On the recovery of a surface with prescribed first and second fundamental forms

On the recovery of a surface with prescribed first and second fundamental forms

Philippe G. Ciarlet

, François Larsonneur

Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, 4 Place Jussieu, 75005 Paris, France

Manuscript received 23 October 2001

Dedicated to Klaus Kirchgässner on the occasion of his seventieth birthday

Abstract

The fundamental theorem of surface theory asserts that, if a field of positive definite symmetric matrices of order two and a field of symmetric matrices of order two together satisfy the Gauss and Codazzi–Mainardi equations in a connected and simply connected open subset ofR², then there exists a surface inR³with these fields as its first and second fundamental forms (global existence theorem) and this surface is unique up to isometries inR³(rigidity theorem).

The aim of this paper is to provide a self-contained and essentially elementary proof of this theorem by showing how it can be established as a simple corollary of another well-known theorem of differential geometry, which asserts that, if the Riemann–

Christoffel tensor associated with a field of positive definite symmetric matrices of order three vanishes in a connected and simply connected open subset ofR³, then this field is the metric tensor field of an open set that can be isometrically imbedded inR³(global existence theorem) and this open set is unique up to isometries inR³(rigidity theorem). For convenience, we also give a self-contained proof of this theorem, as such a proof does not seem to be easy to locate in the existing literature.

In addition to the simplicity of its principle, this approach has the merit to shed light on the analogies existing between these two fundamental theorems of differential geometry.2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.

Résumé

Le théorème fondamental de la théorie des surfaces affirme que, si un champ de matrices symétriques définies positives d’ordre deux et un champ de matrices symétriques d’ordre deux vérifient ensemble les équations de Gauss et de Codazzi–

Mainardi dans un ouvert connexe et simplement connexe deR², alors il existe une surface dansR³dont ces champs sont les première et deuxième formes fondamentales (théorème d’existence globale) et cette surface est unique aux isométries deR³ près (théorème de rigidité).

Le but de cet article est de donner une preuve autosuffisante et essentiellement élémentaire de ce théorème en montrant comment il peut être obtenu comme un simple corollaire d’un autre théorème bien connu de géométrie différentielle, qui affirme que, si le tenseur de Riemann–Christoffel associé à un champ de matrices symétriques définies positives d’ordre trois s’annule sur un ouvert connexe et simplement connexe deR³, alors ce champ est le tenseur métrique d’un ouvert qui peut être plongé isométriquement dansR³(théorème d’existence globale) et cet ouvert est unique aux isométries deR³près (théorème de rigidité). Par commodité, on donne également une preuve autosuffisante de ce théorème, car une telle preuve ne semble pas aisée à localiser dans la littérature.

Outre la simplicité de son principe, cette approche a le mérite d’illustrer les analogies existant entre ces deux théorèmes fondamentaux de la géométrie différentielle.2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.

* Corresponding author

E-mail addresses: [email protected] (P.G. Ciarlet), [email protected] (F. Larsonneur).

0021-7824/02/$ – see front matter 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.

PII: S 0 0 2 1 - 7 8 2 4 ( 0 1 ) 0 1 2 3 6 - 3

Introduction

The two-dimensional equations proposed by Koiter [12] for modeling a nonlinearly elastic shell made with a homogeneous and isotropic material are derived from three-dimensional elasticity on the basis of two a priori assumptions: One assumption, of a geometrical nature, is the Kirchhoff–Love assumption; it asserts that any point situated on a normal to the middle surface remains on the normal to the deformed middle surface after the deformation has taken place and that, in addition, the distance between such a point and the middle surface remains constant. The other assumption, of a mechanical nature, asserts that the state of stress inside the shell is planar and parallel to the middle surface (this second assumption is itself based on delicate a priori estimates due to John [11]).

Taking these a priori assumptions into account, W.T. Koiter then reaches the conclusion that the strain energyV per unit area of the undeformed middle surface of the shell takes the form (cf. [12, equations (4.2), (8.1), and (8.3)]):

V=ε 2a^{αβσ τ}

a_{σ τ}−a_{σ τ}

a_αβ−a_αβ +ε³

6a^{αβσ τ}b_{σ τ}−b_{σ τ}b_αβ−b_αβ , where 2εis the thickness of the shell,

a^{αβσ τ}:= 4λµ

λ+2µa^αβa^{σ τ}+2µ

a^ασa^βτ +a^ατa^βσ ,

λ >0 andµ >0 are the two Lamé constants of the constituting material,a_αβ andb_αβare the covariant components of the first and second fundamental forms of the given undeformed middle surface,(a^αβ)=(a_αβ)⁻¹(details about these notions are provided in Section 3), and finallya_αβandb_αβare the covariant components of the first and second fundamental forms of the unknown deformed middle surface under the action of given applied forces (ad hoc boundary conditions are also specified along the boundary of the middle surface).

