities (4)

m

X

i=1

f_i=Id_{T M},

m

X

i=1

t_i=Id_E,

m

X

i=1

s_i= 0 and

m

X

i=1

h_i= 0.

Moreover, we have the following relations between these operators coming from the fact that π_i◦π_j=δ_i^jπ_i

fi◦fj+si◦hj =δ_i^jfi, (5)

ti◦tj+hi◦sj =δ_i^jti, (6)

f_i◦s_j+s_i◦t_j=δ_i^js_i, (7)

hi◦fj+ti◦hj=δ^j_ihi, (8)

where δ_i^j is the classical Kronecker symbol, that is, 1 ifi=j and 0 ifi6=j. Moreover, from the fact thatπi is parallel, we deduce easily that for anyX, Y ∈T M andξ∈N M, we have

∇_X(f_iY)−f_i(∇_XY) =A_hiYX+s_i(B(X, Y)), (9)

∇^⊥_X(hiY)−hi(∇XY) =ti(B(X, Y))−B(X, fiY), (10)

∇^⊥_X(t_iξ)−t_i(∇^⊥_Xξ) =−B(s_iξ, X)−h_i(A_ξX), (11)

∇X(siξ)−si(∇^⊥_Xξ) =−fi(AξX) +At_iξX.

(12)

Finally, from the expression for the curvature tensorR, we get the following Gauss, Codazzi ande Ricci equations

(G) R(X, Y)Z =

m

X

i=1

c_i

hf_iY, Zif_iX− hf_iX, Zif_iY

+A_B(Y,Z)X−A_B(X,Z)Y,

(C) (∇XB)(Y, Z)−(∇YB)(X, Z) =

m

X

i=1

ci

hfiY, ZihiX− hfiX, ZihiY

,

(R) R^⊥(X, Y)ξ=

m

X

i=1

ci

hhiY, ξihiX− hhiX, ξihiY

+B(AξY, X)−B(AξX, Y).

3. Main result

Now, conversely, consider (Mⁿ, g) a Riemannian manifold andE ad-dimensional vector bundle over M endowed with a metric g and a compatible connection ∇. Moreover, let B : T M × T M −→E be a symmetric (2,1)-tensor andfi:T M −→T M,hi:T M −→E andti:E−→E be some (1,1)-tensors for i ∈ {1,· · ·, m}. We define si as the dual of hi with respect to the metrics eg:=g⊕gonT M⊕E, that is, for anyX ∈TxM andξ∈Ex,

g_x(h_iX, ξ) =g_x(X, s_iξ).

Finally, for anyξ∈Γ(E), we defineAξ by

hAξX, Yi=hB(X, Y), ξi,

for any X, Y ∈Γ(T M). Following the discussions of Section 2, we introduce now the following natural definition.

Definition 3.1. We say that(M, g, E, g,∇, B, f_i, h_i, t_i)satisfies the compatibility equations for the multiproduct Pe=M1× · · · ×Mm if

i) fi andti are symmetric for any i∈ {1,· · · , m},

ii) for any i∈ {1,· · ·, m}, Equations (4)-(11)are satisfied, that is,

m

X

i=1

fi=IdT M,

m

X

i=1

ti=IdE,

m

X

i=1

si= 0 and

m

X

i=1

hi= 0 f_i◦f_j+s_i◦h_j =δ^j_if_i,

ti◦tj+hi◦sj =δ_i^jti, fi◦sj+si◦tj=δ_i^jsi, hi◦fj+ti◦hj=δ^j_ihi,

∇_X(f_iY)−f_i(∇_XY) =A_hiYX+s_i(B(X, Y)),

∇X(hiY)−hi(∇XY) =ti(B(X, Y))−B(X, fiY),

∇X(tiξ)−ti(∇Xξ) =−B(siξ, X)−hi(AξX), iii) the rank ofπi isni for any i∈ {1,· · ·, m} andPm

i=1ni=n+p,, and

iv) the Gauss, Ricci and Codazzi equations (G), (C) and (R) are satisfied. Namely for any X, Y, Z∈Γ(T M)and any ξ∈Γ(E),

R(X, Y)Z =

m

X

i=1

ci

hfiY, ZifiX− hfiX, ZifiY

+A_B(Y,Z)X−A_B(X,Z)Y,

(∇XB)(Y, Z)−(∇YB)(X, Z) =

m

X

i=1

ci

hfiY, ZihiX− hfiX, ZihiY

,

R(X, Y)ξ=

m

X

i=1

c_i

hhiY, ξih_iX− hhiX, ξih_iY

+B(A_ξY, X)−B(A_ξX, Y), whereR is the curvature associated with the connection∇.

We can now state the main result of the paper.

