- Email: [email protected]
Extension of a Riemannian metric with vanishing curvature
C. R. Acad. Sci. Paris, Ser. I 338 (2004) 391–396
Differential Geometry
Extension of a Riemannian metric with vanishing curvature Philippe G. Ciarlet a , Cristinel Mardare b a Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong b Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France
Received 1 December 2003; accepted 15 December 2003 Presented by Robert Dautray
Abstract Let Ω be a connected and simply-connected open subset of Rn such that the geodesic distance in Ω is equivalent to the Euclidean distance. Let there be given a Riemannian metric (gij ) of class C 2 and of vanishing curvature in Ω, such that the Then there exists a connected open functions gij and their partial derivatives of order 2 have continuous extensions to Ω. of Rn containing Ω and a Riemannian metric (g˜ ij ) of class C 2 and of vanishing curvature in Ω that extends the subset Ω metric (gij ). To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved. Résumé Prolongement d’une métrique riemannienne à courbure nulle. Soit Ω un ouvert connexe et simplement connexe de Rn tel que la distance géodésique dans Ω soit équivalente à la distance euclidienne. Soit (gij ) une métrique riemannienne de classe C 2 et de courbure nulle dans Ω, telle que les fonctions gij et leurs dérivées partielles d’ordre 2 aient des extensions continues Alors il existe un ouvert connexe Ω de Rn contenant Ω et une métrique riemannienne (g˜ ij ) de classe C 2 et de courbure à Ω. qui prolonge la métrique (gij ). Pour citer cet article : P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 nulle dans Ω (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.
1. Preliminaries An integer n 2 is given once and for all, Latin indices and exponents vary in the set {1, 2, . . . , n}, and the summation convention with respect to repeated indices and exponents is used. The notations Sn and Sn> designate the space of all symmetric matrices, and the set of all positive-definite symmetric matrices, of order n. If Ω is an open subset of Rn , we define the set C 2 Ω; Sn> := C ∈ C 2 Ω; Sn ; C(x) ∈ Sn> for all x ∈ Ω .
E-mail addresses: [email protected] (P.G. Ciarlet), [email protected] (C. Mardare). 1631-073X/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved. doi:10.1016/j.crma.2003.12.017
392
P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 391–396
We define as follows spaces of functions, vector fields, or matrix fields, “of class C up to the boundary of Ω”: consists of all functions Definition 1.1. Let Ω be an open subset of Rn . For any integer 1, the space C (Ω) f ∈ C (Ω) that, together with all their partial derivatives ∂ α f, 1 |α| , can be extended by continuity to Ω. Rn ) and C (Ω; Sn ). Any continuous extension to Ω will be Analogous definitions hold for the spaces C (Ω; identified by a bar. We also define the set Sn> := C ∈ C 2 Ω; Sn ; C(x) . C 2 Ω; ∈ Sn> for all x ∈ Ω Let Ω be a connected open subset of Rn . Given two points x, y ∈ Ω, a path joining x to y in Ω is any mapping γ ∈ C 1 ([0, 1]; Rn) that satisfies γ (t) ∈ Ω for all t ∈ [0, 1] and γ (0) = x and γ (1) = y. Given a path γ joining x to y in Ω, its length is defined by 1 L(γ ) :=
γ (t) dt,
0
where | · | denotes the Euclidean norm in Rn . Let Ω be a connected open subset of Rn . The geodesic distance between two points x, y ∈ Ω is defined by dΩ (x, y) = inf L(γ ); γ is a path joining x to y in Ω . The following definition is in effect a mild regularity assumption on the boundary of an open subset of Rn : Definition 1.2. An open subset Ω of Rn satisfies the geodesic property if it is connected and, given any point x0 ∈ ∂Ω and any ε > 0, there exists δ = δ(x0 , ε) > 0 such that dΩ (x, y) < ε
for all x, y ∈ Ω ∩ B(x0 ; δ),
where B(x0 ; δ) := {y ∈ Rn ; |y − x| < δ}. Let a Riemannian metric (gij ) ∈ C 2 (Ω; Sn> ) be given over an open subset Ω of Rn . The Christoffel symbols of the second kind associated with this metric are then defined by 1 Γijk := g k (∂i gj + ∂j g i − ∂ gij ), where g k := (gij )−1 , 2 and the mixed components of its associated Riemann curvature tensor are defined by p