In order that they actually define a surface imbedded in a three-dimensional Euclidean space, the unknown components a_αβ andb_αβ must therefore satisfy ad hoc sufficient compatibility conditions, which take the form of the classical Gauss and Codazzi–Mainardi equations. This crucial observation thus explains why “a system of fully consistent equations of compatibility is considered essential in a discussion of the general nonlinear theory of shells” (cf. [12, p. 21]).

Viewed as equality constraints (in the sense of optimization theory) imposed on the unknown functionsa_αβandb_αβ, these compatibility conditions also play a key rôle for finding the expression of the stress resultants and stress couples, by means of ad hoc Lagrange multipliers (cf. [12, equations (7.3)–(7.4)]).

It is this type of consideration that motivated the present work, whose objective is to provide a self-contained and essentially elementary proof (Theorem 5) of the sufficiency of the Gauss and Codazzi–Mainardi equations for the existence of a surface with given first and second fundamental forms in a three-dimensional Euclidean space.

The novelty lies in the proof itself, where this sufficiency is obtained as a simple corollary of another well-known theorem of differential geometry, which asserts that the vanishing of the Riemann–Christoffel tensor is a sufficient condition for the existence of a three-dimensional open set with a given metric tensor in a three-dimensional Euclidean space. For completeness, we also give a self-contained proof of this result (Theorem 2), as such a proof does not seem to be easy to locate in the existing literature.

For completeness again, we also give a self-contained proof of the associated “rigidity theorems”, which assert that, when these sufficient conditions are satisfied, the surface or the three-dimensional open set are unique up to isometries in a three- dimensional Euclidean space (Theorems 3 and 6).

1. The metric tensor of a three-dimensional open set

To begin with, we list some notations and conventions that will be consistently used throughout the article.

All spaces, matrices, etc., considered are real. Forn=2 and 3,Mⁿ,Sⁿ, andSⁿ_>respectively designate the sets of all square matrices of ordern, of all symmetric matrices of ordern, and of all symmetric, positive definite matrices of ordern.

Latin indices and exponents vary in the set{1,2,3}, except when they are used for indexing sequences, and the summation convention with respect to repeated indices or exponents is systematically used in conjunction with this rule. Kronecker’s symbols are designated byδ_i^j, δ_ij, orδ^ijaccording to the context.

Let E³ denote a three-dimensional Euclidean space, let a·b and a∧b denote the Euclidean inner product and exterior product of a,b∈E³, and let |a| =√

a·a denote the Euclidean norm of a∈E³. In addition, let there be given a three- dimensional vector space, identified withR³. Let x_i denote the coordinates of a point x∈R³ and let ∂_i :=∂/∂x_i and

∂_ij:=∂²/∂x_i∂x_j.

LetΩbe an open subset ofR³and letΘ∈C¹(Ω;E³)be an immersion, i.e., a mapping such that the three vectors∂_iΘ(x) are linearly independent at all pointsx∈Ω. Hence the setΘ(Ω)is open in E³(for a proof, see, e.g., [16, Theorem 3.8.10]).

The metric tensor of the setΘ(Ω)is defined by means of its covariant components g_ij(x):=∂_iΘ(x)·∂_jΘ(x), x∈Ω,

which are used in particular for computing lengths of curves inside the setΘ(Ω), considered as being isometrically imbedded in E³. This means that their length is precisely that induced by the Euclidean metric of the Euclidean space E³.

It is also well-known that the matrix field(g_ij):Ω→S³>defined in this fashion cannot be arbitrary. More specifically, its components and some of their partial derivatives must satisfy necessary conditions taking the form of relations (1.3) below (according to our rule governing Latin indices and exponents, relations (1.3) are meant to hold for alli, j, k, q∈ {1,2,3}).

Theorem 1. LetΩbe an open subset ofR³, letΘ∈C³(Ω;E³)be an immersion, and let

g_ij:=∂_iΘ·∂_jΘ (1.1)

denote the covariant components of the metric tensor of the setΘ(Ω). Let the functionsΓ_{ij q}∈C¹(Ω)andΓ_ij^p∈C¹(Ω)be defined by

Γ_{ij q}:=1

2(∂_jg_iq+∂_ig_{j q}−∂_qg_ij) and Γ_ij^p:=g^pqΓ_{ij q}, where g^pq

:=(g_ij)⁻¹. (1.2)

Then, necessarily,

∂_jΓ_ikq−∂_kΓ_{ij q}+Γ_ij^pΓ_kqp−Γ_ik^pΓ_{j qp}=0 inΩ. (1.3)

Proof (provided for completeness). Let g_i:=∂_iΘ. It is then immediately verified that the functionsΓ_{ij q}are also given by

Γ_{ij q}=∂_ig_j·gq. (1.4)

For eachx∈Ω, let the three vectors g^j(x) be defined by the relations g^j(x)·g_i(x)=δ_j^j. Since we also have g^j =g^ijg_i, relations (1.4) imply thatΓ_ij^p=∂_ig_j·g^p. Therefore,

