AN EXISTENCE AND UNIQUENESS RESULT FOR ISOMETRIC IMMERSIONS

on his 70th birthday

AN EXISTENCE AND UNIQUENESS RESULT FOR ISOMETRIC IMMERSIONS

WITH LITTLE REGULARITY

MARCELA SZOPOS

The purpose of this paper is to investigate a classical existence and uniqueness theorem for an isometric immersion of a submanifold in the functional framework of distributions. More specically, we prove that the reconstruction of a sub- manifold in the Euclidean space is still possible under the assumptions that the prescribed geometrical data belong to some Sobolev spaces.

AMS 2000 Subject Classication: 53C42, 53A07.

Key words: isometric immersion, Gauss-Ricci-Codazzi equations, weak derivative, distribution.

1. INTRODUCTION

A classical theorem of dierential geometry states that a simply connected open subset ω of R^p endowed with a Riemannian metric can be isometrically immersed in a Euclidean spaceR^p+qif and only if there exist tensors satisfying the Gauss-Ricci-Codazzi equations. In addition, these immersions are uniquely determined up to a proper isometry of the underlying Euclidean space. Several dierent approaches are proposed in the literature for solving this reconstruc- tion problem: local coordinates (see, for example, [5]), connections in the normal bundle (see [11]), dierential forms (see [12]) or at compatible met- ric connections (see [6], paper which also includes a general discussion on this subject). We emphasize that in all these works, the regularity assumptions on the initial data are considered in a smooth setting, i.e., for derivatives in the usual sense.

Recently, motivated by some problems encountered in nonlinear elastic- ity, dierent extensions have been proposed in the framework of Sobolev-type functions. From this viewpoint, the immersionθ:ω →R^p+qcan be interpreted as a deformation of a solid occupying the set ω under the action of some ap- plied forces, provided that the mapping θ is injective (since we want to avoid

REV. ROUMAINE MATH. PURES APPL., 53 (2008), 56, 555565

the interpenetrability of matter). The casep= 3, q = 0is related to the theory of nonlinear three-dimensional elasticity while case p = 2, q = 1 constitutes a model for the nonlinear shell theory. The corresponding geometrical situa- tions, namely that of an open set in R³ and that of a surface have thus been studied within the theory of distributions (a complete review of these recent results can be found in [3]).

Another eld of interest for the study of isometric immersions in the distributional framework is the theory of relativity. In this direction, the case of immersions of a Lorenztian manifold with distributional curvature tensor was recently considered, since such manifolds represent spacetimes of general relativity that possibly contain gravitational waves, shock waves, and other singular patterns (see [7], [8]).

In this paper, we investigate the case of ap-dimensional submanifold iso- metrically immersed in the Euclidean spaceR^p+q in the functional framework of distributions. Although the class of smooth immersions has been extensively studied within the theory of Riemannian geometry, the Sobolev-type regularity issues do not seem to have been previously considered in the existing litera- ture. More specically, we prove, under suitable assumptions on the prescribed geometrical data, that the reconstruction of an isometric immersion and of the corresponding normal basis is still possible in some specic Sobolev spaces (see Theorem 3.3). In particular, our result generalizes that of [9, Theorem 9], where the recovery of a surface with prescribed rst and second fundamental forms is proved in a Sobolev setting. Moreover, this framework provides a uni- ed description of the geometrical setting for continuum mechanics for strings, membranes and solids (for a more detailed treatment of the geometrical aspects of solid mechanics in higher dimensions, see, for example, [1]).

The paper is organized as follows. In the next section, we dene our notation and provide some known results that will be subsequently needed.

In Section 3, we rst present the classical geometrical framework and then establish our main result.