p
p
p
p
R·ij k := ∂j Γik − ∂k Γij + Γik Γj − Γij Γk . If this tensor vanishes in Ω and Ω is simply-connected, a classical result in differential geometry asserts that (gij ) is the metric tensor field of a manifold Θ(Ω) that is isometrically immersed in Rn . More specifically (see, e.g., Ciarlet and Larsonneur [2, Theorem 2] for an elementary and self-contained proof), there exists an immersion Θ ∈ C 3 (Ω; Rn ) that satisfies ∂i Θ(x) · ∂j Θ(x) = gij (x)
for all x ∈ Ω,
and, if in addition Ω is connected, such an immersion is unique up to isometries in Rn . In [3] (see [4] for a complete proof), we indicated how a manifold with boundary, i.e., a subset of Rn of the form can be likewise recovered from a metric tensor field that, together with its partial derivatives of order 2, Θ(Ω), in such a way that the continuous extensions of the matrices (gij ) can be continuously extended to the closure Ω
P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 391–396
393
More specifically, in [3,4] the above existence and uniqueness result is extended as remain positive-definite in Ω. follows “up to the boundary of Ω”: Theorem 1.3. Let Ω be a simply-connected open subset of Rn that satisfies the geodesic property (see Sn> ) (in the sense of Definition 1.1) that satisfies Definition 1.2). Let there be given a matrix field (gij ) ∈ C 2 (Ω; p
R·ij k = 0 in Ω. Rn ) (again in the sense of Definition 1.1) that satisfies (the notations ∂i Θ Then there exists a mapping Θ ∈ C 3 (Ω; and gij represent the continuous extensions of the fields ∂i Θ and of the functions gij , according to Definition 1.1): ∂i Θ(x) · ∂j Θ(x) = gij (x)
for all x ∈ Ω,
and such a mapping is unique up to isometries in Rn . 2. Another definition of the space C (Ω) The final objective of this Note is to provide sufficient conditions guaranteeing that a Riemannian metric (gij ) ∈ C 2 (Ω; Sn> ) with a Riemann curvature tensor vanishing in an open subset Ω of Rn can be extended to a Sn> ) on a connected open set Ω containing Ω, in such a way that the Riemann Riemannian metric (g˜ij ) ∈ C 2 (Ω; (see Theorem 3.1). curvature tensor associated with this extension still vanishes in Ω is needed (see Theorem 2.2). This is why we first To this end, another characterization of the space C (Ω) introduce another notion of “geodesic property”, stronger than that introduced in Definition 1.2. Definition 2.1. An open subset Ω of Rn satisfies the strong geodesic property if it is connected and there exists a constant CΩ such that dΩ (x, y) CΩ |x − y| for all x, y ∈ Ω, where dΩ designates the geodesic distance in Ω (cf. Section 1). Remarks. (1) Any connected open subset of Rn with a Lipschitz-continuous boundary satisfies the strong geodesic property; for a proof, see, e.g., Proposition 5.1 in Anicic, Le Dret and Raoult [1]. (2) The strong geodesic property clearly implies the geodesic property, but not conversely; consider, e.g., a bounded open subset of R2 whose boundary is a cardioid. The following theorem, which hinges in particular on a profound result of Whitney [7] shows that, when an introduced in Definition 1.1 admits a remarkably open set Ω satisfies the strong geodesic property, the space C (Ω) simple characterization. This result will in turn play a key role in the announced extension theorem. Theorem 2.2. Let Ω be an open subset of Rn that satisfies the strong geodesic property. Then for any integer of Definition 1.1 can be also defined as 1, the space C (Ω) = f |Ω ∈ C (Ω); f ∈ C Rn . C Ω Sketch of proof. The proof, which is only briefly sketched here (see [4] for a complete proof), hinges on the following steps. Note that the assumption that Ω satisfies the strong geodesic property is not needed until part (iv). (i) To begin with, we list some notations used throughout this proof. Given a multi-index α = (α1 , α2 , . . . , αn ) ∈ Nn , we let
394
P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 391–396
|α| :=
αi
and ∂ α :=
i
0 := (0, 0, . . . , 0) 0! := 1
∂ |α| ∂x1α1 ∂x2α2 . . . ∂xnαn
if |α| 1,
and ∂ 0 f := f,
and α! := (α1 !)(α2 !) · · · (αn !).
