By Julian Lowell Coolidge

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E. in Ω, together imply that ∇Φ(x) ∈ O3+ for almost all x ∈ U . By (i), there thus exist c ∈ E3 and Q ∈ O3+ such that Φ(x) = Θ(x) = c + Q ox for almost all x = Θ(x) ∈ U , or equivalently, such that Ξ(x) := ∇Θ(x)∇Θ(x)−1 = Q for almost all x ∈ U. Since the point x0 ∈ Ω is arbitrary, this relation shows that Ξ ∈ L1loc (Ω). By a classical result from distribution theory (cf. 6]), Sect. 8] Uniqueness of immersions with the same metric tensor 37 we conclude from the assumed connectedness of Ω that Ξ(x) = Q for almost all x ∈ Ω, and consequently that Θ(x) = c + QΘ(x) for almost all x ∈ Ω.

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). Three-dimensional diﬀerential geometry 40 [Ch.

7-1), of which Liouville’s theorem is the special case corresponding to Θ(x) = x for all x ∈ Ω. (3) The result established in part (i) of the above proof asserts that, given a connected open subset Ω of R3 , if a mapping Θ ∈ H 1 (Ω; E3 ) is such that ∇Θ(x) ∈ O3+ for almost all x ∈ Ω, then there exist c ∈ E3 and Q ∈ O3+ such that Θ(x) = c + Qox for almost all x ∈ Ω. This result thus constitutes a generalization of Liouville’s theorem. 7-3), an assumption about the sign of det ∇Θ becomes necessary.