Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity

Assuming minimal regularity assumptions on the data, we revisit the classical problem of finding isometric immersions into the Minkowski spacetime for hypersurfaces of a Lorentzian manifold. Our approach encompasses metrics having Sobolev regularity and Riemann curvature defined in the distributional sense, only. It applies to timelike, spacelike, or null hypersurfaces with arbitrary signature that possibly changes from point to point.


I
Given a Lorentzian manifold (M, g) of dimension n + 1 or a hypersurface H ⊂ M, we study here whether it can be immersed isometrically in the Minkowski space M n+1 := (R n+1 , η). While this subject has been extensively studied within the class of smooth immersions in the context of Riemannian geometry, we are interested in the present paper in the case of Lorentzian manifolds and their hypersurfaces of arbitrary signature and in metrics with limited regularity in a Sobolev space. Our analysis will encompass metrics g of class W 1,p loc with p greater than the dimension of the underlying manifold, which, in fact, is the optimal regularity. We prove the existence of a global isometric immersion if the underlying manifold is simply connected. We also prove the uniqueness up to isometries of the Minkowski space and the stability of the immersion with respect to the metric.
One of our results is as follows: The corresponding immersion problem in the case of the Euclidian space has been recently revisited by Ciarlet and his collaborators; see, for instance, [3] and the references therein. Observe that, in the above theorem, the curvature of the manifold is defined in the sense of distributions only; for the definition of covariant derivatives and curvature tensors associated with metrics with limited regularity we rely on LeFloch and C. Mardare [5] and the references cited therein. The proof of Theorem 1.1 (in Section 3 below) will rely on earlier work by S. Mardare on Pfaff-type systems [6,8]. Previous arguments strongly used the assumption that the metric under consideration was Riemannian. To establish Theorem 1.1, we take account that the metric is Lorentzian; in fact, our argument immediately extends also to any pseudo-Riemannian manifold (with arbitrary signature).
Our second contribution concerns the immersion of hypersurfaces within the Minkowski space. We first consider the case of hypersurfaces with general signature, then we specialize our results to spacelike or timelike submanifolds. Consider a hypersurface H ⊂ M in a Lorentzian manifold with dimension n + 1 and a transverse field (henceforth called rigging) ℓ along H , that is, a vector field ℓ ∈ TM that is transversal to H . Then, the Levi-Civita connection of M can be decomposed into "tangent" and "transversal" components as follow: where ∇ , K, L, M are operators defined on TH . We say that an immersion ψ : H → M n+1 and a rigging ℓ ′ : H → TM n+1 preserve the operators ∇ , K, L, M if the Levi-Civita connection ∇ η of the Minkowski spacetime satisfies The main result established in Section 4 below is as follows. Observe that no assumption is made on the signature of the hypersurface. In particular, this theorem applies to hypersurfaces that are nowhere null. For such hypersurfaces we obtain in Section 5 a simpler derivation by choosing the rigging to be the unit normal vector field to H . Recall that the pull-back on H of the forms g and ∇n, denoted by g and K, are the first and second fundamental forms of H ⊂ M, respectively. In Section 5 below we will prove: Theorem 1.3 (Immersion of spacelike or timelike hypersurfaces). Suppose that H is simply connected and nowhere null and that (g , K) is of class W there exists proper isometries π,π of the Minkowski space such that The paper is organized as follows. In Section 2, we introduce our notation and provide some preliminary results. Sections 3, 4, and 5 are devoted to the proof of Theorems 1.1, 1.2, and 1.3, respectively, and contain slightly more general conclusions.

N  
For background on the analysis techniques (Sobolev spaces on manifolds, etc), we refer to [1,2].