∂_ig_j=Γ_ij^pgp (1.5)

since∂_ig_j=(∂_ig_j·g^p)g_p. Differentiating relations (1.4) yields

∂_kΓ_{ij q}=∂_ikg_j·g_q+∂_ig_j·∂_kg_q, so that relations (1.4) and (1.5) together give

∂_ig_j·∂_kg_q=Γ_ij^pg_p·∂_kg_q=Γ_ij^pΓ_kqp. Consequently,

∂_ikg_j·gq=∂_kΓ_{ij q}−Γ_ij^pΓ_kqp. (1.6)

Since∂_ikg_j=∂_ijg_k, we also have

∂_ikg_j·gq=∂_jΓ_ikq−Γ_ik^pΓ_{j qp}, (1.7)

so that relations (1.3) are simply obtained by subtracting (1.6) from (1.7). ✷

Remarks. (1) The vectors g_i and g^j introduced above form the covariant and contravariant bases, the functiong^ij are the contravariant components of the metric tensor, the functionsΓ_ij^pandΓ_{ij q}are the Christoffel symbols of the first, and second, kind and finally, the functions

R_{qij k}:=∂_jΓ_ikq−∂_kΓ_{ij q}+Γ_ij^pΓ_kqp−Γ_ik^pΓ_{j qp}

are the covariant components of the Riemann–Christoffel curvature tensor, of the setΘ(Ω). The relationsR_{qij k}=0 found in Theorem 1 thus express that the Riemann–Christoffel tensor of the setΘ(Ω)(equipped with the metric tensor with covariant componentsg_ij) vanishes. For details, see, e.g., [5, p. 303].

(2) The necessary conditionsR_{qij k}=0 of Theorem 1 thus simply constitute a re-writing of the relations∂_ikg_j=∂_ijg_k in the form of the equivalent relations∂_ikg_j·gq=∂_ijg_k·gq.

2. Existence and uniqueness of a mapping defined on an open set inR³that gives rise to a prescribed metric tensor We now turn to the reciprocal questions:

Given an open subsetΩofR³and a smooth enough matrix field(g_ij):Ω→S³_>, when is it the metric tensor field of an open setΘ(Ω)⊂E³, i.e., when does there exist an immersionΘ:Ω→E³such thatg_ij=∂_iΘ·∂_jΘinΩ?

If such an immersion exists, to what extent is it unique?

The answers turn out to be remarkably simple: Under the additional assumptions thatΩis connected and simply connected, the necessary conditions (1.3) of Theorem 1 are also sufficient for the existence of such an immersion, and this immersion is unique up to isometries inR³.

As it does not seem easy to locate a self-contained, elementary, and complete proof of this well-known result of differential geometry in the existing literature, we provide one here for completeness. Its outline follows, with some modifications and simplifications, that of Blume [1]. In addition, we have included and adapted to our present purposes the proof of a crucial global existence theorem for systems of partial differential equations due to Cartan [4]. “Local” versions of the next theorem, based on the theory of locally integrable Pfaff systems and on the Frobenius theorem, are found by Choquet-Bruhat, Dewitt- Morette, Dillard-Bleick [5, p. 303] and Malliavin [14, p. 133].

This result comprises two essentially distinct parts, a global existence result (Theorem 2) and a uniqueness result (Theorem 3), the latter being called the rigidity theorem. Note that these two results are established under different assumptions on the setΩand on the smoothness of the field(g_ij).

Theorem 2 (global existence theorem). LetΩbe a connected and simply connected open subset ofR³and(g_ij)∈C²(Ω;S³>) be a matrix field that satisfies

∂_jΓ_ikq−∂_kΓ_{ij q}+Γ_ij^pΓ_kqp−Γ_ik^pΓ_{j qp}=0 inΩ, (2.1)

where

Γ_{ij q}:=1

2(∂_jg_iq+∂_ig_{j q}−∂_qg_ij) and Γ_ij^p:=g^pqΓ_{ij q}, where g^pq

:=(g_ij)⁻¹. (2.2)

Then there exists an immersionΘ∈C³(Ω;E³)such that

g_ij=∂_iΘ·∂_jΘ inΩ. (2.3)

The proof of Theorem 2 relies on a simple, yet crucial, observation. When the mappingΘ=(Θ)is a priori given (as in Section 1), its componentsΘ satisfy the relations∂_ijΘ=Γ_ij^p∂pΘ, which are nothing but another way of writing the relations∂_ig_j=Γ_ij^pg_p(see the proof of Theorem 1). This observation thus suggests to begin by solving (Lemma 2) the linear system of partial differential equations

∂_iF_j=Γ_ij^pF_p inΩ, (2.4)

whose solutionsF_j:Ω→Rthen constitute natural candidates for the partial derivatives ∂_jΘ of the unknown mapping Θ=(Θ)(Lemma 3).

To begin with, we prove the following lemma, which will in turn allow us to re-write relations (2.1) in a slightly different form (cf. equations (2.8)), more appropriate for the existence result of Lemma 2.