Theorem 3.2. Let(Mⁿ, g)be a simply connected Riemannian manifold andE ad-dimensional vector bundle over M endowed with a metric g and a compatible connection ∇^⊥. Moreover, let B :T M ×T M −→E be a symmetric (2,1)-tensor and fi :T M −→T M,hi:T M −→E and ti:E−→Ebe some(1,1)-tensors fori∈ {1,· · ·, m}. If(M, g, E, g,∇, B, fi, hi, ti)satisfies the compatibility equations for the multiproduct Pe=M1× · · · ×Mmthen, there exists an isometric immersion ϕ: M −→ Pe such that the normal bundle of M for this immersion is isomorphic toE and such that the second fundamental form, the normal connection and the projections on each factor of TPe_|M are given by B , ∇ and (fi, hi, ti) respectively. Precisely, there exists a vector bundle isometry ϕe:E−→(ϕ(M))^⊥ so that

π_i(ϕ_∗X) =ϕ_∗(f_iX) + Φ(h_iX),

πi(ϕξ) =e ϕ∗(siX) + Φ(tiξ), IIf =ϕe◦B,

∇^⊥ϕe=ϕ∇.e

Moreover, this isometric immersion is unique up to an isometry of P.e

Our approach to prove this theorem is not based on the moving frame technique, but is in the spirit of [6] and uses techniques introduced in [4] and [5].

Proof: We give the proof for the case cm 6= 0, the case cm = 0 can be proved analo- gously with minor changes. First, for any i ∈ {1, . . . , m}, let us denote by Ei a trivial line bundle over M equipped with the Euclidean metric, ifci>0, and minus the Euclidean metric, ifci<0. We consider the vector bundleF over M,

F=T M⊕E ⊕^m

i=1

E_i,

defined by the orthogonal Withney sum of Riemannian vector bundles. We denote by eg the metric over F obtained from g, g and the metrics on each E_i. For any i ∈ {1, . . . , m}, we consider a section ξ_i ofE_i such thateg(ξ_i, ξ_i) = _c¹

i. We introduce now the following connection onF, denoted byD

DXY =∇XY +B(X, Y)−

m

X

i=1

cig(fiX, Y)ξi,

D_Xν=∇^⊥_Xν−A_νX−

m

X

i=1

c_ig(h_iX, ν)ξ_i, DXξi=fiX+hiX,

for any vector fields X, Y tangent toM and any sectionν ofE.

Lemma 3.3. The connectionD is compatible with the metriceg.

Proof: This comes easily from the definition. LetX, Y, Z∈Γ(T M),ν, η∈Γ(E). We have Xeg(Y, Z) = Xg(Y, Z)

= g(∇_XY, Z) +g(Y,∇_XZ)

= eg(DXY, Z) +eg(Y, DXZ),

since the tangential parts of D_XY and D_XZ are ∇_XY and ∇_XZ respectively. Similarly, we have

Xeg(ν, η) = Xg(ν, η)

= g(∇^⊥_Xν, η) +g(ν,∇^⊥_Xη)

= eg(D_Xν, η) +eg(ν, D_Xη),

since the normal parts of DXν andDXη are∇^⊥_Xν and∇^⊥_Xη respectively. Moreover, we have Xeg(ξi, ξj) = 0

= eg(fiX+hiX, ξj) +eg(ξi, fjX+hjX)

= eg(D_Xξ_i, ξ_j) +eg(ξ_i, D_Xξ_j).

Finally for mixed terms, we have Xeg(ξ_i, Y) = 0

= g(fiX, Y)−cig(ξe i, ξi)g(fiX, Y)

= eg(f_iX+h_iX, Y) +ge ξ_i,∇XY +B(X, Y)−

m

X

i=1

c_ig(f_iX, Y)ξ_i

!

= eg(DXξi, ξj) +eg(ξi, DXξj), and

Xeg(ξi, ν) = 0

= g(h_iX, ν)−c_ieg(ξ_i, ξ_i)g(h_iX, ν)

= eg(f_iX+h_iX, ν) +eg ξ_i,∇^⊥_Xν−A_νX−

m

X

i=1

c_ig(h_iX, ν)ξ_i

!

= eg(D_Xξ_i, ν) +eg(ξ_i, D_Xν).

By bilinearity, we get the property for any sections αandβ ofF. Now, we consider the curvature tensor associated with the connection D, denoted by R^D and defined byR^D(X, Y) =DXDY −DYDX−D_[X,Y]. We can prove the following Lemma 3.4. The connectionD is flat, that is,R^D= 0.

Proof: LetX, Y, Z∈Γ(T M) andν ∈Γ(E). We will prove thatR^D(X, Y)Z= 0,R^D(X, Y)ν = 0 and R^D(X, Y)ξi = 0 for any i ∈ {1,· · ·, m}. Then by linearity of the curvature R^D in its third argument, we will get thatR^D= 0. First, we have

D_XD_YZ = ∇_X∇_YZ+B(X,∇_YZ)−

m

X

i=1

c_ig(f_iX,∇_YZ)ξ_i+∇^⊥_XB(Y, Z)−A_B(Y,Z)X

−

m

X

i=1

ci

h

g(B(Y, Z), hiX)ξi+ (fiX+hiX) +g(∇XfiY, Z)ξi+g(fiY,∇XZ)ξi

i .