Throughout this paper, all Greek indices and exponents vary in the set {0, 1, . . . , n}, while Latin indices, save for n, p, m, ℓ and q, vary in the set {1, . . . , n}. Einstein summation convention for repeated indices is used. A pseudo-Riemannian manifold is a smooth manifold M endowed with a metric, that is, a symmetric non-degenerate (0, 2)-tensor field g of constant index. A pseudo-Riemannian manifold is called Riemannian if its index is zero and Lorentzian if its index is one. The Minkowski spacetime M n+1 is the vector space R n+1 endowed with the Minkowski metric An isometric immersion of a pseudo-Riemannian manifold (M, g) into another pseudo-Riemannian manifold (M ′ , g ′ ) is an immersion ψ : M → M ′ that preserves the metric tensor, in the sense that ψ * g ′ = g. Here and in the sequel, notation such as ψ * g ′ and ψ * X denotes respectively the pull-back of g ′ and the push-forward of X by ψ; in particular, ψ * g ′ is the (0, 2)-tensor field on M defined by Related to the Minkowski space, we define the (n + 1) × (n + 1) matrix and the sets The matrices in O η (n + 1) and O η + (n + 1) are respectively called Minkowskiorthogonal and proper Minkowski-orthogonal matrices. By contrast with the set of (usual) orthogonal matrices, i.e., those matrices P that satisfy P T P = I, where I = diag(1, ..., 1) denotes the identity matrix of order n + 1, the set O η (n + 1) is not bounded. An isometry of the Minkowski spacetime M n+1 is a mapping where v ∈ R n+1 and Q ∈ O η (n + 1). Such an isometry π is called proper if Q ∈ O η + (n + 1). Let S(n + 1) denote the space of all symmetric real matrices. For any matrix G ∈ S(n + 1), let λ 0 ≤ λ 1 ≤ ... ≤ λ n denote its (real) eigenvalues. Let L(n + 1) := {G ∈ S(n + 1); λ 0 < 0 < λ 1 ≤ ... ≤ λ n } denote the set of all Lorentz matrices of order n+1 and define for all 0 < ε ≤ 1 the subsets L ε (n + 1) := {G ∈ L(n + 1); | det G| > ε and |G| < ε −1 }.
Note that L(n + 1) = lim ε→0 L ε (n + 1). It is easy to show that any matrix G ∈ L(n + 1) has a decomposition G = F T I η F for some invertible matrix F of order n + 1 (see the beginning of the proof of Lemma 2.1 below). But such a decomposition is not unique, for the matrix QF with Q ∈ O η (n + 1) also satisfies G = (QF) T I η (QF) (the converse is also true, i.e., if G =F T I ηF theñ F = QF for some Q ∈ O η (n + 1)). Since the set O(n + 1) is not bounded, this shows in particular that the norm of the matrix F in the above decomposition is not controlled by the norm of G. This is one of the reasons we need to prove the following lemma about the decomposition of Lorentz matrices: The mapping F depends on G but the constant C(ε, n) does not.
The following results on systems of first-order partial differential equations are due to S. Mardare [8, Theorems 1.1 and 4.1] and will be used throughout the article.

Theorem 2.2 (Existence and uniqueness for Pfaff-type systems).
Let Ω be a connected and simply connected open subset of R m , let p > m ≥ 2, let q ≥ 1 and ℓ ≥ 1, and let x 0 ∈ Ω and Y 0 ∈ R q×ℓ . Then, the system of matrix equations Although the above result was stated in [8, Theorem 1.1] for systems of the form ∂ α Y = YA α + C α , it is a simple matter to check that the technique therein extends and provides our more general result stated above. On the other hand, following closely the proof in [8, Theorems 4.1 and 6.8] we can also check the continuous dependence property stated now.

I   L 
Let (M, g) be a Lorentzian manifold of dimension n + 1. We want to investigate whether (M, g) can be immersed isometrically in the Minkowski space of the same dimension, i.e., whether there exists an immersion ψ : M → R n+1 such that ψ * η = g. In coordinates, this condition asserts that, for every local chart ϕ : U ⊂ M → Ω ⊂ R n+1 , the composite mapping f := ψ • ϕ −1 : Ω → R n+1 satisfies the following two conditions where g αβ = g(∂ α , ∂ β ) and I η is the matrix defined by (2.1). Here, ∂ α denote the tangent vector fields on M along the given coordinates x α . Observe that the matrix (g αβ ) is symmetric, invertible, and has exactly one negative eigenvalue at every point of Ω, since g is assumed to be Lorentzian. First, we prove that the Riemann curvature of a spacetime of dimension n + 1 must vanish if it is isometrically immersed in the Minkowski space of the same dimension. Proof. For smooth immersions, this is a classical result. We need to check that the classical arguments carry over to an immersion f that is only of class W 2,p loc with p > n + 1. This is in fact the minimal regularity for which the Riemann curvature tensor of g is well-defined as a distribution.