Lemma 1. LetΩbe an open subset ofR³and let there be given a field(g_ij)∈C²(Ω;S³)of symmetric invertible matrices.

The functionsΓ_{ij q}, Γ_ij^p, andg^pq being defined as in (2.2), define the functions

R_{qij k} := ∂_jΓ_ikq−∂_kΓ_{ij q}+Γ_ij^pΓ_kqp−Γ_ik^pΓ_{j qp}, (2.5)

R_·^p_{ij k} := ∂_jΓ_ik^p−∂_kΓ_ij^p+Γ_ikΓ_j ^p−Γ_ijΓ_k^p. (2.6)

Then

R_·^p_{ij k}=g^pqR_{qij k} and R_{qij k}=gpqR_·^p_{ij k}. (2.7)

Proof. Let us establish relations (2.7)a. Using the relations Γ_{j q}+Γ_{j q}=∂_jg_q and Γ_ikq=g_qΓ_ik,

which themselves follow from definitions (2.2)aand (2.2)_b, and noting that g^pq∂_jg_q+g_q∂_jg^pq

=∂_j g^pqg_q

=0, we obtain

g^pq

∂_jΓ_ikq−Γ_ik^rΓ_{j qr}

= ∂_jΓ_ik^p−Γ_ikq∂_jg^pq−Γ_ikg^pq(∂_jg_q−Γ_{j q})

= ∂_jΓ_ik^p+Γ_ikΓ_j ^p−Γ_ik

g^pq∂_jg_q+g_q∂_jg^pq

=∂_jΓ_ik^p+Γ_ikΓ_j ^p. Likewise,

g^pq

∂_kΓ_{ij q}−Γ_ij^rΓ_kqr

=∂_kΓ_ij^p+Γ_ijΓ_kl^p,

and thus relations (2.7)_aare established. Relations (2.7)_aand (2.7)_bare clearly equivalent. ✷

Following Cartan [4], we now establish the existence of solutions to the linear system of partial differential equations (2.4).

Note that Cartan’s approach was later adapted to nonlinear systems of partial differential equations by Thomas [19].

Lemma 2. LetΩbe a connected and simply connected open subset ofR³and let there be given functionsΓ_ij^p=Γ_{j i}^p∈C¹(Ω) satisfying the relations

∂_jΓ_ik^p−∂_kΓ_ij^p+Γ_ikΓ_j ^p−Γ_ijΓ_k^p=0 inΩ, (2.8)

which, by Lemma 1, are equivalent to relations (2.1) when the functionsΓ_ij^pare defined as in Theorem 2. Let a pointx⁰∈Ω and a matrix(F_j⁰)∈M³be given. Then there exists one, and only one, field(F_j)∈C²(Ω;M³)that satisfies

∂_iF_j(x) = Γ_ij^p(x)F_p(x), x∈Ω, (2.9)

F_j x⁰

= F_j⁰. (2.10)

Proof. For clarity, the proof is broken into three parts, numbered (i) to (iii).

(i) Letx¹ be an arbitrary point in the setΩ, distinct fromx⁰. SinceΩ is connected, there exists a path γ=(γⁱ)∈ C¹([0,1];R³)joiningx⁰tox¹inΩ; this means that

γ(0)=x⁰, γ(1)=x¹, and γ(t )∈Ω for all 0t1.

Lettingx=γ(t ),0t1, in equations (2.9), we conclude that, if a matrix field(F_j)∈C¹(Ω;M³)satisfies equations (2.9), then, for each integer∈ {1,2,3}, the three functionsζ_j∈C¹([0,1])defined by (for simplicity, the dependence onis dropped in what follows)

ζ_j(t ):=F_j γ(t )

, 0t1, (2.11)

satisfy the following Cauchy problem for a linear system of three ordinary differential equations with respect to three unknowns:

dζ_j

dt (t ) = Γ_ij^p

γ(t )dγⁱ

dt (t )ζp(t ), 0t1, (2.12)

ζ_j(0) = ζ_j⁰, (2.13)

where the “initial” values appearing in the Cauchy conditions (2.13) are given by

ζ_j⁰:=F_j⁰. (2.14)

Note in passing that the three Cauchy problems (2.12)–(2.13) obtained by letting=1,2, or 3 only differ by their initial valuesζ_j⁰.

It is well known that a Cauchy problem of the form (with self-explanatory notations) dζ

dt(t )=A(t )ζ(t ), 0t1, ζ(0)=ζ⁰,

has one and only one solutionζ∈C¹([0,1];R³)if A∈C⁰([0,1];M³)(see, e.g., [16, Theorem 4.3.1, p. 388]). Hence each Cauchy problem (2.12)–(2.13) has one and only one solution.

Incidentally, this result already shows that, if the system (2.9)–(2.10) has a solution, it is unique.

(ii) In order that the three values ζ_j(1) found by solving (2.12)–(2.13) for a given integer ∈ {1,2,3} be acceptable candidates for the three unknown valuesF_j(x¹), they must be of course independent of the path chosen for joiningx⁰tox¹.