Therefore, we get

R^D(X, Y)Z = R(X, Y)Z−

m

X

i=1

c_ih

g(f_iY, Z)f_iX−g(f_iX, Z)f_iYi

−AB(Y,Z)X+A_B(X,Z)Y

+(∇XB)(Y, Z)−(∇YB)(X, Z)−

m

X

i=1

ci

h

g(fiY, Z)hiX−g(fiX, Z)hiYi

−

m

X

i=1

ci

h

g (∇Xfi)Y −Ah_iYX+siB(X, Y), Zi

−

m

X

i=1

ci

h

g (∇Yfi)X−Ah_iXY +siB(X, Y), Zi

= 0

by using the Gauss equation (first line), the Codazzi equation (second line) and equation (9) (third and fourth lines). Similarly, we have

D_XD_YZν = ∇^⊥_X∇^⊥_Yν− ∇_XA_νY −B(X, A_νY)−A_∇⊥ XνY +

m

X

i=1

ci

h

g(fiX, AνY)ξi+g(ν, hiY)(fiX+hiX)i

−

m

X

i=1

ci

hg(∇^⊥_Yν, hiX)−g(∇^⊥_Xν, hiY)g(ν,∇^⊥_XhiY)i . And hence,

R^D(X, Y)ν = R^⊥(X, Y)ν−B(X, AνY) +B(Y, AνX)−

m

X

i=1

ci

h

g(hiY, ν)hiX−g(hiX, ν)hiYi

+∇YAνX+A_∇⊥

YνX− ∇XAνY −A_∇⊥ XνY −

m

X

i=1

ci

h

g(hiY, ν)fiX−g(hiX, ν)fiYi

+

m

X

i=1

ci

h

g(fiX, AνY)−g(ν,(∇Yhi)X) +g(tiB(X, Y), ν)i

+

m

X

i=1

c_ih

g(f_iY, A_νX)−g(ν,(∇Xh_i)Y) +g(t_iB(X, Y), ν)i .

The first line in the right hand side vanishes due to Ricci equation. The second line vanishes by Codazzi equation and the third and fourth lines vanish by using equation (10). Finally, for any i∈ {1,· · · , m}, we have

D_XD_Yξ_j = ∇Xf_jY +B(X, f_jY)−

m

X

i=1

c_ig(f_iX, f_iY)ξ_i

+∇^⊥_XhjY −Ah_jYX−

m

X

i=1

cig(hiX, hiY)ξi. Hence,

R^D(X, Y)ξj = (∇Xfj)Y −Ah_jYX−sjB(X, Y)−(∇Yfj)X+Ah_jXY +sjB(X, Y)

∇^⊥_XhjY +B(X, fjY)−tjB(X, Y)− ∇^⊥_YhjX−B(Y, fjX)−tjB(X, Y)

= 0

by equations (9) and (10). Thus, we get that the connectionD is flat.

We define now for any i∈ {1,· · ·, m}the mapπi:F −→F by πiX=fiX+hiX,

πiν =siν+tiν, πiξj=δ^j_iξj,

for any X∈Γ(T M) and any ν∈Γ(E). We have the following properties

Lemma 3.5. For any i∈ {1,· · · , m}, the map πi is symmetric with respect toeg, parallel with respect to D and

(1) π_i◦π_i=π_i,

(2) Pm

i=1πi=IdF.

Proof: The symmetry is clear because of the symmetry offi,ti and the fact thatsiis the dual of hi. The fact thatπi isD-parallel comes from the definition ofD and equations (5) to (12).

Indeed, we have for X, Y ∈Γ(T M),

(DXπi)Y = DX(πiY)−πi(DXY)

= DX(fiY +hiY)−πi ∇XY +B(X, Y)−

m

X

k=1

ckg(fkX, Y)ξk

!

= ∇X(fiY) +B(X, fiY)−

m

X

k=1

ckg(fkX, fiY)ξk

+∇_X(h_iY)−A_hiYX−

m

X

k=1

c_kg(h_kX, h_iY)ξ_k

−fi(∇XY) +hi(∇XY)−siB(X, Y)−tiB(X, Y) +cig(fiX, Y)ξi. By the use of equations (9) and (10), we get

(D_Xπ_i)Y = c_ig(f_iX, Y)ξ_i−

m

X

k=1

c_kh

g(f_kX, f_iY) +g(h_kX, h_iY)ξ_ki

= c_ig(f_iX, Y)ξ_i−

m

X

k=1

c_kh

g(f_i◦f_kX+s_i◦h_kX, Y)i

= 0,

since, by (5), we havefi◦fk+si◦hk =δ_i^kfi. The computations are analogous for a sectionν of Eor for one of the ξi.

The relation πi◦πi =πi is obvious from the definition of πi and relations (5) to (8). Finally, we get immediately that Pm

i=1π_i=Id_F from the definition and assumption (4).