The assumptions of the lemma show that the mapping f = ψ • ϕ −1 belongs to the space W 2,p loc (Ω, R n+1 ) and that the covariant components g αβ of the metric g satisfy the relations where f α = f α denote the covariant and contravariant components of f with respect to a given Cartesian basis in R n+1 . Since ∂ f α ∂x β ∈ W 1,p loc (Ω) and this space is in fact an algebra (we use here the assumption p > n + 1), the above relation implies that g αβ ∈ W 1,p loc (Ω). In view of the definitions of the inverse of a matrix and of the Christoffel symbols, this implies that Hence, the Riemann curvature tensor of the metric g, defined by is well defined as the sum of a distribution in W −1,p loc (Ω) and a function L We will now show that R τ σαβ = 0 in the distributional sense. Define Since these vectors form a basis in R n+1 for every x ∈ Ω, the vector ∂F σ ∂x α (x) can be decomposed over this basis and so there exist coefficients C τ ασ such that ατ .
Solving this system shows that Hence, the partial derivatives which can be rewritten as matrix equations, that is, where Γ α := (Γ β ασ ) denote the matrix field with Γ β ασ as its component at the β-row and σ-column. In particular, we see that the mapping ψ preserves the connection.
To conclude, we now rely on the commutativity property for second derivatives of the fields F α and obtain: Combined with the relation ∂F σ ∂x α = Γ β ασ F β , this implies that, for every test function v ∈ D(Ω), Since the vectors fields F 0 , F 1 , ..., F n form a basis in W 1,p loc (Ω; R n+1 ), we can define the dual basis F 0 , F 1 , ..., F n and use the components of the vector field F γ w, where w is any test function in D(Ω), in lieu of v in the above equations. This implies that Since the test function w was arbitrary, the above equation is equivalent with the following equation between distributions in D ′ (Ω): The next lemma establishes a partial converse to Lemma 3.1, that is, if the Riemann curvature of the metric g vanishes, then the Lorentzian manifold (M, g) can be locally isometrically immersed in the Minkowski space of the same dimension. We recall that an isometry π : M n+1 → M n+1 of the Minkowski spacetime M n+1 was defined in (2.2). loc (U) such that ψ * η = g. Moreover, if π is an isometry of the Minkowski spacetime M n+1 , then π • ψ also satisfy (π • ψ) * η = g.
Proof. Let ϕ : U ⊂ M → Ω ⊂ R n+1 be a local chart such that Ω := ϕ(U) is connected and simply connected. The components g αβ of the metric with respect to this chart belongs to W 1,p loc (Ω). This space being an algebra, the inverse of matrix field (g αβ ) and the Christoffel symbols Γ σ αβ satisfy Finding an isometric immersion ψ : where d f denotes the gradient of f (the set Ω and the vectorial space R n+1 are equipped with the usual Cartesian coordinates and bases). Thus we are left with solving a nonlinear matrix equation in the unknown f . We are going to show that the equation (3.4) can be solved if the components of the matrix (g στ ) are the covariant components of a metric whose Riemann curvature tensor vanishes in the distributional sense. We first observe that the equation (3.4) can be solved "at one point" ). Solving the nonlinear equation (3.4) is now reduced to solving two linear systems. The first one is the Pfaff system where, for every α, Γ α := (Γ σ αβ ) denotes the matrix field with Γ σ αβ as its element at the σ-row and β-column. This system has a solution F ∈ W 1,p are equivalent to the fact that the Riemann curvature tensor of g vanishes in the distributional sense.
The second system is the Poincaré system where F is the solution of the first system (3.5). The above Poincaré system has a solution f ∈ W 2,p loc (Ω; R n+1 ) since the compatibility conditions that F must satisfy, namely where F α denotes the α-column of the matrix field F, are equivalent to relations Γ σ αβ = Γ σ βα in view of (3.5). Or these relations are indeed satisfied by the Christoffel symbols.
It remains now to prove that the solution f of the system (3.6) satisfies the equation Combining relations (3.5) and (3.6) shows But the matrix field (g στ ) also satisfies this equation, i.e., since this is simply a rewriting of the definition of the Christoffel symbols.