So, letγ₀∈C¹([0,1];R³)and γ₁∈C¹([0,1];R³)be two paths joiningx⁰tox¹ inΩ. The open setΩ being simply connected, there exists a homotopy G=(Gⁱ):[0,1] × [0,1] →R³joiningγ₀toγ₁inΩ, i.e., such that

G(·,0)=γ₀, G(·,1)=γ₁, G(t, λ)∈Ω for all 0t1,0λ1, G(0, λ)=x⁰ and G(1, λ)=x¹ for all 0λ1,

and that is smooth enough, in the sense that G∈C¹

[0,1] × [0,1];R³

and ∂

∂t ∂G

∂λ

= ∂

∂λ ∂G

∂t

∈C⁰

[0,1] × [0,1];R³ .

Let ζ(·, λ)=(ζ_j(·, λ))∈C¹([0,1];R³) denote for each 0λ1 the solution of the Cauchy problem (2.12)–(2.13) corresponding to the path G(·, λ)joiningx⁰tox¹. We thus have

∂ζ_j

∂t (t, λ) = Γ_ij^p

G(t, λ)∂Gⁱ

∂t (t, λ)ζ_p(t, λ) for all 0t1, 0λ1, (2.15)

ζ_j(0, λ) = ζ_j⁰ for all 0λ1. (2.16)

Our objective is to show that

∂ζ_j

∂λ(1, λ)=0 for all 0λ1, (2.17)

as this relation will imply thatζ_j(1,0)=ζ_j(1,1)as desired. For this purpose, a direct differentiation of (2.15) shows that, for all 0t1, 0λ1,

∂

∂λ ∂ζ_j

∂t

=

Γ_ij^qΓ_kq^p +∂_kΓ_ij^p ζ_p∂G^k

∂λ

∂Gⁱ

∂t +Γ_ij^pζ_p ∂

∂λ ∂Gⁱ

∂t

+σ_qΓ_ij^q∂Gⁱ

∂t , (2.18)

where

σ_j:=∂ζ_j

∂λ −Γ_kj^pζ_p∂G^k

∂λ , (2.19)

on the one hand (in (2.18), (2.19), and in (2.20) below,Γ_ij^q, ∂_kΓ_ij^p, etc., stand forΓ_ij^q(G(·,·)), ∂_kΓ_ij^p(G(·,·)), etc.). On the other, a direct differentiation of (2.19) shows that, for all 0t1,0λ1,

∂

∂t ∂ζ_j

∂λ

=∂σ_j

∂t +

∂_iΓ_kj^p∂Gⁱ

∂t ζp+Γ_kj^q∂ζq

∂t

∂G^k

∂λ +Γ_ij^pζp

∂

∂t ∂Gⁱ

∂λ

, so that, by (2.15),

∂

∂t ∂ζ_j

∂λ

=∂σ_j

∂t +

∂_iΓ_kj^p+Γ_kj^qΓ_iq^p ζp

∂Gⁱ

∂t

∂G^k

∂λ +Γ_ij^pζp

∂

∂t ∂Gⁱ

∂λ

. (2.20)

Since

∂

∂λ ∂ζ_j

∂t

= ∂

∂t ∂ζ_j

∂λ

and ∂

∂λ ∂Gⁱ

∂t

= ∂

∂t ∂Gⁱ

∂λ

by assumption, subtracting (2.20) from (2.18) yields

∂σ_j

∂t +

∂_iΓ_kj^p−∂_kΓ_ij^p+Γ_kj^qΓ_iq^p−Γ_ij^qΓ_kq^p ζp

∂G^k

∂λ

∂Gⁱ

∂t −Γ_ij^q∂Gⁱ

∂t σq=0.

The assumed symmetriesΓ_ij^p=Γ_{j i}^pcombined with the assumed relations (2.8) show that

∂_iΓ_kj^p−∂_kΓ_ij^p+Γ_kj^qΓ_iq^p−Γ_ij^qΓ_kq^p =0, on the one hand. On the other,

σ_j(0, λ)=∂ζ_j

∂λ(0, λ)−Γ_kj^p G(0, λ)

ζ_p(0, λ)∂G^k

∂λ (0, λ)=0,

sinceζ_j⁰(0, λ)=ζ_j⁰(cf. (2.16)) and G(0, λ)=x⁰for all 0λ1. Therefore, for any fixed value of the parameterλ∈ [0,1], each functionσ_j(·, λ)defined by (2.19) satisfies a Cauchy problem for an ordinary differential equation, viz.,

dσ_j

dt (t, λ) = Γ_ij^q

G(t, λ)∂Gⁱ

∂t (t, λ)σq(t, λ), 0t1, (2.21)

σ_j(0, λ) = 0. (2.22)

But the solution of (2.21)–(2.22) is unique; consequentlyσ_j(t, λ)=0 for all 0t1. In particular then, σ_j(1, λ)=∂ζ_j

∂λ(1, λ)−Γ_kj^p G(1, λ)

ζp(1, λ)∂G^k

∂λ (1, λ)=0 for all 0λ1, and thus relations (2.17) hold since G(1, λ)=x¹for all 0λ1.