If x = (xi ) and y = (yi ) are two points in Rn , we let (y − x)0 := 1 and (y − x)α := (y1 − x1 )α1 (y2 − x2 )α2 · · · (yn − xn )αn . Concurrently with the multi-index notation ∂ α f for partial derivatives we also use the notations ∂i1 f :=
∂f ∂ mf , . . . , ∂i1 i2 ...im f := , ∂xi1 ∂xi1 ∂xi2 · · · ∂xim
with the understanding that, whenever a summation involves such indices i1 , i2 , . . . , im , then they range in the set {1, 2, . . . , n} independently of each other. (ii) Let Ω be a connected open subset of Rn , let x and y be two points in Ω, and let γ = (γi ) ∈ C 1 ([0, 1]; Rn) be a path joining x to y in Ω. Then any function f ∈ C m (Ω), m 1, satisfies the following identity, which may be viewed as a Taylor formula along a path: 1 1 ∂i1 f (x)(yi1 − xi1 ) + · · · + ∂i ...i f (x)(yi1 − xi1 ) · · · (yim−1 − xim−1 ) 1! (m − 1)! 1 m−1
1 t1 tm−2 tm−1 + ··· ∂i1 ...im f γ (tm ) γim (tm ) dtm γim−1 (tm−1 ) dtm−1 · · · dt2 γi 1 (t1 ) dt1 .
f (y) = f (x) +
0
0
0
0
(iii) The identity established in (ii) in turn implies the following estimate, which may be viewed as a generalized mean-valued theorem along a path: Let Ω be a connected open subset of Rn , let x and y be two points in Ω, let γ ∈ C 1 ([0, 1]; Rn) be a path joining x to y in Ω, and let a function f ∈ C m (Ω), m 1, be given. Then 1 α 1/2 1 β β m f (y) − ∂ f (z) − ∂ α f (x)2 L(γ ) ∂ sup f (x)(y − x) , β! α! z∈γ ([0,1]) |α|=m
|β|m
where L(γ ) denotes the length of the path γ . (iv) The strong geodesic property then allows to get rid of the dependence on the path γ in the estimate found in (iii), according to the following sharpened estimate: Let Ω be an open subset of Rn that satisfies the m 1, be given, the space C m (Ω) being defined as in strong geodesic property and let a function f ∈ C m (Ω), and any number ε > 0, there exists δ = δ(x0 , ε) such that Definition 1.1. Then, given any point x0 ∈ Ω 1 β f (x)(y − x)β ε|y − x|m for all x, y ∈ Ω f(y) − ∩ B(x0 ; δ), ∂ β! |β|m
and ∂ β f ∈ C 0 (Ω), 1 |β| m, denote the continuous extensions of the functions f ∈ C 0 (Ω) where f∈ C 0 (Ω) and ∂ β f ∈ C 0 (Ω). 1, according to Definition 1.1. According to a deep (v) Let there be given a function f in the space C (Ω), result of Whitney [7], f is also the restriction to Ω of a function in the space C (Rn ) if, for each multi-index α with the following property: For any points x, y ∈ Ω and satisfying 0 |α| , there exist functions fα ∈ C 0 (Ω) any multi-index α satisfying 0 |α| , let 1 Rα (y; x) := fα (y) − fα+β (x)(y − x)β . β! |β| −|α|
P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 391–396
395
and any number ε > 0, there exists δ = δ(x0 , ε) such that Then, given any point x0 ∈ Ω Rα (y; x) ε|y − x| −|α| for all x, y ∈ Ω ∩ B(x0 ; δ) and 0 |α| . and ε > 0 be given. Then the estimate of part (iv) applied to To verify that this is indeed the case, let x0 ∈ Ω each function ∂ α f , 0 |α| , shows that there exists δα = δα (x0 , ε) such that 1 β β α ∂αf − ∩ B(x0 ; δα ). ∂ (∂ f )(x)(y − x) ε|y − x| −|α| for all x, y ∈ Ω β! |β| −|α|
