By Philippe G. Ciarlet

An advent to Differential Geometry with functions to Elasticity

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To see this, let for instance Ω be an open ball centered at the origin in R3 , let Θ(x) = (x1 x22 , x2 , x3 ) and let Θ(x) = Θ(x) if x2 ≥ 0 and Θ(x) = (−x1 x22 , −x2 , x3 ) if x2 < 0 (this counterexample was kindly communicated to the author by Herv´e Le Dret). (6) If a mapping Θ ∈ C 1 (Ω; E3 ) satisﬁes det ∇Θ > 0 in Ω, then Θ is an immersion. Conversely, if Ω is a connected open set and Θ ∈ C 1 (Ω; E3 ) is an immersion, then either det ∇Θ > 0 in Ω or det ∇Θ < 0 in Ω. 7-3 is simply intended to ﬁx ideas (a similar result clearly holds under the other assumption).

1 CONTINUITY OF AN IMMERSION AS A FUNCTION OF ITS METRIC TENSOR Let Ω be a connected and simply-connected open subset of R3 . 6-1, a well-deﬁned element F (C) in the quotient set C 3 (Ω; E3 )/R, where (Θ, Θ) ∈ R means that there exist a vector a ∈ E3 and a matrix Q ∈ O3 such that Θ(x) = a + QΘ(x) for all x ∈ Ω. A natural question thus arises as to whether there exist natural topologies on the space C 2 (Ω; S3 ) and on the quotient set C 3 (Ω; E3 )/R such that the mapping F deﬁned in this fashion is continuous.

At a mapping Θ that is isometrically equivalent to the identity mapping of Ω. The recent and noteworthy continuity result of Reshetnyak [2003] for quasi-isometric mappings is in a sense complementary to the above one (it also deals with Sobolev type norms). Chapter 2 DIFFERENTIAL GEOMETRY OF SURFACES INTRODUCTION We saw in Chapter 1 that an open set Θ(Ω) in E3 , where Ω is an open set in R3 and Θ : Ω → E3 is a smooth injective immersion, is unambiguously deﬁned (up to isometries of E3 ) by a single tensor ﬁeld, the metric tensor ﬁeld, whose covariant components gij = gji : Ω → R are given by gij = ∂i Θ · ∂j Θ.