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T ∂λ ∂λ ∂t Let ζ(·, λ) = (ζj (·, λ)) ∈ C 1 ([0, 1]; R3 ) denote for each 0 ≤ λ ≤ 1 the solution of the Cauchy problem corresponding to the path G(·, λ) joining x0 to x1 . We thus have ∂ζj ∂Gi (t, λ) = Γpij (G(t, λ)) (t, λ)ζp (t, λ) for all 0 ≤ t ≤ 1, 0 ≤ λ ≤ 1, ∂t ∂t ζj (0, λ) = ζj0 for all 0 ≤ λ ≤ 1. Sect. 6] Existence of an immersion with a prescribed metric tensor 31 Our objective is to show that ∂ζj (1, λ) = 0 for all 0 ≤ λ ≤ 1, ∂λ as this relation will imply that ζj (1, 0) = ζj (1, 1), as desired.

6-1). Then there exist immersions Θn ∈ C 3 (Ω; E3 ) satisfying (∇Θn )T ∇Θn = Cn in Ω, n ≥ 0, such that lim n→∞ Θn − Θ 3,K = 0 for all K Ω. Proof. The proof is broken into four parts. In what follows, C and Cn designate matrix ﬁelds possessing the properties listed in the statement of the theorem. (i) Let Θ ∈ C 3 (Ω; E3 ) be any mapping that satisﬁes ∇ΘT ∇Θ = C in Ω. Then there exist a countable number of open balls Br ⊂ Ω, r ≥ 1, such that ∞ r Ω = r=1 Br and such that, for each r ≥ 1, the set s=1 Bs is connected and the restriction of Θ to Br is injective.

For each integer , we may thus unambiguously deﬁne a vector ﬁeld (F j ) : Ω → R3 by letting F j (x1 ) := ζj (1) for any x1 ∈ Ω, where γ ∈ C 1 ([0, 1]; R3 ) is any path joining x0 to x1 in Ω and the vector ﬁeld (ζj ) ∈ C 1 ([0, 1]) is the solution to the Cauchy problem dγ i dζj (t) = Γpij (γ(t)) (t)ζp (t), 0 ≤ t ≤ 1, dt dt ζj (0) = ζj0 , corresponding to such a path. To establish that such a vector ﬁeld is indeed the -th row-vector ﬁeld of the unknown matrix ﬁeld we are seeking, we need to show that (F j )3j=1 ∈ C 1 (Ω; R3 ) and that this ﬁeld does satisfy the partial diﬀerential equations ∂i F j = Γpij F p in Ω corresponding to the ﬁxed integer used in the above Cauchy problem.