Since ∂ β (∂ α f )(x) = ∂ β+α f (x) for all x ∈ Ω, it also follows that ∂ β (∂ α f )(x) = ∂ β+α f (x) for all x ∈ Ω. α Therefore Whitney’s theorem can be applied, with fα := ∂ f and δ := min{δα ; 0 |α| }. ✷ 3. Extension of a Riemannian metric with a vanishing curvature We are now in a position to prove the announced extension result. The notations are the same as in Theorem 1.3. Theorem 3.1. Let Ω be a simply-connected open subset of Rn that satisfies the strong geodesic property and let Sn> ) that satisfies there be given a matrix field (gij ) ∈ C 2 (Ω; p
R·ij k = 0 in Ω. of Rn containing Ω and a matrix field (g˜ij ) ∈ C 2 (Ω; Sn> ) such that Then there exist a connected open subset Ω p = 0 in Ω, g˜ ij (x) = gij (x) for all x ∈ Ω and R ·ij k
∈ C 0 (Ω) denote the mixed components of the Riemann curvature tensor associated with where the functions R ·ij k the field (g˜ij ). p
Proof. Since Ω a fortiori satisfies the geodesic property and Ω is simply-connected, there exists by Theorem 1.3 Rn ) that satisfies a mapping Θ ∈ C 3 (Ω; ∂i Θ(x) · ∂j Θ(x) = gij (x) for all x ∈ Ω. ∈ C 3 (Rn ; Rn ) that Since Ω satisfies the strong geodesic property, there in turn exists by Theorem 2.2 a mapping Θ satisfies Θ(x) = Θ(x) for all x ∈ Ω. Let then · ∂j Θ(x) g˜ ij (x) := ∂i Θ(x)
for all x ∈ Rn ,
and define the set U := x ∈ Rn ; g˜ij (x) ∈ Sn> , (since g˜ij (x) = gij (x) for all x ∈ Ω). Finally, define the set Ω as the connected which is open in Rn and contains Ω hence the set Ω is open and connected. component of U that contains Ω; p of the Riemann curvature tensor associated with the field (g˜ij ) are Furthermore, the mixed components R ·ij k ⊂ U. since the matrices (g˜ij (x)) are by construction invertible for all x ∈ Ω well defined in the set Ω 3 n Because g˜ij (x) = ∂i Θ(x) · ∂j Θ(x) for all x ∈ Ω and the restriction Θ|Ω ∈ C (Ω; R ) is an immersion, the are simply the well-known necessary conditions that a Riemannian metric induced by an p = 0 in Ω relations R ·ij k immersion satisfies. ✷ A similar extension theorem for surfaces in R3 can be likewise established; see [5,6].
396
P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 391–396
Acknowledgement The work described in this Note was supported by a grant from the Research Grants Council of the Hong Kong Special Admimistrative Region, China [Project No. 9040869, CityU 100803].
References [1] S. Anicic, H. Le Dret, A. Raoult, The infinitesimal rigid displacement lemma in Lipschitz coordinates and application to shells with minimal regularity, in press. [2] P.G. Ciarlet, F. Larsonneur, On the recovery of a surface with prescribed first and second fundamental forms, J. Math. Pures Appl. 81 (2002) 167–185. [3] P.G. Ciarlet, C. Mardare, On the recovery of a manifold with boundary in Rn , C. R. Acad. Sci. Paris, Ser. I, in press. [4] P.G. Ciarlet, C. Mardare, Recovery of a manifold with boundary and its continuity as a function of its metric tensor, in press. [5] P.G. Ciarlet, C. Mardare, Extension of a surface in R3 , in press. [6] P.G. Ciarlet, C. Mardare, Recovery of a surface with boundary and its continuity as a function of its two fundamental forms, in press. [7] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934) 63–89.