2. NOTATION AND PRELIMINARIES

To begin with, we introduce some conventions and notation that will be used throughout the article. For any N ≥ 1, the N-dimensional Euclidean space E^N will be identied with R^N and will be endowed with the Euclidean norm dened by |a|=p

ha, ai, where ha, bi denotes the Euclidean inner prod- uct of a, b ∈ R^N. Next, M^N, O^N, S^N, A^N and S^N_> designate, respectively, the set of all real square matrices, of all orthogonal matrices, of all symmetric matrices, of all anti-symmetric matrices and of all positive denite symmetric

matrices, of order N. The identity matrix in M^N is denoted IN and M^M×N designates the space of all real matrices with M rows andN columns.

Let two integersp≥2, q≥1be xed throughout the paper. We denote by y = (y1, . . . , yp) a generic point of R^p and by {e₁, . . . , ep} the canonical basis of R^p. In what follows, Greek indices vary in the set {1, . . . , p}, Latin indices i, j, k vary in the set {1, . . . , q}, Latin indices m, n, s vary in the set {1, . . . , p+q}and the summation convention with respect to repeated indices is systematically used.

Partial derivative operators of order l ≥ 1, in the usual sense or in the sense of distributions, are denoted ∂^αf, where α= (α_i)∈N^p is a multi-index such that |α| := Pp

i=1αi = l. Partial derivative operators of the rst and second order are also denoted ∂α := _∂y^∂

α and ∂αβ := _∂y^∂2

α∂y_β.

Let r,r˜be two integers, let X be a nite dimensional space, such as R, R^N, M^N etc. and let ω be an open subset of R^p. The space of all innitely dierentiableX-valued functions with compact support inωis denotedD(ω,X) and the space of X-valued distributions overω is denotedD⁰(ω,X). The usual Lebesgue, respectively Sobolev, spaces being denoted L^r(ω,X), respectively W^r,r˜ (ω,X), we let

W_loc^˜r,r(ω,X) ={Y ∈ D⁰(ω,X), Y ∈W^˜r,r(U,X) for each open subsetU bω}, whereU bω means that the closure ofU inR^p is a compact subset of ω.

Remark 2.1. Forr > p, the equivalence classes of functions inW^1,r(ω,R) will be identied with their continuous representative, as allowed by the Sobolev embedding theorem.

Remark 2.2. We also note that, for any r > p, f g ∈ L^r_loc(ω,R) if f ∈ W_loc^1,r(ω,R) and g ∈ L^r_loc(ω,R), and that f g ∈ W_loc^1,r(ω,R) if f ∈ W_loc^1,r(ω,R) and g∈W_loc^1,r(ω,R), sinceW^1,r(U,R) is an algebra for all open balls U bω.

We next present some existence and uniqueness results for linear dier- ential systems with little regularity, that will be used in the proof of our main result. The rst theorem (see [10, Theorem 6.4]) ensures the global existence of the solution to a Pfa system, in the W_loc^1,r(ω,R)-setting for any dimension p≥2, whereω is a simply connected open subset of R^p and r > p:

Theorem 2.3. Let ω ⊂ R^p be a simply connected open set, let x⁰ ∈ω, let M ≥ 1 and N ≥ 1 be integers and let Y⁰ ∈M^M×N. Let there be given p matrix elds Aα ∈L^r_loc(ω,M^N), r > p, satisfying for all α, β the relations

∂_βA_α−∂_αA_β =A_αA_β−A_αA_β

in the space of distributions D⁰(ω,M^N). Then the Pfa system

∂αY =Y Aα in D⁰(ω,M^M×N), Y(x⁰) =Y⁰,

has one and only one solution Y ∈W_loc^1,r(ω,M^M×N).

The second result (see [10, Theorem 6.5]) is a generalization of the clas- sical Poincaré theorem:

Theorem 2.4. Let ω be a simply connected open subset of R^p. Let f1, . . . , fp ∈L^r_loc(ω,R), where r≥1, satisfy the compatibility conditions

∂_αf_β =∂_βf_α in D⁰(ω,R),

for all α, β ∈ {1, . . . , p}. Then there exists ψ ∈ W_loc^1,r(ω,R), unique up to an additive constant, such that

∂αψ=fα in L^r_loc(ω,R), for all α∈ {1, . . . , p}.