We consider the subsets Fⁱ ofF defined by Fⁱ=n

α∈F |π_jα=δ^j_iαfor anyj∈ {1,· · · , m}o .

Note that, since theπiare symmetric, then the subbundlesFⁱare orthogonal with respect toeg.

We finally need a last lemma

Lemma 3.6. For any i ∈ {1,· · · , m}, there exists orthonormal n_i + 1 parallel sections σ₁ⁱ,· · · , σ_nⁱ

i+1 of Fⁱ.

Proof: Letpbe a point ofM. For any i∈ {1,· · ·, m}, let{vⁱ₁,· · ·, v_nⁱ₁₊₁} be an orthonormal basis ofF_i^p, which is of dimensionn_i+ 1 by the assumption on the rank ofπ_i. Moreover, since ξi clearly belongs toFi, we can choosev¹_i =p

|ci|ξ1(p). Thus, we haveeg(vⁱ₁, v₁ⁱ) = sign(ci) and g(ve _kⁱ, v_kⁱ) = 1 for k ∈ {2,· · · , ni+ 1}. Since the Fⁱ are orthogonal, the set of allv^k_i forms an orthogonal basis ofFp. Now, since the connectionDis flat andM is simply connected, then for anyi∈ {1,· · · , m} there exists a family of parallel sectionsσⁱ₁,· · · , σⁱ_ni+1, such thatσ_kⁱ(p) =v_kⁱ. Moreover, sinceD is compatible with the metriceg, then the sections are orthonormal. Finally, since the maps π are D-parallel, then, for any i ∈ {1,· · ·, m} and any k ∈ {1,· · ·, ni+ 1}, π_i(σ_kⁱ) =σⁱ_k, that isσ_kⁱ is a section ofFⁱ. This concludes the proof of the lemma.

We will construct now the isometric immersion from M into Pe. For this, we consider the following functions. Fori∈ {1,· · · , m}andk∈ {1,· · · , ni+ 1}, letϕⁱ_k be defined by

ϕⁱ_k=eg(σⁱ_k, ξi).

The candidate for the isometric immersion is

ϕ:M −→E1× · · · ×Em,

where Ei is the Euclidean space Rⁿⁱ⁺¹ ifci >0 and the Minkowski spaceLⁿⁱ⁺¹ ifci <0. We will show that the mapϕgoes into Pe⊂E1× · · · ×Em and satisfies all the properties stated in Theorem 3.2.

First, we have

1 ci

=eg(ξ_i.ξ_i) =

ni+1

X

k=1

eg(ξ_i, σ_kⁱ)²eg(σ_kⁱ, σⁱ_k)

= sign(ci)eg(ξi, σ₁ⁱ)²+

n_i+1

X

k=2

eg(ξi, σⁱ_k)²

= sign(ci)(ϕⁱ₁)²+

ni+1

X

k=2

(ϕⁱ_k)².

Thus, we get that (ϕⁱ₁,· · ·, ϕⁱ_n

i+1) ∈ M_i, the n_i-dimensional simply connected space form of curvatureci, and so,ϕ(M) lies inPe.

Now, we will show that ϕis an immersion. For this, letp∈M andv∈T_pM so thatϕ_∗(v) = 0.

From the definition of ϕ, the fact thatϕ_∗(v) = 0 implies eg(σ_kⁱ(p), πiv) =g(ve _kⁱ, πiv) = 0,

for any i∈ {1,· · · , m} and k∈ {1,· · ·, n_i+ 1}. Since, for anyi, {vⁱ_k} is an orthonormal basis of F_pⁱ, we get that π_iv= 0. Moreover, from Lemma 3.5,Pm

i=1π_i=Id_F, thenv= 0. This holds for any pand anyv, so we get thatϕis an immersion.

Moreover, forv, w∈T_pM, we have

hϕ∗(v), ϕ∗(w)i =

m

X

i=1

sign(ci)eg(σ₁ⁱ, πiv)g(σe ₁ⁱ, πiw) +

ni+1

X

k=2

eg(σ_kⁱ, πiv)eg(σ_kⁱ, πiw)

!

=

m

X

i=1

eg(πiv, πiw)

= eg(v, w),

whereh·,·iis the (pseudo)-Euclidean metric onE1×· · ·×Em. Hence,ϕis an isometric immersion from M intoPe.

Now, we define the following bundle isomorphism

Φ :F −→T(E1× · · · ×Em)_|ϕ(M),

by Φ(σⁱ_k) = eⁱ_k, where {eⁱ₁,· · ·, eⁱ_ni+1} is the canonical frame of TEi restricted to ϕ(M). For X ∈Γ(T M), we have

Φ(X) =

m

X

i=1 ni+1

X

k=1

g(X, σe ⁱ_k)eⁱ_k

=

m

X

i=1 n_i+1

X

k=1

g(πe iX, σ_kⁱ)eⁱ_k

= ϕ_∗(X).