Differential Geometry
Extension of a Riemannian metric with vanishing curvature Philippe G. Ciarlet a , Cristinel Mardare b a Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong b Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France
Received 1 December 2003; accepted 15 December 2003 Presented by Robert Dautray
Abstract Let Ω be a connected and simply-connected open subset of Rn such that the geodesic distance in Ω is equivalent to the Euclidean distance. Let there be given a Riemannian metric (gij ) of class C 2 and of vanishing curvature in Ω, such that the Then there exists a connected open functions gij and their partial derivatives of order 2 have continuous extensions to Ω. of Rn containing Ω and a Riemannian metric (g˜ ij ) of class C 2 and of vanishing curvature in Ω that extends the subset Ω metric (gij ). To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved. Résumé Prolongement d’une métrique riemannienne à courbure nulle. Soit Ω un ouvert connexe et simplement connexe de Rn tel que la distance géodésique dans Ω soit équivalente à la distance euclidienne. Soit (gij ) une métrique riemannienne de classe C 2 et de courbure nulle dans Ω, telle que les fonctions gij et leurs dérivées partielles d’ordre 2 aient des extensions continues Alors il existe un ouvert connexe Ω de Rn contenant Ω et une métrique riemannienne (g˜ ij ) de classe C 2 et de courbure à Ω. qui prolonge la métrique (gij ). Pour citer cet article : P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 nulle dans Ω (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.
1. Preliminaries An integer n 2 is given once and for all, Latin indices and exponents vary in the set {1, 2, . . . , n}, and the summation convention with respect to repeated indices and exponents is used. The notations Sn and Sn> designate the space of all symmetric matrices, and the set of all positive-definite symmetric matrices, of order n. If Ω is an open subset of Rn , we define the set C 2 Ω; Sn> := C ∈ C 2 Ω; Sn ; C(x) ∈ Sn> for all x ∈ Ω .
E-mail addresses: [email protected] (P.G. Ciarlet), [email protected] (C. Mardare). 1631-073X/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved. doi:10.1016/j.crma.2003.12.017
392
P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 391–396
We define as follows spaces of functions, vector fields, or matrix fields, “of class C up to the boundary of Ω”: consists of all functions Definition 1.1. Let Ω be an open subset of Rn . For any integer 1, the space C (Ω) f ∈ C (Ω) that, together with all their partial derivatives ∂ α f, 1 |α| , can be extended by continuity to Ω. Rn ) and C (Ω; Sn ). Any continuous extension to Ω will be Analogous definitions hold for the spaces C (Ω; identified by a bar. We also define the set Sn> := C ∈ C 2 Ω; Sn ; C(x) . C 2 Ω; ∈ Sn> for all x ∈ Ω Let Ω be a connected open subset of Rn . Given two points x, y ∈ Ω, a path joining x to y in Ω is any mapping γ ∈ C 1 ([0, 1]; Rn) that satisfies γ (t) ∈ Ω for all t ∈ [0, 1] and γ (0) = x and γ (1) = y. Given a path γ joining x to y in Ω, its length is defined by 1 L(γ ) :=
γ (t) dt,
0
where | · | denotes the Euclidean norm in Rn . Let Ω be a connected open subset of Rn . The geodesic distance between two points x, y ∈ Ω is defined by dΩ (x, y) = inf L(γ ); γ is a path joining x to y in Ω . The following definition is in effect a mild regularity assumption on the boundary of an open subset of Rn : Definition 1.2. An open subset Ω of Rn satisfies the geodesic property if it is connected and, given any point x0 ∈ ∂Ω and any ε > 0, there exists δ = δ(x0 , ε) > 0 such that dΩ (x, y) < ε
for all x, y ∈ Ω ∩ B(x0 ; δ),
where B(x0 ; δ) := {y ∈ Rn ; |y − x| < δ}. Let a Riemannian metric (gij ) ∈ C 2 (Ω; Sn> ) be given over an open subset Ω of Rn . The Christoffel symbols of the second kind associated with this metric are then defined by 1 Γijk := g k (∂i gj + ∂j g i − ∂ gij ), where g k := (gij )−1 , 2 and the mixed components of its associated Riemann curvature tensor are defined by p
p
p
p
p