Remark 2.5. Note that, as observed in [10], the regularity assumptions on the coecients in the previous systems are optimal.

Finally, the third theorem is a uniqueness result that complements that of Theorem 2.3. The proof of this extension is a straightforward adaptation in any dimension p≥2 of [4, Theorem 6]:

Theorem 2.6. Let ω be a connected open subset of R^p, let r > p, let N ≥1 be an integer, letBα andCα be matrix elds in the space L^r_loc(ω,M^N), and let a point y⁰ ∈ω and a matrix F⁰ ∈ M^N be given. Then there exists at most one matrix eld F ∈W_loc^1,r(ω,M^N) that satises the Pfa system

∂_αF =F B_α+C_αF in D⁰(ω,M^N), F(y⁰) =F⁰. 3. MAIN RESULT

In order to state our main result, we start with a presentation of the geometrical framework of the problem. Let θ:ω → R^p+q be an immersion of the open set ω in the (p+q)-dimensional Euclidean space, identied with R^p+q. Then, for each y∈ω, there exists a neighborhood U ⊂ω ofy such that θ(U)⊂R^p+q is a submanifold ofR^p+q (for details, see [2, Ch. 6]). A basis in the tangent space T_θ(y)θ(ω)is given by

∂₁θ(y) :=dθ_y(e₁), . . . , ∂_pθ(y) :=dθ_y(e_p),

a vector eld N is called normal if hN(y), ∂_αθ(y)i = 0 for all y ∈ ω and for all α ∈ {1, . . . , p}. The derivatives of N are given by ∂αN(y) := dNy(eα).

We denote by X(ω) and X(ω)^⊥ the set of dierentiable vector elds that are tangent, respectively normal, toθ(ω).

The Euclidean metric of R^p+q induces a Riemannian metric on ω, also called the rst fundamental form of the immersion, dened by its covariant components

aαβ(y) :=h∂_αθ(y), ∂βθ(y)i, ∀y ∈ω, ∀α, β∈ {1, . . . , p}.

If ω is endowed with this metric, θ becomes an isometric immersion of ω intoR^p+q.

Let ∇ denote the usual Riemann connection on R^p+q (see [2, Ch. 2]).

If X and Y are vector elds on ω and X, Y are local extensions toR^p+q, we dene the induced Riemann connection on θ(ω) by ∇_XY := (∇_XY)^T, where by V^T we denote the tangential part of a vector V. The remaining normal part is denoted B(X, Y). The mapping B:X(ω)× X(ω) → X^⊥(ω) dened in this fashion is a symmetric bilinear mapping, which is called the second fundamental form of the immersion θ:ω →R^p+q. We thus have

(3.1) ∇_XY =∇_XY +B(X, Y)

for all tangent vector elds X and Y.

To dene the induced normal connection of the immersion, we take the projection of ∇_XN onto the normal space, where X is a tangent vector eld and N is a normal vector eld. Denoting this induced connection by ∇^⊥_XN, we thus have

(3.2) ∇_XN =A(X, N) +∇^⊥_XN,

where the mapping A:X(ω)× X^⊥(ω)→ X(ω)is related to the mappingB by the relation hA(X, N), Yi=− hB(X, Y), Nifor all Y ∈ X(ω).

We now write these expressions in the local coordinates(y₁, . . . , y_p)onω. Denote by Xα =∂αθ, for all 1 ≤α ≤p, the basis in the tangent bundle and x an orthogonal basis, denoted {N¹, . . . , N^q}, in the normal bundle. Hence Nⁱ, N^j

=δ^ij and Nⁱ, Xα

= 0 for all1≤i, j ≤q and 1≤α≤p.

The Riemann connection induced onθ(ω)and the normal connection of the immersion dene the Christoel symbols by the relations∇_XαXβ = Γ^τ_αβXτ

and ∇^⊥_X

βNⁱ =T_β^ijN^j.