Moreover, for anyi∈ {1,· · · , m}, we have Φ(ξ_i) =

ni+1

X

k=1

eg(ξ_i, σ_kⁱ)eⁱ_k.=

ni+1

X

k=1

ϕⁱ_keⁱ_k.

Hence, Φ(ξ_i) is the normal direction ofM_iinEi. Since Φ is an isometry of the fibers, we deduce that Φ(E) is the normal bundleT^⊥ϕ(M) ofϕ(M) inPe. We denote byϕethe restriction of Φ to E. It is clear that ϕeis an isomorphism of vector bundles betweenE andT^⊥ϕ(M).

Since Φ sends the orthonormal parallel sections{σ_kⁱ}ofF onto the orthonormal parallel sections {eⁱ_k} ofE1× · · · ×Em, we have

(13) Φ(D_XY) =∇⁰_ϕ

∗(X)ϕ_∗(Y),

(14) Φ(DXν) =∇⁰_ϕ∗(X)ϕ(νe ),

(15) Φ(DXξi) =∇⁰_ϕ∗(X)Φ(ξi), where ∇⁰ is the Levi-Civita connection ofE¹× · · · ×E^m.

For any i∈ {1,· · · , m}, we define the map eπi = Φ◦πi◦Φ⁻¹. From this definition, it is clear that πe_i(e^j_k) =δ^j_ie^j_k. Then, it follows that the mapseπ_i are symmetric, parallel along ϕ(M) and satisfy eπ_i◦πe_j =δ_i^jeπ_i and Pm

i=1πe_i = Id_{T(E1×···×Em)}. Thus, it is clear that these maps are the restrictions on ϕ(M) of the projections on each factorTEi ofT(E1× · · · ×Em).

Moreover, from the definition of eπ_i, we deduce immediately that πei(ϕ_∗X) =ϕ_∗(fiX) + Φ(hiX), and

πei(Φξ) =ϕ_∗(siX) + Φ(tiξ).

Indeed, we have

eπi(ϕ_∗X) = Φ(πi(X)) = Φ(fiX+hiX) =ϕ_∗(fiX) +ϕ(he iX), and

eπi(ϕ(ν)) = Φ(πe i(ν)) = Φ(siν+hiν) =ϕ_∗(siν) +ϕ(te iν).

Finally, we will prove that the second fundamental form is given byBand the normal connection is given by∇. From Equation (13), we have

∇⁰_ϕ∗(X)ϕ_∗(Y) = Φ(D_XY)

= Φ(∇XY +B(X, Y)−

m

X

i=1

c_ig(f_iX, Y)ξ_i)

= ϕ_∗(∇_XY) +ϕ(B(X, Ye ))−

m

X

i=1

c_ig(f_iX, Y)Φ(ξ_i).

Then the normal part inTPe isϕ(B(X, Ye )), which implies that the second fundamental form of the immersion ϕisϕe◦B. Moreover, from Equation (14), we have

∇⁰_ϕ∗(X)ϕ(νe ) = Φ(D_Xν)

= Φ(∇Xν−AνX−

m

X

i=1

cig(hiX, ν)ξi)

= ϕ(∇e Xν) +ϕ_∗(A_νX)−

m

X

i=1

c_ig(h_iX, ν)Φ(ξ_i).

Thus, the normal part inTPeisϕ(∇e _Xν) and we deduce that∇^⊥_Xϕ(νe ) =ϕ(∇e _Xν). Then, we get

∇^⊥ϕe=ϕ∇. This concludes the proof of the existence in Theorem 3.2.e

Now, we will prove the uniqueness of this isometric immersion up to an isometry of P.e This follows directly from the following proposition.

Proposition 3.7. Let ϕ, ϕ⁰ : M −→ Pe be two isometric immersions with respective normal bundles E, E⁰ and second fundamental forms B, B⁰. Let fi, hi andf_i⁰, h⁰_i be the (1,1)-tensors defined by (2)forϕandϕ⁰ respectively. Assume that

i) fiX =f_i⁰X for anyi∈ {1,· · · , m}andX ∈Γ(T M),

ii) there exists an isometry of vector bundlesφ:E−→E⁰ so that φ(hiX) =h⁰_iX,

φ(B(X, Y)) =B⁰(X, Y), φ(∇Xν) =∇⁰_Xφ(ν), for any i∈ {1,· · ·, m},X, Y ∈Γ(T M)andν ∈E.

Then, there exists an isometry αofPe such that ϕ⁰=α◦ϕandα_∗|E=φ.

Proof: We give the complete proof for cm 6= 0, the case cm can be proven with a minor modification.

