By Ciarlet P.G.

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N Given x = (xi ) ∈ Ω, let ∂i = ∂/∂xi . Let Rqijk denote the functions conn structed from the functions gij in the same way that the functions Rqijk are 50 Three-dimensional diﬀerential geometry [Ch. 1 constructed from the functions gij . 8-1 applied over the set Ω shows that n there exist mappings Θ ∈ C 3 (Ω; E3 ) satisfying n n n in Ω, n ≥ 0, ∂i Θ · ∂j Θ = gij such that lim n→∞ n Θ − id b 3,K = 0 for all K Ω, where id denotes the identity mapping of the set Ω, the space E3 being identiﬁed here with R3 .

6). Another extension, again motivated by nonlinear three-dimensional elasticity, is the following: Let Ω be a bounded and connected subset of R3 , and let B be an elastic body with Ω as its reference conﬁguration. e. in Ω as a possible deformation Sect. 8] An immersion as a function of its metric tensor 57 of B when B is subjected to ad hoc applied forces and boundary conditions. The almost-everywhere injectivity of Θ (understood in the sense of Ciarlet & Neˇcas [1987]) and the restriction on the sign of det ∇Θ mathematically express (in an arguably weak way) the non-interpenetrability and orientation-preserving conditions that any physically realistic deformation should satisfy.

Note that the functions Rqijk occurring in their statements are n in the same way that the funcmeant to be constructed from the functions gij tions Rqijk are constructed from the functions gij . To begin with, we establish the sequential continuity of the mapping F at C = I. 8-1. Let Ω be a connected and simply-connected open subset of n n ) ∈ C 2 (Ω; S3> ), n ≥ 0, be matrix ﬁelds satisfying Rqijk = 0 in R3 . Let Cn = (gij Ω, n ≥ 0, such that lim n→∞ Cn − I 2,K = 0 for all K Ω. Then there exist immersions Θn ∈ C 3 (Ω; E3 ) satisfying (∇Θn )T ∇Θn = Cn in Ω, n ≥ 0, such that lim n→∞ Θn − id 3,K = 0 for all K Ω where id denotes the identity mapping of the set Ω, the space R3 being identiﬁed here with E3 (in other words, id(x) = x for all x ∈ Ω).