R·ij k := ∂j Γik − ∂k Γij + Γik Γj − Γij Γk . If this tensor vanishes in Ω and Ω is simply-connected, a classical result in differential geometry asserts that (gij ) is the metric tensor field of a manifold Θ(Ω) that is isometrically immersed in Rn . More specifically (see, e.g., Ciarlet and Larsonneur [2, Theorem 2] for an elementary and self-contained proof), there exists an immersion Θ ∈ C 3 (Ω; Rn ) that satisfies ∂i Θ(x) · ∂j Θ(x) = gij (x)
for all x ∈ Ω,
and, if in addition Ω is connected, such an immersion is unique up to isometries in Rn . In [3] (see [4] for a complete proof), we indicated how a manifold with boundary, i.e., a subset of Rn of the form can be likewise recovered from a metric tensor field that, together with its partial derivatives of order 2, Θ(Ω), in such a way that the continuous extensions of the matrices (gij ) can be continuously extended to the closure Ω
P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 391–396
393
More specifically, in [3,4] the above existence and uniqueness result is extended as remain positive-definite in Ω. follows “up to the boundary of Ω”: Theorem 1.3. Let Ω be a simply-connected open subset of Rn that satisfies the geodesic property (see Sn> ) (in the sense of Definition 1.1) that satisfies Definition 1.2). Let there be given a matrix field (gij ) ∈ C 2 (Ω; p
R·ij k = 0 in Ω. Rn ) (again in the sense of Definition 1.1) that satisfies (the notations ∂i Θ Then there exists a mapping Θ ∈ C 3 (Ω; and gij represent the continuous extensions of the fields ∂i Θ and of the functions gij , according to Definition 1.1): ∂i Θ(x) · ∂j Θ(x) = gij (x)
for all x ∈ Ω,
and such a mapping is unique up to isometries in Rn . 2. Another definition of the space C (Ω) The final objective of this Note is to provide sufficient conditions guaranteeing that a Riemannian metric (gij ) ∈ C 2 (Ω; Sn> ) with a Riemann curvature tensor vanishing in an open subset Ω of Rn can be extended to a Sn> ) on a connected open set Ω containing Ω, in such a way that the Riemann Riemannian metric (g˜ij ) ∈ C 2 (Ω; (see Theorem 3.1). curvature tensor associated with this extension still vanishes in Ω is needed (see Theorem 2.2). This is why we first To this end, another characterization of the space C (Ω) introduce another notion of “geodesic property”, stronger than that introduced in Definition 1.2. Definition 2.1. An open subset Ω of Rn satisfies the strong geodesic property if it is connected and there exists a constant CΩ such that dΩ (x, y) CΩ |x − y| for all x, y ∈ Ω, where dΩ designates the geodesic distance in Ω (cf. Section 1). Remarks. (1) Any connected open subset of Rn with a Lipschitz-continuous boundary satisfies the strong geodesic property; for a proof, see, e.g., Proposition 5.1 in Anicic, Le Dret and Raoult [1]. (2) The strong geodesic property clearly implies the geodesic property, but not conversely; consider, e.g., a bounded open subset of R2 whose boundary is a cardioid. The following theorem, which hinges in particular on a profound result of Whitney [7] shows that, when an introduced in Definition 1.1 admits a remarkably open set Ω satisfies the strong geodesic property, the space C (Ω) simple characterization. This result will in turn play a key role in the announced extension theorem. Theorem 2.2. Let Ω be an open subset of Rn that satisfies the strong geodesic property. Then for any integer of Definition 1.1 can be also defined as 1, the space C (Ω) = f |Ω ∈ C (Ω); f ∈ C Rn . C Ω Sketch of proof. The proof, which is only briefly sketched here (see [4] for a complete proof), hinges on the following steps. Note that the assumption that Ω satisfies the strong geodesic property is not needed until part (iv). (i) To begin with, we list some notations used throughout this proof. Given a multi-index α = (α1 , α2 , . . . , αn ) ∈ Nn , we let
394
P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 391–396
|α| :=
αi
and ∂ α :=
i
0 := (0, 0, . . . , 0) 0! := 1
∂ |α| ∂x1α1 ∂x2α2 . . . ∂xnαn
if |α| 1,
and ∂ 0 f := f,
and α! := (α1 !)(α2 !) · · · (αn !).