As previously, we denote by{eⁱ_k}, fori∈ {1,· · · , m}andk∈ {1,· · · , ni+ 1}the canonical frame ofE1× · · · ×Em. Hence, we denote byϕⁱ_k and (ϕ⁰)ⁱ_k the components ofϕandϕ⁰ respectively in the frame{eⁱ_k}. We consider the mapG:M −→GL(E1× · · · ×Em) defined by

Gp(ϕ∗(X)) =ϕ⁰_∗(X), Gp(ν) =φ(ν),

G_p(ξ_i) =ξ⁰_i,

for any i∈ {1,· · ·, m},X, Y ∈Γ(T M) andν∈E, and where ξi andξ_i⁰ are defined by ξi=

n_i+1

X

k=1

ϕⁱ_keⁱ_k and

ξ_i⁰ =

n_i+1

X

k=1

(ϕ⁰)ⁱ_keⁱ_k.

We will show that the mapGis constant, that is, that it does not depend on the pointp. First of all, we remark that for any i and any X ∈ Γ(T M), we have ∇⁰_Xξ_i = π_i(ϕ_∗(X)), where π_i is the projection on TEi and ∇⁰ is the Levi-Civita connection of E1× · · · ×Em. Now, we will

show that ∇⁰G= 0, or equivalently that∇⁰_X(G(V))−G(∇⁰_XV) = 0 for anyX ∈Γ(T M) and V ∈Γ(T(E1× · · · ×Em)_|ϕ(M)).

First, for V ∈ϕ∗(T M), that is,V =ϕ∗(Y) withY tangent toM, we have

∇⁰_X(G(ϕ_∗(Y)))−G(∇⁰_Xϕ_∗(Y)) = ∇⁰_Xϕ⁰_∗(Y)−G(ϕ_∗(∇XY) +B(X, Y)) +

m

X

i=1

cig(fi(X), Y)ξ_i⁰

= ϕ⁰_∗(∇XY) +B⁰(X, Y)−

m

X

i=1

cig(f_i⁰(X), Y)ξ⁰_i

−G(ϕ_∗(∇_XY) +B(X, Y)) +

m

X

i=1

c_ig(f_i(X), Y)ξ⁰_i

= 0,

since, by assumption,fiX =f_i⁰X andφ(B(X, Y)) =B⁰(X, Y). Now, ifν ∈E, we have

∇⁰_X(G(ν))−G(∇⁰_Xν) = −A⁰_φ(ν)X+∇⁰_Xφ(ν)−

m

X

i=1

c_ieg(φ(ν), h⁰_i(X))ξ⁰

−G −AνX+∇Xν−

m

X

i=1

c_ieg(φ(ν), h_i(X))ξ

!

= 0,

since by assumption,φ(B(X, Y)) =B⁰(X, Y),φ(hiX) =h⁰_iX andφ(∇Xν) =∇⁰_Xφ(ν).

Finally, we have

∇⁰_X(G(ξ))−G(∇⁰_Xξ) = πi(ϕ⁰_∗(X))−G(πi(ϕ_∗(X)))

= ϕ⁰_∗(f_i⁰X) +h⁰_iX−G(ϕ_∗(f_iX) +h_iX)

= 0, sincef_iX =f_i⁰X andφ(h_iX) =h⁰_iX.

Hence, we get that the mapGis constant alongM.

Uniqueness up to rigid motion is proved, which concludes the proof of Theorem 3.2.

4. Associated families of minimal surfaces and pluriminimal K¨alher hypersufaces In this section, we use Theorem 3.2 to prove the existence of associated families of minimal surfaces into the multiproductPe.

Let (Σ, g) be an oriented Riemannian surface. We denote by J its complex structure, that is, the rotation of angle ^π₂ onT M. For any θ∈R, we setRθ= cos(θ)I+ sin(θ)J. Remark, that Rθ is parallel. First, we have the following proposition.

Proposition 4.1. Assume that(Σ, g, E, g,∇, B, fi, hi, ti)satisfies the compatibility equation for Pe and that B is trace-free for anyξ ∈E, then(Σ, g, E, g,∇, Bθ, f_i,θ, h_i,θ, t_,θ)also satisfies the

compatibility equations forPe, where

Bθ(X, Y) =B(RθX, Y), fi,θ =Rθ◦fi◦ R⁻¹_θ , hi,θ=hi◦ R⁻¹_θ , ti,θ =ti.

Moreover, Bθ is also trace-free for anyξ∈E.

Proof: First, from the definition off_i,θ,h_i,θ andt_i,θ and the fact that f_i◦f_j+s_i◦h_j =δ_i^jf_i,

t_i◦t_j+h_i◦s_j =δ_i^jt_i, fi◦sj+si◦tj=δ_i^jsi, hi◦fj+ti◦hj=δ^j_ihi,

we get immediately that fi,θ andti,θ are symmetric and fi,θ◦fj,θ+si,θ◦hj,θ=δ_i^jfi,θ, ti,θ◦tj,θ+hi,θ◦sj,θ=δ^j_iti,θ, f_i,θ◦s_j,θ+s_i,θ◦t_j,θ=δ^j_is_i,θ, h_i,θ◦f_j,θ+t_i,θ◦h_j,θ =δ_i^jh_i,θ.