In addition, the definition of the matrix . Therefore the uniqueness part of Theorem 2.2 shows that the two solutions coincide, i.e., (d f Finally, if π is any isometry of the Minkowski spacetime (R n+1 , η), it is easily checked that π • ψ is also an isometric immersion of (U, g) into (R n+1 , η). Lemma 3.2 is a local existence result, in the sense that the isometric immersion ψ is defined only on a subset U ⊂ M. But we have a good control of the size of this neighborhood, since U only needs to be connected, simply connected, and defined by a single local chart. This fact, together with the uniqueness result of Lemma 3.5 below, allow us to establish a global existence result (i.e., the isometric immersion ψ is defined over the entire manifold M) when M is simply connected (see Lemma 3.6 below).
We next prove a stability result for the isometric immersion ψ defined by Lemma 3.2.
then there exists isometries π andπ of the Minkowski space M n+1 such that Proof. Let ϕ : U ⊂ M → Ω ⊂ R n+1 be a local chart such that U is connected with a smooth boundary. Let f := ψ • ϕ −1 andf :=ψ • ϕ −1 . In coordinates, the relations g = ψ * η andg =ψ * η read As in the proof of Lemma 3.1 (see (3.3)), these relations imply that f,f ∈ W 2,p loc (Ω; R n+1 ) and that the matrix fields d f and df satisfy the equations where Γ α is the matrix field defined from the metric g as in the proof of Lemma 3.2 andΓ α is defined in the same way but with g replaced byg.
Fix any point x ⋆ ∈ Ω. By Lemma 2.1, there exist matrices F andF such that (g στ (x ⋆ )) = F T I η F and (g στ (x ⋆ )) =F T I ηF , for some constant C depending on ε, n. Combined with Equations (3.7) (applied at the point x ⋆ ), the first relations above imply that the matrices Q = F(d f (x ⋆ )) −1 andQ =F(df (x ⋆ )) −1 (the matrices (d f (x ⋆ )) and (df (x ⋆ )) are invertible since ψ andψ are immersions) satisfy the relations Since the matrix fields (Q d f ) and (Q df ) satisfy the equations the stability property for Pfaff's systems stated in Section 2 implies that where the constant C depends only on n, ε and Ω.
On the other hand, the definition of the Christoffel symbols Γ σ αβ andΓ σ αβ shows that Usig this inequality and the second inequality of (3.8) in the previous inequality yields This inequality in turn implies (thanks to Poincaré-Wirtinger's inequality) that Since the matrices Q andQ are Minkowski-orthogonal, the mappings π : y ∈ R n+1 → v + Qy ∈ R n+1 and π : y ∈ R n+1 →ṽ +Qy ∈ R n+1 are isometries of the Minkowski spacetime M n+1 . Letting x = ϕ(p), p ∈ U, in the above inequality shows that This inequality still holds when U is replaced with the possibly larger set A since A is connected and A is compact.
An immediate consequence of the previous lemma is the following uniqueness result.  To begin with, we fix a point p 0 ∈ M and a local chart (ϕ 0 , U 0 ) at p 0 , where U 0 ⊂ M is a connected and simply connected neighborhood of p 0 and ϕ 0 : U 0 ⊂ M → ϕ 0 (U 0 ) ⊂ R n+1 . Then, Lemma 3.2 shows that there exists an isometric immersion ψ 0 : (U 0 , g) → (R n+1 , η).
Consider the set A of all pairs ∆, (ϕ m , U m ) K m=1 , with K = |∆| − 2 (where |∆| denotes the cardinality of the set ∆), that satisfies all the properties of the above paragraph with the only difference that now t K+1 ≤ 1 only (in other words, the last element of ∆ need not be equal to 1). It is enough to prove that sup Obviously, A is not empty and sup A t K+1 > 0. Assume on the contrary that s := sup A t K+1 < 1 and consider a local chart (ϕ, U) at γ(s) such that U is connected and simply connected. Since γ is continuous and U is a neighbourhood of γ(s), there exists 0 < ε < min{s, 1−s} such that γ([s − ε, s + ε]) ⊂ U. On the other hand, since s is a supremum, there exists a pair ∆, (ϕ m , U m ) L m=1 , where L = |∆| − 2, such that t L+1 ≥ s − ε (t L+1 being the last element of the set ∆). It is easy to check that the pair belongs to A. Hence, sup A t K+1 ≥ s + ε, which contradicts the definition of s. Therefore, s = 1.