(iii) For each integer, we may thus unambiguously define a vector field(F_j):Ω→R³by letting F_j

x¹

:=ζ_j(1) for anyx¹∈Ω,

whereγ∈C¹([0,1];R³)is any path joiningx⁰tox¹inΩand the vector field(ζ_j)∈C¹([0,1])is the solution of the Cauchy problem (2.12)–(2.13) corresponding to such a path.

To establish that such a vector field is indeed theth row-vector field of the unknown matrix field we are seeking, we need to show that(F_j)³_j=1∈C¹(Ω;R³)and that this field does satisfy the partial differential equations (2.9) corresponding to the fixed integerused in the Cauchy problem (2.12)–(2.13).

Letxbe an arbitrary point inΩ and let the integeri∈ {1,2,3}be fixed in what follows. Then there existx¹∈Ω, a path γ∈C¹([0,1];R³)joiningx⁰tox¹inΩ,τ∈ ]0,1[, and an open intervalI⊂ [0,1]containingτsuch that

γ(t )=x+(t−τ )e_i fort∈I,

where e_iis theith basis vector inR³. Since each functionζ_jis continuously differentiable in[0,1]and satisfies the differential equation (2.12),

ζ_j(t )=ζ_j(τ )+(t−τ )dζ_j

dt (τ )+o(t−τ )=ζ_j(τ )+(t−τ )Γ_ij^p γ(τ )

ζ_p(τ )+o(t−τ ) for allt∈I. Equivalently,

F_j

x+(t−τ )e_i

=F_j(x)+(t−τ )Γ_ij^p(x)F_p(x)+o(t−x).

This relation shows that each functionF_j possesses partial derivatives in the setΩ, given at eachx∈Ωby

∂_iF_p(x)=Γ_ij^p(x)F_p(x).

Consequently, the matrix field(F_j)is of classC¹inΩ(its partial derivatives are continuous inΩ) and it satisfies the partial differential equations (2.9), as desired. Differentiating equations (2.9) shows that the matrix field(F_j)is in fact of classC² inΩ. ✷

Remark. The assumptions (2.8) made in Lemma 2 on the functionsΓ_ij^p=Γ_{j i}^pare thus sufficient conditions for equations (2.9) to have solutions. Conversely, a simple computation shows that they may be also viewed as necessary conditions, simply expressing that, if equations (2.9) have a solution, then necessarily∂_ikF_j=∂_kiF_j inΩ.

It is no surprise that these necessary conditions are of the same nature as those of Theorem 1, viz.,∂_ikg_j=∂_ijg_kinΩ. In order to conclude the proof of Theorem 1, it remains to adequately choose the “initial” conditionsF_j⁰ appearing in equations (2.10):

Lemma 3. LetΩbe a connected and simply connected subset ofR³and let(g_ij)∈C²(Ω;S³_>)be a matrix field satisfying equations (2.1), the functionsΓ_{ij q}, Γ_ij^p, andg^pqbeing defined as in (2.2).

Given an arbitrary pointx⁰∈Ω, let(F_j⁰)∈S³>denote the square root of the matrix(g⁰_ij):=(g_ij(x⁰))∈S³>.

Let(F_j)∈C²(Ω;M³)denote the solution of the corresponding system (2.9)–(2.10); this solution exists and is unique by Lemmas 1 and 2. Then there exists an immersionΘ=(Θ)∈C³(Ω;E³)such that

∂_jΘ=F_j and g_ij=∂_iΘ·∂_jΘ inΩ. (2.23)

Proof. (i) We first show that the three vector fields defined by g_j:=(F_j)³₌₁∈C²

Ω;R³

(2.24) satisfy

g_i·g_j=g_ij inΩ. (2.25)

To this end, we note that, by construction, these fields satisfy

∂_ig_j = Γ_ij^pgp inΩ, (2.26)

g_j(x⁰) = g⁰_j, (2.27)

where g⁰_jis thejth column vector of the matrix

g_ij(x⁰)∈S³_>. Hence the matrix field(g_i·g_j)∈C²(Ω;M³)satisfies

∂_k(g_i·g_j) = Γ_kj^m(g_m·g_i)+Γ_ki^m(g_m·g_j) inΩ, (2.28)

(g_i·g_j) x⁰

= g_ij⁰. (2.29)

By (2.2)a,∂_kg_ij=Γ_ikj+Γ_{j ki}, and by (2.2)_b–(2.2)c,Γ_{ij q}=gpqΓ_ij^p. Hence the matrix field(g_ij)∈C²(Ω;S³_>)satisfies

∂_kg_ij = Γ_kj^mg_mi+Γ_ki^mg_mj inΩ, (2.30)

g_ij x⁰

= g⁰_ij. (2.31)

Viewed as a system of partial differential equations, together with “initial” conditions, with respect to the matrix field (g_ij):Ω→M³, the system (2.30)–(2.31) can have at most one solution in the spaceC²(Ω;M³). To see this, letx¹∈Ω be distinct fromx⁰and letγ∈C¹([0,1];R³)be any path joiningx⁰tox¹inΩ(as in part (i) of the proof of Lemma 2). Then the nine functionsg_ij(γ(t )),0t1, satisfy a Cauchy problem for a linear system of nine ordinary differential equations and this Cauchy problem has at most one solution. By inspecting (2.28)–(2.29) and (2.30)–(2.31), we thus conclude that the fields g_i∈C²(Ω;R³)defined in (2.24) satisfy (2.25).