If x = (xi ) and y = (yi ) are two points in Rn , we let (y − x)0 := 1 and (y − x)α := (y1 − x1 )α1 (y2 − x2 )α2 · · · (yn − xn )αn . Concurrently with the multi-index notation ∂ α f for partial derivatives we also use the notations ∂i1 f :=
∂f ∂ mf , . . . , ∂i1 i2 ...im f := , ∂xi1 ∂xi1 ∂xi2 · · · ∂xim
with the understanding that, whenever a summation involves such indices i1 , i2 , . . . , im , then they range in the set {1, 2, . . . , n} independently of each other. (ii) Let Ω be a connected open subset of Rn , let x and y be two points in Ω, and let γ = (γi ) ∈ C 1 ([0, 1]; Rn) be a path joining x to y in Ω. Then any function f ∈ C m (Ω), m 1, satisfies the following identity, which may be viewed as a Taylor formula along a path: 1 1 ∂i1 f (x)(yi1 − xi1 ) + · · · + ∂i ...i f (x)(yi1 − xi1 ) · · · (yim−1 − xim−1 ) 1! (m − 1)! 1 m−1
1 t1 tm−2 tm−1 + ··· ∂i1 ...im f γ (tm ) γim (tm ) dtm γim−1 (tm−1 ) dtm−1 · · · dt2 γi 1 (t1 ) dt1 .
f (y) = f (x) +
0
0
0
0
(iii) The identity established in (ii) in turn implies the following estimate, which may be viewed as a generalized mean-valued theorem along a path: Let Ω be a connected open subset of Rn , let x and y be two points in Ω, let γ ∈ C 1 ([0, 1]; Rn) be a path joining x to y in Ω, and let a function f ∈ C m (Ω), m 1, be given. Then 1 α 1/2 1 β β m f (y) − ∂ f (z) − ∂ α f (x)2 L(γ ) ∂ sup f (x)(y − x) , β! α! z∈γ ([0,1]) |α|=m
|β|m
where L(γ ) denotes the length of the path γ . (iv) The strong geodesic property then allows to get rid of the dependence on the path γ in the estimate found in (iii), according to the following sharpened estimate: Let Ω be an open subset of Rn that satisfies the m 1, be given, the space C m (Ω) being defined as in strong geodesic property and let a function f ∈ C m (Ω), and any number ε > 0, there exists δ = δ(x0 , ε) such that Definition 1.1. Then, given any point x0 ∈ Ω 1 β f (x)(y − x)β ε|y − x|m for all x, y ∈ Ω f(y) − ∩ B(x0 ; δ), ∂ β! |β|m
and ∂ β f ∈ C 0 (Ω), 1 |β| m, denote the continuous extensions of the functions f ∈ C 0 (Ω) where f∈ C 0 (Ω) and ∂ β f ∈ C 0 (Ω). 1, according to Definition 1.1. According to a deep (v) Let there be given a function f in the space C (Ω), result of Whitney [7], f is also the restriction to Ω of a function in the space C (Rn ) if, for each multi-index α with the following property: For any points x, y ∈ Ω and satisfying 0 |α| , there exist functions fα ∈ C 0 (Ω) any multi-index α satisfying 0 |α| , let 1 Rα (y; x) := fα (y) − fα+β (x)(y − x)β . β! |β| −|α|
P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 391–396
395
and any number ε > 0, there exists δ = δ(x0 , ε) such that Then, given any point x0 ∈ Ω Rα (y; x) ε|y − x| −|α| for all x, y ∈ Ω ∩ B(x0 ; δ) and 0 |α| . and ε > 0 be given. Then the estimate of part (iv) applied to To verify that this is indeed the case, let x0 ∈ Ω each function ∂ α f , 0 |α| , shows that there exists δα = δα (x0 , ε) such that 1 β β α ∂αf − ∩ B(x0 ; δα ). ∂ (∂ f )(x)(y − x) ε|y − x| −|α| for all x, y ∈ Ω β! |β| −|α|