In order to prove that s = 1 is in fact a maximum (i.e., sup A t K+1 = max A t K+1 ), we consider a local chart (ϕ, U) at p = γ(1) and we repeat the argument above with the only difference that now ε is chosen such that 0 < ε < 1 and γ([1 − ε, 1]) ⊂ U.
With γ, ∆, and (ϕ m , U m ) K m=1 constructed as above, Lemmas 3.2 and 3.5 allow us to successively choose isometric immersions in such a way that ψ m = ψ m−1 on the connected component containing γ(t m ) of U m ∩ U m−1 , for all m = 1, 2, ..., K.
Finally, we define a mapping ψ : (M, g) → (R n+1 , η) by ψ(p) := ψ K (p), where ψ K is defined as above. Indeed, one can see that this definition is independent on the choice of ψ K by using the simple-connectedness of the manifold M (a similar argument was used in [8]). Then, ψ is clearly an immersion since this property is local.

I      
We now turn our attention to hypersurfaces H ⊂ M in a Lorentzian manifold M of dimension n + 1. The basic question we addressed here is whether such a hypersurface can be immersed in the Minkowski spacetime (R n+1 , η) by means of an immersion ψ : H → R n+1 that preserves the geometry of the hypersurface.
To begin with, we recall the corresponding results in Riemannian geometry. Let H ⊂ M be a hypersurface in a Riemannian manifold (M, g) of dimension n + 1. Then, H is endowed with first and second fundamental forms, which together characterize the geometry of the hypersurface. The first fundamental form g is the pull-back of g on H , while the second fundamental form K is the pull-back of ∇ν on H , where ν is a normal form to the hypersurface H and ∇ is the Levi-Civita connection induced by g on M. Then, the question is whether there exists an immersion ψ : H → E n+1 , where E n+1 denotes the Euclidean space of dimension n + 1, such that g and K are the first and second fundamental forms induced by ψ. In the classical setting (i.e., all data are smooth), Bonnet's theorem asserts that such an immersion exists locally if and only if the fundamental forms satisfy the Gauss and Codazzi equations. Subsequently, this theorem was generalized by Hartman & Wintner [4] to fundamental forms (g , K) that are only in C 1 × C 0 , and finally by S. Mardare [7,8] to the case (g , K) ∈ W 1,p × L p with p > n, the latter regularity being optimal.
If (M, g) is now a Lorentzian manifold, the first fundamental form of a hypersurface H ⊂ M need not provide useful information about the geometry of H : If the hypersurface is null, then its normal vector field is a null vector lying in the tangent bundle to the hypersurface and the first fundamental form is degenerate. For this reason, the first fundamental form is replaced in the Lorentzian setting with a connection ∇ on H , defined by projecting the Levi-Civita connection ∇ (associated with the Lorentzian metric g) along a prescribed vector field ℓ transversal to H . Such a vector field ℓ ∈ TM is called a rigging and must satisfy It is convenient to normalize ℓ by imposing ν p , ℓ p = 1 for all p ∈ H , where ν denotes as usual a normal form on H chosen once and for all. For a mathematical presentation of the notion of rigging, we refer to LeFloch and C. Mardare [5]. Given a hypersurface H ⊂ M and a rigging ℓ ∈ TM, the rigging projection on H is denoted X ∈ TM → X ∈ TH and is defined by setting Then the connection ∇ on H is defined by and the second fundamental form K of H is defined by Beside the connection ∇ and the second fundamental form K which in a sense characterize the geometry of the hypersurface H , we must introduce additional operators characterizing the rigging vector ℓ. Guided by the decomposition we define the operators L : TH → TH and M : TH → R by We note that the operators ∇ , K, L, M can be also introduced via the decompositions: Our principal objective in this section is to prove that the operators ∇ , K, L, M characterize the pair (H , ℓ) formed by the hypersurface and the rigging vector field. We are going to generalize Bonnet's theorem in the Lorentzian setting to a pair (H , ℓ). Let M n+1 = (R n+1 , η) be the Minkowski spacetime of dimension n + 1 and let ∇ η be the Levi-Civita connection induced by η. If ψ : H → R n+1 is an immersion and ℓ ′ : H → TR n+1 is a rigging along H ′ := ψ(H ) (this definition makes sense since H ′ is locally a hypersurface of R n+1 ), then we define the operators ∇ ′ , K ′ , L ′ , M ′ on H ′ via the decompositions (similar to (4.2)): Equivalently, this means that Throughout this section, we assume that ψ ∈ W

(4.4)
Proof. Let Γ k ij denote the Christoffel symbols associated with the connection ∇ , so that Then, the definition of the operators ∇ , K, L, M shows that the assumption that the immersion ψ and rigging ℓ ′ preserve the operators ∇ , K, L, M shows that the function f : Since the second derivatives of ∂ f ∂x h and ℓ ′ commute, the above relations imply that Hence, we find (the relations below should be understood in the distributional sense, against test functions in D(Ω), as in the proof of Lemma 3.1) Using the fact that ℓ ′ , ∂x n is a basis in the tangent space TR n+1 , it is easily seen that the last two equations are equivalent to the generalized Gauss and Codazzi equations of the lemma.