It is also clear that with this definition, the rank ofπi,θ is the same that the rank ofπiand that

m

X

i=1

fi,θ=IdT M,

m

X

i=1

ti,θ=IdE and

m

X

i=1

hi,θ= 0.

Now, we will show that analogues of Equations (5)-(7) are satisfied for (Σ, g, E, g,∇, Bθ, f_i,θ, h_i,θ, t_,θ). First, we have forX, Y tangent to Σ

∇Xf_i,θY −f_i,θ(∇XY) = ∇X(Rθf_iR⁻¹_θ Y)− Rθf_iR⁻¹_θ (∇XY)

= Rθ∇X(fiR⁻¹_θ Y)− RθfiR⁻¹_θ (∇XY), sinceRθ is parallel. Moreover, using (5), we get

∇Xf_i,θY −f_i,θ(∇XY) = RθA_h

i(R⁻¹_θ Y)X+Rθs_i B(X,R⁻¹_θ Y) +RθfiR⁻¹_θ (∇XY)− Rθfi(∇XR⁻¹_θ Y)

= A^θ_hi,θYX+si,θ(Bθ(X, Y)),

which is the desired equation. The two other equations can be shown in a similar way.

Finally, we prove that Gauss, Codazzi and Ricci equations are also fulfilled.

First we consider the Gauss equation. We notice that, for a surface, we have

m

X

i=1

ci

hfi,θY, Zifi,θX− hfi,θX, Zifi,θY

+A_{θ Bθ(Y,Z)}X−A_{θ Bθ(X,Z)}Y

=

m

X

i=1

ci

RθfiR⁻¹_θ X∧ RθfiR⁻¹_θ Y

Z+ detAθ

=

m

X

i=1

c_idetR_θf_iR⁻¹_θ + detA_θ=

m

X

i=1

c_idetf_i+ detA=R(X, Y)Z since determinants are invariant under rotations. Hence Gauss equation is satisfied.

Let∇eXA^ν_θ=∇XA^ν_θY−A^ν_θ∇XY−A^ν_θ∇⊥

XνY. Considering now Codazzi equation, we have, using the property ofhi,

(∇eXA^ν_θ)Y −(∇eYA^ν_θ)X = Rθ

h(∇eXA^ν)Y −(∇eYA^ν)Xi

= Rθ m

X

i=1

ci

fiYhX, siνi −fiXhY, siνi

= Rθ m

X

i=1

cifi(X∧Y)siν =

m

X

i=1

ciRθfi(X∧Y)R⁻¹_θ Rθsiν

=

m

X

i=1

c_i(R_θf_iR⁻¹_θ X∧Y)s_i,θν =

m

X

i=1

c_i(f_i,θX∧Y)s_i,θν

=

m

X

i=1

ci

fi,θY hX, si,θνi −fi,θXhY, si,θνi

=

m

X

i=1

ci

fi,θYhhi,θX, νi −fi,θXhhi,θY, νi

, and Codazzi is satisfied.

Similarly we get for the Ricci equation, using the properties of the wedge product R^⊥(X, Y)ξ =

m

X

i=1

c_i

h_iR⁻¹_θ X∧h_iR⁻¹_θ Y

ξ+B_θ(A_{θ ξ}Y, X)−B_θ(A_{θ ξ}X, Y)

=

m

X

i=1

ci

hiX∧hiY

ξ+B(AξY, X)−B(AξX, Y)

Since the surface is minimal, the shape operator anti-commutes withJ and we have indeed B_θ^ν(A_{θ ξ}Y, X)−B^ν_θ(A_{θ ξ}X, Y) = h[A_{θ ν}, A_{θ ξ}]X, Yi=h(RθAνRθAξ− RθAξRθAν)X, Yi

= h(A_νR⁻¹_θ R_θA_ξ−A_ξR⁻¹_θ R_θA_ν)X, Yi=h[A_ν, A_ξ]X, Yi.

Finally, let (e₁, e₂=J e₁) be a local orthonromal frame of Σ. We have tr(B_θ) =B_θ(e₁, e₁) +B_θ(e₂, e₂) = B(R_θe₁, e₁) +B(R_θe₂, e₂)

= cosθ[B(e1, e1) +B(e2, e2)] = 0,

sinceB is trace-free.

From this proposition, we can prove easily the following theorem about associated fami- lies of minimal surfaces in multiproducts. Namely, we get the following statement.

Theorem 4.2. Let Σ be a simply connected surface and x:M −→Pe be a conformal minimal immersion with normal bundle E, second fundamental form B and normal connection ∇^⊥. Let fi, hi, si and ti be the (1,1)-tensors induced by the projections πi. Let p0 ∈ Σ. Then, there exists a unique family (x_θ)_θ∈R of conformal minimal immersionsx_θ: Σ−→Pe so that

i) xθ(p0) =x(p0)andd(xθ)p₀ = (dx)p₀,

ii) the metric induced byX andXθ are the same,

iii) the second fundamental form fox_θ(Σ)in Pe is given byB_θ(X, Y) =B(R_θX, R_θY), for any X, Y ∈Γ(TΣ).

iv) for any i∈ {1,· · ·, m},X ∈Γ(TΣ)andξ∈Γ(E),

πi(dxθX) =dxθ(fi,θX) +hi,θX and πi(ξ) =dxθ(si,θX) +ti,θX, Moreover, x₀=xand the family(x_θ)_θ∈R is continuous with respect toθ.