Remark 3.1. We have Γ^γ_βα = Γ^γ_αβ since the connection ∇ is symmetric, T_β^ij =−T_β^ji, since the basis {Nⁱ}_i is orthonormal, and

Γ^τ_αβ = 1

2a^στ(∂αa_βσ+∂_βaασ −∂σa_αβ),

where (a^στ) = (a_αβ)⁻¹, since the connection ∇is compatible with the metric induced by θ.

The coecients of the second fundamental form are dened by bⁱ_αβ :=−

B(X_α, X_β), Nⁱ .

Remark 3.2. Note thatbⁱ_αβ =bⁱ_βα since the bilinear mapping B is sym- metric and that the mapping A can be expressed in local coordinates as A(X_α, Nⁱ) = b^iτ_αX_τ, where b^iτ_α := a^στbⁱ_ασ denote the mixed components of the second fundamental form.

Since the Euclidean space R^p+q is at, the Riemann curvature tensor associated with the connection ∇on R^p+q vanishes. Using relations (3.1) and (3.2) and expressing the fact that its tangential part and its normal part vanish, we deduce the following relations, written in local coordinates:

(∂_αΓ^τ_βδ−∂_βΓ^τ_αδ+ Γ^σ_βδΓ^τ_ασ−Γ^σ_αδΓ^τ_βσ)a_{τ γ}=bⁱ_γαbⁱ_δβ−bⁱ_γβbⁱ_δα, (3.3)

∂_αT_β^ij−∂_βT_α^ij+T_β^kjT_α^ik−T_α^kjT_β^ik+a^στ(b^j_ατbⁱ_βσ−b^j_βτbⁱ_ασ) = 0, (3.4)

∂_αb^j_γβ −∂_βb^j_γα = Γ^τ_αγb^j_{τ β}−Γ^τ_βγb^j_{τ α}+bⁱ_γβT_α^ij−bⁱ_γαT_β^ij, (3.5)

whereα, β, γ, δ, σ, τ ∈ {1, . . . , p}andi, j, k ∈ {1, . . . , q}. These are the classical Gauss-Ricci-Codazzi equations satised for a submanifold of Euclidean space, as given, for example, in [6].

Our purpose is to establish that the reconstruction of a submanifold of R^p+qcan be done under some weaker regularity assumptions than those previ- ously found in the literature. More precisely, the following theorem states that the result still holds if the matrix elds A= (a_αβ),Bⁱ= (bⁱ_αβ)andT_α= (Tα^ij) are respectively of class W_loc^1,r(ω,S^p>), L^r_loc(ω,S^p) and L^r_loc(ω,A^q), for r > p, and the compatibility conditions are satised in the distributional sense, i.e.,

Z

ω

(Γ^τ_αδ∂_βφ−Γ^τ_βδ∂_αφ+Γ^σ_βδΓ^τ_ασφ−Γ^σ_αδΓ^τ_βσφ)a_{τ γ}dy= Z

ω

(bⁱ_γαbⁱ_δβ−bⁱ_γβbⁱ_δα)φdy, (3.6)

Z

ω

(T_α^ij∂_βφ−T_β^ij∂αφ+T_β^kjT_α^ikφ−T_α^kjT_β^ikφ)dy= Z

ω

a^στ(b^j_βτbⁱ_ασ−b^j_ατbⁱ_βσ)φdy, (3.7)

Z

ω

(b^j_γα∂_βφ−b^j_γβ∂_αφ)dy = Z

ω

(Γ^τ_αγb^j_{τ β}−Γ^τ_βγb^j_{τ α}+bⁱ_γβT_α^ij−bⁱ_γαT_β^ij)φdy, (3.8)

for all test functions φ∈ D(ω,R). Our result is as follows.