We showed in the previous lemma that the generalized Gauss and Codazzi equations are necessary for the existence of an immersion ψ and rigging ℓ ′ preserving the operators ∇ , K, L, M. We now show that these equations are also sufficient, at least as far as the local existence of ψ and ℓ ′ is concerned. Proof. Let ϕ : U ⊂ H → Ω ⊂ R n be any local chart so that Ω := ϕ(U) is connected and simply connected. Let Γ k ij , K ij , L k j , M i denote the components of the operators ∇ , K, L, M with respect to this chart and note that all these components belong to the space L p loc (Ω). Finding an immersion ψ : U → M n+1 and a rigging ℓ ′ : U → TM n+1 that preserve the operators ∇ , K, L, M reduces to finding an immersion f : Ω → R n+1 and a rigging ℓ ′ : This will be done in two stages. First, we solve the Pfaff system where in the definition of the matrices A i the row index is k and the column index is h. In view of Theorem 2.2, this system has a solution F ∈ W 1,p loc (Ω; R (n+1)×(n+1) ), since the compatibility conditions ∂A j are precisely equivalent to the Gauss and Codazzi equations (4.4). Moreover, the solution to this system is unique provided we impose an initial condition, say F(x ⋆ ) = F ⋆ ∈ R (n+1)×(n+1) for some x ⋆ ∈ Ω (F ⋆ will be chosen later). Then, we solve the Poincaré system where F i is the i-th column vector of the matrix field F that satisfies the system (4.6). This Poincaré system has a solution f ∈ W 2,p loc (Ω, R n+1 ) since the compatibility conditions are satisfied. Indeed, they are equivalent to the equations Γ k ij = Γ k ji and K ij = K ji in view of equation (4.6); or the Christoffel symbols and the covariant components of the second fundamental form clearly satisfy these symmetry properties.
To conclude the proof, we remark that f is an immersion and the vector field ℓ ′ := F n+1 (that is, ℓ ′ is the (n + 1)-th column vector of the matrix field F that satisfies the system (4.6)) is transversal to f (Ω) provided we choose the matrix F ⋆ to be invertible. This is a consequence of the fact that the solution F of the Pfaff system (4.6) in invertible at every x ∈ Ω if and only if F is invertible at one single point; see S. Mardare [8, Lemma 6.1]. The desired immersion is then defined by ψ : Finally, if σ is any affine bijection of the Minkowski spacetime (R n+1 , η), it is easily checked (in view of Equations (4.6)) that σ • ψ and σ * ℓ ′ also satisfy the conclusions of the lemma.
Let x ⋆ ∈ Ω. Since the matrices F(x * ) andF(x * ) are invertible, the matrix Q :=F(x * )(F(x * )) −1 ∈ R (n+1)×(n+1) is well defined and is also invertible. Then the matrix fields (QF) andF satisfy (QF)(x * ) =F(x * ) and Then, the stability property for Pfaff systems stated in Section 2 shows that there exists a constant C = C(ε, Ω) such that This inequality in turn implies (thanks to Poincaré-Wirtinger's inequality) that there exists a vector v ∈ R n+1 such that Noting thatF = [(df )l ′ ] and QF = [Q(d f ) Qℓ ′ ] (the notation [...] designates the matrix obtained by adjoining the columns of the matrices listed inside the brackets), we deduce from the last two inequalities that Using the change of variables x = ϕ(p) in the last inequality above yields the inequality of the lemma over the set A = U, with a mapping σ : M n+1 → M n+1 defined by σ(y) = v + Qy for all y ∈ R n+1 . Finally, to derive the desired inequality over the possibly larger set A it suffices to use the connectedness of A and the compactness of A.
