By Ciarlet P.G.

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**Additional resources for An introduction to differential geometry with applications to elasticity (lecture notes)**

**Sample text**

Then there exist a vector c ∈ E3 and an orthogonal matrix Q ∈ O3 such that Θ(x) = c + QΘ(x) for all x ∈ Ω. Sect. 7] Uniqueness of immersions with the same metric tensor 37 Proof. For convenience, the three-dimensional vector space R3 is identiﬁed throughout this proof with the Euclidean space E3 . In particular then, R3 inherits the inner product and norm of E3 . The spectral norm of a matrix A ∈ M3 is denoted |A| := sup{|Ab|; b ∈ R3 , |b| = 1}. To begin with, we consider the special case where Θ : Ω → E3 = R3 is the identity mapping.

7-1) c ∈ R3 and Q ∈ O3 such that Φ(x) = c + QΘ(x) for all x ∈ Ω, so that ∇Φ(x) = Q∇Θ(x) for all x ∈ Ω. The relation ∇Θ(x0 ) = ∇Φ(x0 ) then implies that Q = I and the relation Θ(x0 ) = Φ(x0 ) in turn implies that c = 0. Remark. One possible choice for the matrix F0 is the square root of the symmetric positive-deﬁnite matrix C(x0 ). 7-1 constitutes the “classical” rigidity theorem, in that both immersions Θ and Θ are assumed to be in the space C 1 (Ω; E3 ). The next theorem is an extension, due to Ciarlet & C.

As Ω {−Γikq ∂j ϕ + Γijq ∂k ϕ + Γpij Γkqp ϕ − Γpik Γjqp ϕ} dx = 0 for all ϕ ∈ D(Ω). The existence result has also been extended “up to the boundary of the set Ω” by Ciarlet & C. Mardare [2004a]. More speciﬁcally, assume that the set Ω 36 Three-dimensional diﬀerential geometry [Ch. 1 satisﬁes the “geodesic property” (in eﬀect, a mild smoothness assumption on the boundary ∂Ω, satisﬁed in particular if ∂Ω is Lipschitz-continuous) and that the functions gij and their partial derivatives of order ≤ 2 can be extended by continuity to the closure Ω, the symmetric matrix ﬁeld extended in this fashion remaining positive-deﬁnite over the set Ω.