Theorem 3.3. Let ω be a simply connected open subset of R^p. Let r > p and let there be given a positive denite symmetric matrix eld A = (a_αβ) ∈ W_loc^1,r(ω,S^p>), q symmetric matrix elds Bⁱ = (bⁱ_αβ) ∈ L^r_loc(ω,S^p), i ∈ {1, . . . , q}, and p anti-symmetric matrix elds T_α = (Tα^ij) ∈ L^r_loc(ω,A^q), α ∈ {1, . . . , p}, satisfying the Gauss-Ricci-Codazzi equations inD⁰(ω,R). Then

(I) There exist an immersion θ ∈ W_loc^2,r(ω,R^p+q) and an orthonormal family ofqvector eldsN¹, . . . , N^q ∈W_loc^1,r(ω,R^p+q), normal toθ(ω), satisfying

h∂_αθ, ∂_βθi=a_αβ in W_loc^1,r(ω,R), ∀α, β∈ {1, . . . , p}, (3.9)

∂_αβθ, Nⁱ

=−bⁱ_αβ in L^r_loc(ω,R), ∀α, β∈ {1, . . . , p},∀i∈ {1, . . . , q}, (3.10)

∂_αNⁱ, N^j

=T_α^ij in L^r_loc(ω,R), ∀α∈ {1, . . . , p},∀i, j ∈ {1, . . . , q}.

(3.11)

(II) Moreover, if a mappingθ˜and an orthonormal family of vector elds {N˜¹, . . . ,N˜^q}, normal to θ(ω)˜ satisfy the same relations onω, then there exist a vector a∈R^p+q and a matrix Q∈O^p+q such thatθ(y) =˜ a+Qθ(y) for all y ∈ω andN˜ⁱ(y) =QNⁱ(y) for all y∈ω, andi∈ {1, . . . , q}.

Proof. For clarity, the theorem is established in a series of six steps.

Step 1. We introduce the matrix eld Γ_α:ω→M^p+q dened by

(3.12) Γα:=































Γ¹_1α Γ¹_2α . . . Γ¹_pα b¹¹_α b²¹_α . . . b^q1_α Γ²_1α Γ²_2α . . . Γ²_pα b¹²_α b²²_α . . . b^q2α

... ... ... ... ... ...

Γ^p_1α Γ^p_2α . . . Γ^ppα b^1pα b^2pα . . . b^qpα

−b¹_1α −b¹_2α . . . −b¹_pα T_α¹¹ T_α¹² . . . T_α^1q

−b²_1α −b²_2α . . . −b²_pα T_α²¹ T_α²² . . . T_α^2q

... ... ... ... ... ...

−b^q_1α −b^q_2α . . . −b^q_pα Tα^q1 Tα^q2 . . . Tα^qq





























 ,

whereΓ^τ_αβ = ¹₂a^στ(∂_αa_βσ+∂_βa_ασ−∂_σa_αβ), (a^στ) = (a_αβ)⁻¹ andb^iτ_α :=a^στbⁱ_ασ. By the denition of the inverse of a matrix, it follows from Remarks 2.1 and 2.2 that (a^στ) ∈ W_loc^1,r(ω,S^p>), hence Γ^τ_αβ and b^iτ_α are in L^r_loc(ω,R). Consequently, by using the regularity assumptions made on the initial data, we deduce that Γ_α ∈L^r_loc(ω,M^p+q) for all α∈ {1, . . . , p}.

Step 2. We show that the matrix elds Γα satisfy the compatibility con- ditions

∂βΓα−∂αΓβ+ ΓβΓα−ΓαΓβ =0 in D⁰(ω,M^p+q), for all α, β ∈ {1, . . . , p}.

First, we note that sinceΓ_α∈L^r_loc(ω,M^p+q)⊂ D⁰(ω,M^p+q), the products and the derivatives are well-dened distributions, hence the relations above make sense.

Second, we deduce from the Gauss-Ricci-Codazzi equations and a formal calculus that these compatibility conditions are satised inω. More precisely, if L_mndenotes the coecient of the matrix appearing in the left-hand side of the above relation, where m is the row index, then Lmn = 0for m, n∈ {1, . . . , p}

is a consequence of relation (3.3), the conditions Lmn = 0for m ∈ {1, . . . , p}, n ∈ {p+ 1, . . . , p+q} and for m ∈ {p+ 1, . . . , p+q}, n ∈ {1, . . . , p} are satised because of relation (3.5), and the conditions Lmn = 0 for m, n ∈ {p+ 1, . . . , p+q}are satised since relation (3.4) holds.