Since ∂ β (∂ α f )(x) = ∂ β+α f (x) for all x ∈ Ω, it also follows that ∂ β (∂ α f )(x) = ∂ β+α f (x) for all x ∈ Ω. α Therefore Whitney’s theorem can be applied, with fα := ∂ f and δ := min{δα ; 0 |α| }. ✷ 3. Extension of a Riemannian metric with a vanishing curvature We are now in a position to prove the announced extension result. The notations are the same as in Theorem 1.3. Theorem 3.1. Let Ω be a simply-connected open subset of Rn that satisfies the strong geodesic property and let Sn> ) that satisfies there be given a matrix field (gij ) ∈ C 2 (Ω; p
R·ij k = 0 in Ω. of Rn containing Ω and a matrix field (g˜ij ) ∈ C 2 (Ω; Sn> ) such that Then there exist a connected open subset Ω p = 0 in Ω, g˜ ij (x) = gij (x) for all x ∈ Ω and R ·ij k
∈ C 0 (Ω) denote the mixed components of the Riemann curvature tensor associated with where the functions R ·ij k the field (g˜ij ). p
Proof. Since Ω a fortiori satisfies the geodesic property and Ω is simply-connected, there exists by Theorem 1.3 Rn ) that satisfies a mapping Θ ∈ C 3 (Ω; ∂i Θ(x) · ∂j Θ(x) = gij (x) for all x ∈ Ω. ∈ C 3 (Rn ; Rn ) that Since Ω satisfies the strong geodesic property, there in turn exists by Theorem 2.2 a mapping Θ satisfies Θ(x) = Θ(x) for all x ∈ Ω. Let then · ∂j Θ(x) g˜ ij (x) := ∂i Θ(x)
for all x ∈ Rn ,
and define the set U := x ∈ Rn ; g˜ij (x) ∈ Sn> , (since g˜ij (x) = gij (x) for all x ∈ Ω). Finally, define the set Ω as the connected which is open in Rn and contains Ω hence the set Ω is open and connected. component of U that contains Ω; p of the Riemann curvature tensor associated with the field (g˜ij ) are Furthermore, the mixed components R ·ij k ⊂ U. since the matrices (g˜ij (x)) are by construction invertible for all x ∈ Ω well defined in the set Ω 3 n Because g˜ij (x) = ∂i Θ(x) · ∂j Θ(x) for all x ∈ Ω and the restriction Θ|Ω ∈ C (Ω; R ) is an immersion, the are simply the well-known necessary conditions that a Riemannian metric induced by an p = 0 in Ω relations R ·ij k immersion satisfies. ✷ A similar extension theorem for surfaces in R3 can be likewise established; see [5,6].
396
P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 391–396
Acknowledgement The work described in this Note was supported by a grant from the Research Grants Council of the Hong Kong Special Admimistrative Region, China [Project No. 9040869, CityU 100803].
References [1] S. Anicic, H. Le Dret, A. Raoult, The infinitesimal rigid displacement lemma in Lipschitz coordinates and application to shells with minimal regularity, in press. [2] P.G. Ciarlet, F. Larsonneur, On the recovery of a surface with prescribed first and second fundamental forms, J. Math. Pures Appl. 81 (2002) 167–185. [3] P.G. Ciarlet, C. Mardare, On the recovery of a manifold with boundary in Rn , C. R. Acad. Sci. Paris, Ser. I, in press. [4] P.G. Ciarlet, C. Mardare, Recovery of a manifold with boundary and its continuity as a function of its metric tensor, in press. [5] P.G. Ciarlet, C. Mardare, Extension of a surface in R3 , in press. [6] P.G. Ciarlet, C. Mardare, Recovery of a surface with boundary and its continuity as a function of its two fundamental forms, in press. [7] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934) 63–89.