(ii) It thus remains to show that there exists an immersionΘ∈C³(Ω;E³)such that

∂_iΘ=g_i inΩ, (2.32)

where g_i are the vector fields defined in (2.24).

By (2.2),Γ_ij^p=Γ_{j i}^p; hence any solution(F_j)∈C²(Ω;M³)of the system (2.9)–(2.10) satisfies

∂_iF_j=∂_jF_i inΩ.

The open setΩbeing simply connected, Poincaré’s theorem shows that, for each integer, there exists a functionΘ∈C³(Ω) such that

∂_iΘ=F_i inΩ,

or, equivalently, such that the mappingΘ:=(Θ)∈C³(Ω;E³)satisfies equations (2.32). ThatΘ is an immersion follows from the assumed invertibility of the matrices(g_ij). The proof of Lemma 3, and consequently that of Theorem 2, are thus complete. ✷

Remark. The assumed positive definiteness of the matrices(g_ij)is used only in Lemma 3, for defining an ad hoc “initial”

vector g⁰_i in the system (2.26)–(2.27).

We now turn to the question of uniqueness. The proof given here is adapted from Ciarlet [6, Theorems 1.8–1 and 1.8–2] and Blume [1, Theorem 2.1].

Theorem 3 (rigidity theorem). LetΩ be a connected open subset ofR³and letΘ∈C¹(Ω;E³)andΘ∈C¹(Ω;E³)be two immersions whose associated metric tensors satisfy (with self-explanatory notations)

g_ij=g_ij inΩ. (2.33)

Then there exist a vector c∈E³and an orthogonal matrix Q of order three such that

Θ(x)=c+QΘ(x) for allx∈Ω. (2.34)

Proof. For convenience, the three-dimensional vector spaceR³is identified throughout this proof with the Euclidean space E³. In particular then,R³inherits the inner product and norm of E³. We also use the following notation: The matrix representing the Fréchet derivative atx∈Ωof a differentiable mappingΘ=(Θ):Ω→E³is denoted

∇Θ(x):=

∂_jΘ(x)

∈M³,

whereis the row index andjthe column index (equivalently,∇Θ(x)is the matrix of order three whosejth column vector is

∂_jΘ), and the spectral norm of a matrix A∈M³is denoted

|A| :=sup

|Av|;v∈R³,|v| =1 .

To begin with, we consider the special case whereΘ:Ω→E³=R³ is the identity mapping. Solving equations (2.33) reduces in this case to findingΘ∈C¹(Ω;E³)such that

∇Θ(x)^T∇Θ(x)=I for allx∈Ω. (2.35)

How to solve equations (2.35) is the object of parts (i) to (iii).

(i) We first establish that a mappingΘ∈C¹(Ω;E³)that satisfies (2.35) is locally an isometry: Given any pointx⁰∈Ω, there exists an open neighborhoodV ofx⁰contained inΩsuch that

Θ(y)−Θ(x)= |y−x| for allx, y∈V . (2.36)

LetBbe an open ball centered atx⁰and contained inΩ. Since the setBis convex, the mean-value theorem shows that Θ(y)−Θ(x) sup

z∈]x,y[∇Θ(z)|y−x| for allx, y∈B.

Since the spectral norm of an orthogonal matrix is one, we thus have

Θ(y)−Θ(x)|y−x| for allx, y∈B. (2.37)

Since the matrix∇Θ(x⁰)is invertible, the local inversion theorem shows that there exist an open neighborhoodV ofx⁰ contained inΩand an open neighborhoodVofΘ(x⁰)in E³such that the restriction ofΘtoVis aC¹-diffeomorphism fromV ontoV. Besides, there is no loss of generality in assuming thatV is contained inBand thatVis convex (to see this, apply the local inversion theorem first to the restriction ofΘtoB, thus producing a “first” neighborhoodVofx⁰contained inB, then to the restriction of the inverse mapping obtained in this fashion to an open ballVcentered atΘ(x⁰)and contained inΘ(V)).

LetΘ⁻¹:V→V denote the inverse mapping ofΘ:V →V. The chain rule applied to the relationΘ⁻¹(Θ(x))=xfor all x∈V then shows that

∇Θ⁻¹(x )=∇Θ(x)⁻¹ for allx=Θ(x), x∈V .