Third, the above computations are still valid under the regularity as- sumptions given in the statement of the theorem, since it suces to check that

Z

ω

(ΓαΓβ −ΓβΓα)φdy= Z

ω

(Γβ∂αφ−Γα∂βφ)dy

for all φ∈ D(ω,M^p+q). This assertion is a direct consequence of the previous formal computations, of relations (3.6), (3.8), (3.7) and a density argument.

Step 3. We dene the matrix eld F ∈ W_loc^1,r(ω,M^p+q) as the unique solution to the problem

∂_αF =FΓ_α inD⁰(ω,M^p+q), (3.13)

F(y₀) =G^1/2₀ , (3.14)

where the matrix G₀ is given by

(3.15) G0:=

A(y0) 0

0 I_q

, and y0∈ω is xed.

We emphasize that this system is obtained from equations (3.1) and (3.2) written in local coordinates, i.e.,

∂_αX_β = Γ^τ_αβX_τ−bⁱ_αβNⁱ, (3.16)

∂αNⁱ =T_α^ijN^j+b^iτ_αXτ. (3.17)

As for the initial condition, since A = (a_αβ) is a positive denite symmetric matrix eld, G^1/2₀ ∈S^p> is well dened. In order to prove that this overdeter- mined system of partial dierential equations can be solved, we rst note that the compatibility conditions are veried thanks to Step 2. Then, Theorem 2.3 applied to the above Cauchy problem provides the existence and uniqueness of a solution F ∈W_loc^1,r(ω,M^p+q).

Step 4. We dene the mapping θ∈W_loc^2,r(ω,R^p+q) as the unique solution to the problem

∂αθ=fα in D⁰(ω,R^p+q), (3.18)

θ(y₀) =θ₀, (3.19)

wheref_α(y)is theα-th column of the matrixF(y)found in Step3andθ₀ ∈R^p+q is xed.

We also deneNⁱ ∈W_loc^1,r(ω,R^p+q) by the relation

(3.20) Nⁱ =fp+i,

for alli∈ {1, . . . , q}, wherefp+i(y)is the(p+i)-th column of the matrixF(y). We rst need to check the compatibility conditions ∂_βf_α = ∂_αf_β for all α, β ∈ {1, . . . , p} that have to be veried in order to obtain a solution to the problem (3.18)(3.19). A straightforward computation shows that these equalities hold as a consequence of the symmetry properties of the Christoel coecients Γ^τ_αβ and of the coecients of the second fundamental form bⁱ_αβ. The existence of a unique solution θ ∈W_loc^2,r(ω,R^p+q) then follows from The- orem 2.4. Moreover, since ∂αθ = fα and fα ∈ W_loc^1,r(ω,R^p+q) from Step 3, it follows that θ∈W_loc^2,r(ω,R^p+q).

Step 5. We show that the eldsθ and N¹, . . . , N^p found in the previous step satisfy conditions (3.9), (3.10) and (3.11) of the theorem.

To begin with, we dene the matrix eldG= (gmn) :ω→M^p+q by

(3.21) G(y) :=

A(y) 0 0 I_q

, and the matrix eld H = (hmn) :ω→M^p+q by (3.22) H(y) := (hf_m(y),f_n(y)i)_mn,

for all m, n∈ {1, . . . , p+q}. Next, we will show that the matrix eld Y:ω→ M^p+q,Y =G−H∈W_loc^1,r(ω;M^p+q) satises the Cauchy problem

∂_αY =YΓ_α+ Γ^T_αY inD⁰(ω,M^p+q), (3.23)

Y(y₀) =0.

(3.24)

Since Y˜ = 0 also veries system (3.23)(3.24), by using Theorem 2.6, we infer that G=H inW_loc^1,r(ω;M^p+q), which implies in particular relation (3.9).