Proof: We just proved that (Σ, g, E, g,∇, Bθ, f_i,θ, h_i,θ, t_,θ) satisfies the compatibility equations for eachθ. The theorem is then a direct consequence of theorem 3.2. The continuity is ensured

by the construction of Theorem 3.2.

5. Surfaces inS²×S²

LetJ be the complex structure onS². We consider the following complex structures onS²×S² J1= (J, J), J2= (J,−J).

Obviously J₁ and J₂ commute with each other and the projection π₁ and π₂ on each of the factors are given by

π1=Id +J1J2

2 , π2= Id−J1J2

2 .

From equation (1) we get R(X, Ye )Z=

hY, ZiX− hX, ZiY +

hJ₁Y, J₂ZiJ₁J₂X− hJ₁X, J₂ZiJ₁J₂Y . (16)

Let now Σ be a surface isometrically immersed into S²×S². For i ∈ {1,2}, we define four operators ji : TΣ → TΣ, ki : TΣ → NΣ ,li : NΣ → TΣ and mi : NΣ → NΣ such that Ji=ji+ki+li+mi.

From J1J2=J2J1 we get the following equations

j1j2+l1k2=j2j1+l2k1, (17)

k1j2+m1k2=k2j1+m2k1, (18)

j1l2+l1m2=j2l1+l2m1, (19)

k1l2+m1m2=k2l1+m2m1. (20)

The propertyJ_i²=−Id yields

j_i²+liki=−IdTΣ, (21)

k_ij_i+m_ik_i= 0, (22)

jili+limi= 0, (23)

k_il_i+m²_i =−Id_NΣ. (24)

Moreover the fact that the operators Ji are antisymmetric implies the antisymmetry of the operators ji as well as the propertyhkiX, νi=−hX, liνi.

The parallelity of Ji gives

∇X(jiY)−ji(∇XY) =Ak_iYX+li(B(X, Y)), (25)

∇^⊥_X(k_iY)−k_i(∇_XY) =m_i(B(X, Y))−B(X, j_iY), (26)

∇^⊥_X(miξ)−mi(∇^⊥_Xξ) =−B(liξ, X)−ki(AξX), (27)

∇X(l_iξ)−l_i(∇^⊥_Xξ) =−ji(A_ξX) +A_miξX.

(28)

Finally, from (16), we get the Gauss equation

K = 1

2

1 +

hj1e1, j2e2i+hk1e1, k2e2i

hj1e2, j2e1i+hk1e2, k2e1i (29)

−

hj1e₁, j₂e₁i+hk1e₁, k₂e₁i

hj1e₂, j₂e₂i − hk1e₂, k₂e₂i +2|H|²−|B|²

2 , the Codazzi equation

(∇XB)(Y, Z)−(∇YB)(X, Z) = 1

2

hY,(j1j2+l1k2)Zi(k1j2+m1k2)X− hX,(j1j2+l1k2)Zi(k1j2+m1k2)Y

, (30)

and the Ricci equation K^⊥ =

hk1j2+m1k2)e2, ν1i h(k1j2+m1k2)e1, ν2i (31)

hk1j₂+m₁k₂)e₁, ν₁i h(k1j₂+m₁k₂)e₂, ν₂i

+h[Aν₂, Aν₁]e1, e2]i.

Remark 5.1. In[11]the Gauss, Codazzi and Ricci equations are expressed with the help of the two K¨ahler functions C₁ and C₂ : Σ →R defined by ϕ^∗ω_i =C_iω_Σ, i= 1,2, with ω_Σ the area form on Σ. A tidious but straightforward computation shows that those two formulations are equivalent.

Now, we are able to reformulate the main theorem in the case of S²×S² in terms of complex structures instead of projections on each factor.

Corollary 5.2. Let (Σ², g)be a Riemannian surface and(E,h·,·iE,∇^E)a rank2vector bundle over Σendowed with a scalar product and a compatible connection. Suppose that there exists a symmetric(2,1)-tensor fieldB:TΣ×TΣ→E and eight operatorsji:TΣ→TΣ,ki:TΣ→E, li:E→TΣandmi:E→E,i= 1,2 satisfying conditions (17)to(28)and the Gauss, Codazzi and Ricci equations (29),(30)and (31). Then, there exists a unique (up to isometries of S²×S²) isometric immersion from Σinto S²×S² with E as normal bundle, B as second fundamental form and such that the restrictions of the complex structuresJ_i overΣare given byj_i,k_i,l_i and m_i.