The matrix∇Θ⁻¹(x )being thus orthogonal for allx∈V, the mean-value theorem applied in the convex setVshows that Θ⁻¹(y )−Θ⁻¹(x )|y−x| for allx,y∈V ,

or equivalently, that

|y−x|Θ(y)−Θ(x) for allx, y∈V . (2.38)

SinceV ⊂B, inequalities (2.37) and (2.38) together yield the desired relation (2.36).

(ii) We next establish that, if a mappingΘ∈C¹(Ω;E³)is locally an isometry, in the sense that, given anyx⁰∈Ω, there exists an open neighborhoodV ofx⁰contained inΩsuch that relation (2.36) is satisfied, then its derivative is locally constant, in the sense that

∇Θ(x)=∇Θ x⁰

for allx∈V . (2.39)

The setV being that found in (i), let the differentiable functionF:V×V →Rbe defined for allx=(x_p)∈V and all y=(yp)∈V by

F(x, y):=

Θ(y)−Θ(x)

Θ(y)−Θ(x)

−(y−x)(y−x).

ThenF(x, y)=0 for allx, y∈V by (i). Hence G_i(x, y):=1

2

∂F

∂y_i(x, y)=∂Θ

∂y_i (y)

Θ(y)−Θ(x)

−δ_i(y−x)=0

for allx, y∈V. For a fixedy∈V, each functionG_i(·, y):V →Ris differentiable and its derivative vanishes. Consequently,

∂G_i

∂x_j(x, y)= −∂Θ

∂y_i (y)∂Θ

∂x_j (x)+δ_ij=0 for allx, y∈V , or equivalently, in matrix form,

∇Θ(y)^T∇Θ(x)=I for allx, y∈V . Lettingy=x⁰in this relation yields relation (2.39).

(iii) By (ii), the mapping ∇Θ:Ω →M³ is differentiable and its derivative vanishes in Ω. Therefore the mapping Θ:Ω→E³is twice differentiable and its second Fréchet derivative vanishes inΩ. The open setΩbeing connected, a classical result from differential calculus (see, e.g., [16, Theorem 3.7.10]) shows that the mappingΘis affine inΩ, i.e., there exists a vector c∈E³and a matrix Q∈M³such that

Θ(x)=c+Qox for allx∈Ω.

Besides, Q=∇Θ(x⁰)is an orthogonal matrix by (2.35).

(iv) We now consider the “general” equations (2.33), noting that they equivalently read

∇Θ(x)^T∇Θ(x)=∇Θ(x) ^T∇Θ(x) for allx∈Ω. (2.40)

Given any pointx⁰∈Ω, let the neighborhoodsV ofx⁰andVofΘ(x⁰)and the mappingΘ⁻¹:V→V be defined as in part (i) (by assumption, the mappingΘis an immersion; hence the matrix∇Θ(x⁰)is invertible).

Consider the composite mapping Φ:=Θ◦Θ⁻¹:V→E³. Clearly,Φ∈C¹(V;E³)and

∇Φ(x )=∇Θ(x) ∇Θ⁻¹(x )=∇Θ(x) ∇Θ(x)⁻¹ for allx=Θ(x), x∈V , so that, by relations (2.40),

∇Φ(x )^T∇Φ( x )=I for allx∈V .

By parts (i) to (iii), there thus exist a vector c∈R³and an orthogonal matrix Q of order three such that Φ(x )=Θ(x)=c+QΘ(x) for allx=Θ(x), x∈V ,

hence such that

Ξ(x):=∇Θ(x)∇Θ(x)⁻¹=Q for allx∈V .

The continuous mappingΞ:V →M³defined in this fashion is thus locally constant inΩ. As in part (iii), we conclude from the assumed connectedness ofΩthat the mappingΞis constant inΩ. Thus the proof is complete. ✷

Remarks. (1) In terms of metric tensors, parts (i) to (iii) of the above proof provide the solution to the equationsg_ij=δ_ijinΩ, while part (iv) provides the solution to the equationsg_ij=∂_iΘ·∂_jΘinΩ, whereΘ∈C¹(Ω;E³)is a given immersion.

(2) The classical Mazur–Ulam theorem asserts the following: LetΩ be a connected subset inRⁿ, and letΘ:Ω→Rⁿbe a mapping that satisfies

Θ(y)−Θ(x)= |y−x| for allx, y∈Ω.

Then there exist a vector c∈Rⁿand an orthogonal matrix Q of ordernsuch that Θ(x)=c+Qox for allx∈Ω.

Parts (ii) and (iii) of the above proof thus provide a proof of this theorem under the additional assumption that the mappingΘ is of classC¹.

(3) WhenR³is identified with E³as in the proof of Theorem 3, immersions such asΘ∈C¹(Ω;E³)may be thought of as deformations in the sense of “geometrically exact” three-dimensional elasticity (although they should then be in addition injective and orientation-preserving in order to qualify for this definition; for details, see, e.g., [6, Section 1.4]).