Relations (3.10) and (3.11) are then obtained by combining relations (3.16) and (3.17) with the fact that hf_m,fni=gmn for all m, n∈ {1, . . . , p+q}.

It remains to show that the matrix eld Y veries the previous system.

We rst claim that

(3.25) ∂αgmn= (Γα)smgsn+ (Γα)sngms,

where we use the notation(Γα)mnfor the coecients of the matrix eldΓα de- ned in Step 1. In order to establish this relation, we distinguish the following four possible situations: ifm, n∈ {1, . . . , p}, relation (3.25) is a consequence of

∂_αa_βτ = Γ^σ_βαa_στ + Γ^σ_{τ α}a_βσ,

which is obtained from the denition of the Christoel symbols; ifm∈ {1, . . . , p}, n∈ {p+ 1, . . . , p+q} orm∈ {p+ 1, . . . , p+q},n∈ {1, . . . , p}, relation (3.25)

holds because of the symmetry bⁱ_αβ = bⁱ_βα; if m, n ∈ {p+ 1, . . . , p+q}, and relation (3.25) holds since T_α^ij +T_α^ji = 0.

Next, we want to prove that:

(3.26) ∂_αh_mn= (Γ_α)_smh_sn+ (Γ_α)_snh_ms.

To this end, we compute ∂_αh_mn =h∂_αf_m,f_ni+hf_m, ∂_αf_ni by replacing ∂_αf_m by the expressions given in (3.16)(3.17): if m=β∈ {1, . . . , p}then

∂_αf_β = Γ^σ_αβ∂_σθ−bⁱ_αβNⁱ and if m=p+i∈ {p+ 1, . . . , p+q} then

∂_αf_p+i =T_α^ijN^j+b^iσ_α∂_σθ.

Finally, we conclude from (3.25), (3.26) and from the initial condition (3.14) that Y satises system (3.23)(3.24).

Step 6. We establish part (II) of the theorem.

We assume that θ˜and the vector elds N˜¹, . . . ,N˜^q normal to θ(ω)˜ also satisfy the conditions of the theorem. We construct the matrix G˜₀ in the following manner: the αth column is given by ∂_αθ(y˜ ₀), forα ∈ {1, . . . , p} and the(p+i)th column is given byN˜ⁱ(y0), fori∈ {1, . . . , q}. Next, we dene the mapping θˆ:ω →R^p+q by

(3.27) θ(y) =ˆ θ0+G^1/2₀ G˜⁻¹₀ θ(y)˜ −θ(y˜ 0) , and the vector elds Nˆ¹, . . . ,Nˆ^q:ω→R^p+q by

(3.28) Nˆⁱ(y) =G^1/2₀ G˜⁻¹₀ N˜ⁱ(y),

for all i∈ {1, . . . , q}. Note that the denitions ofG^1/2₀ andG˜₀ imply that the matrix Q=G^1/2₀ G˜⁻¹₀ is orthogonal.

A direct computation based on these expressions then shows that the mapping θˆand the vector elds Nˆ¹, . . . ,Nˆ^q also satisfy the conditions of the theorem. Hence, as in the above proof, the corresponding Gauss-Ricci-Codazzi equations are satised in D⁰(ω,R). But, as we have already seen, these equa- tions lead to the matrix relation∂αFˆ = ˆFΓα, together with the initial condition Fˆ(y0) =G^1/2₀ . Consequently, the uniqueness part of Theorem 2.3 implies that F = ˆF inω. In other words, ∂αθ =∂αθˆ, for all α ∈ {1, . . . , p}, hence θ= ˆθ, since θ(y₀) = ˆθ(y₀) =θ₀, and Nⁱ = ˆNⁱ for alli∈ {1, . . . , q}. The conclusion then follows from relations (3.27) and (3.28).

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Received 15 July 2008 Université Paul Sabatier

Institut de Mathématiques de Toulouse(IMT) UMR 5219(CNRS-UPS-INSA-UT1-UT2)

118, route de Narbonne 31062 Toulouse Cedex, France marcela.szopos@math.univ-toulouse.fr