An immediate consequence of the previous lemma is the following uniqueness result.

I      
In this section, we specialize the results of the previous section to hypersurfaces that are nowhere null. If H ⊂ M is such a hypersurface, the normal vector field ν ♯ is transversal to H and therefore the metric g , induced on H by the metric g of the surrounding manifold M, is non-degenerate (see, e.g., LeFloch & Mardare [5, Theorem 6.1]). For this reason, we need not prescribe a rigging along H , the projection on H being made along ν ♯ . With the notation of the previous section, this is equivalent to choosing ℓ = ν ♯ . Therefore, the results of the previous section apply to a spacelike or timelike hypersurface H , but can be simplified since the operators ∇ , K, L, M are now uniquely determined by the fundamental forms of the hypersurface H . How this simplification can be achieved is the subject of this section.
Let (M, g) be a Lorentzian manifold of dimension n + 1. Denote by [X, Y] the Lie bracket of two vector fields, and by ∇ the Levi-Civita connection induced by g. Define the operator ♯ : θ ∈ T * M → θ ♯ ∈ TM by Consider an oriented non-null hypersurface in H ⊂ M and fix a unit normal form ν on H . Since H is non-null, the metric field g : TH × TH → R, also known as the first fundamental form of H , has either index zero (Riemannian metric) or index one (Lorentzian metric). The second fundamental form on H is defined as in the previous section by Then, the operators ∇ , K, L, M associated with rigging ℓ = ν ♯ are defined in terms of g , K as stated in the following lemma. The proof is omitted.
An immediate consequence of this lemma is that the generalized Gauss and Codazzi equations (see (4.4)) reduce to the classical equations, as follows: the generalized Gauss equations coincide with the Gauss equations, the Codazzi-1 equations coincide with Codazzi equations, Codazzi-2 equations are equivalent to the Codazzi equations, and Codazzi-3 equations are equivalent to the equations g hk K ih K jk = g hk K jh K ik expressing the symmetry of the "third" fundamental form of H .
If H is a hypersurface in M and ψ : (H , g ) → (R n+1 , η) is an immersion into the Minkowski spacetime, then the image H ′ = ψ(H ) is locally a hypersurface in R n+1 . Therefore, there exists a smooth unit normal form ν ′ : H → TR n+1 to the hypersurface H ′ (uniquely defined up to its sign); this means that ν ′ p , ψ * X p = 0 for all p ∈ H and X ∈ TH . For definiteness, we choose the sign of ν ′ to be that for which the immersion ψ preserve the orientation. The second fundamental form of H ′ is then defined as the pull-back on H ′ of the two-covariant tensor field ∇ η ν ′ .
Our objective in this section is to study whether there exists an immersion ψ : H → M n+1 that preserves the fundamental forms g , K of the hypersurface, that is, an immersion that satisfies ψ * η = g and ψ * K ′ = K.
Let ψ : (H , g ) → (R n+1 , η) be an isometric immersion, that is, an immersion that satisfies ψ * η = g . If H is nowhere null, then the hypersurface H ′ = ψ(H ) is also nowhere null (since the metric induced by η on H ′ is non-degenerate). Therefore, the unit normal vector field ν ′♯ : H → TR n+1 is transversal to H ′ , hence ℓ ′ := ν ′♯ is a rigging in the sense stated in Section 4. Then the operators ∇ ′ , K ′ , L ′ , M ′ associated with the immersion ψ and rigging ν ′♯ are well defined (see Section 4) and they satisfy the conclusions of Lemma 5.1. As a consequence, an immersion ψ : H → M n+1 preserves the fundamental forms of H if and only if (ψ, ν ′♯ ) preserves the operators In what follows, we may use local coordinates on the hypersurface: if ϕ : U ⊂ H → Ω ⊂ R n denotes a local chart at p ∈ H , then x i denotes a set of Cartesian coordinates in Ω and ∂ i denotes the vector field in TH tangent to the coordinate line x i . Note that the vector fields {∂ 1 , ..., ∂ n } form a basis of the tangent space TH , while {ν ♯ , ∂ 1 , ..., ∂ n } form a basis of the space TM. We denote g ij , K ij , Γ k ij , R k hij respectively the components in the local coordinates x i of g , K, ∇ , Riem g , where Riem g is the Riemann curvature tensor field associated with the metric g . Finally, let (g hk ) := (g ij ) −1 and K h i := g hk K ki . From Lemmas 5.1 and 4.2, we immediately deduce the following necessary conditions for the existence of an immersion preserving the fundamental forms.
We next show that these equations are sufficient for the existence of a local immersion ψ. Proof. The metric g may be either Riemannian or Lorentzian. The proof is the same in both cases, except for the value of the parameter λ = g(ν, ν) appearing below, which is equal to −1 if g is Riemannian and to 1 if g is Lorentzian.
Let ϕ : U ⊂ H → Ω ⊂ R n be any local chart, so that Ω := ϕ(U) is connected and simply connected. Fix a point x ⋆ ∈ Ω and an invertible matrix F ⋆ ∈ R (n+1)×(n+1) that satisfies the relation (this choice will be explained later) where I η was defined in (2.1). The existence of such a matrix F ⋆ is proved by Lemma 2.1. Since the matrix (−F ⋆ ) also satisfies the equation above, we may assume that det F ⋆ > 0. As in the proof of Lemma 4.3, there exists a unique matrix field F belonging to W 1,p loc (Ω; R (n+1)×(n+1) ) that satisfies the Pfaff system ∂F ∂x i = F C i a.e. in Ω, Then one can see that f is an immersion and satisfies (see (4.5)): where ℓ ′ denotes the (n + 1)-column vector field of the matrix field F.
We now prove that F satisfies By construction, this relation is satisfied at x ⋆ . Furthermore, on one hand, equation (5.2) implies that and, on the other hand, the definition of the Christoffel symbols Γ k ij shows that Therefore the uniqueness part of Theorem 2.2 shows that since both satisfies the same Cauchy problem.
Let us now prove that ψ := f • ϕ : U → M n+1 satisfies the conclusions of the lemma. First, the above equation shows that f is an isometric immersion; in other words, ψ preserves the first fundamental form of the hypersurface H . Second, it shows that the (n + 1)-column vector of F, denoted ℓ ′ , is orthogonal to the tangent space of the hypersurface f (Ω) in the Minkowski spacetime and that η(ℓ ′ , ℓ ′ ) = λ. This implies that either ℓ ′ = ν ′♯ , or ℓ ′ = −ν ′♯ . In fact, since det F(x ⋆ ) > 0 and F is continuous (thanks to the Sobolev embedding W 1,p loc (Ω) ⊂ C 0 (Ω) for p > n) on the connected set Ω, we have det F > 0 at every point of Ω; therefore ℓ ′ = ν ′♯ . Combined with the equations (5.4), which are nothing but the classical Gauss and Weingarten equations on the hypersurface f (Ω), this implies that K ij are the covariant components of the second fundamental form of f (Ω). In terms of the immersion ψ, this means that ψ preserves the second fundamental form of the hypersurface H .
Before extending the local immersion of Lemma 5.3 to a global one, we need prove the uniqueness of such an immersion. In fact, we will establish a stronger result, namely that the immersion ψ : H → M n+1 depends continuously on its fundamental forms. where v = −Q f (x ⋆ ) andṽ = −Qf (x ⋆ ). Since the matrices Q andQ are proper Minkowski-orthogonal, the mappings π : y ∈ R n+1 → v + Qy ∈ R n+1 and π : y ∈ R n+1 →ṽ +Qy ∈ R n+1 are proper isometries of the Minkowski spacetime M n+1 . Finally, letting x = ϕ(p), p ∈ U, in the above inequality shows that π •ψ − π • ψ W 2,p (U) ≤ C g − g W 1,p (U) + K − g L p (U) .
This inequality still holds when U is replaced with the possibly larger set A since A is connected and A is compact.
An immediate consequence of the previous lemma is the following uniqueness result. We are now in a position to establish a global version of Lemma 4.3. The proof is similar to that of Lemma 3.6 and is omitted.