An introduction to differential geometry with applications by Ciarlet P.G. By Ciarlet P.G.

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6] Existence of an immersion with a prescribed metric tensor 33 Consequently, the matrix ﬁeld (F j ) is of class C 1 in Ω (its partial derivatives are continuous in Ω) and it satisﬁes the partial diﬀerential equations ∂i F j = Γpij F p in Ω, as desired. Diﬀerentiating these equations shows that the matrix ﬁeld (F j ) is in fact of class C 2 in Ω. In order to conclude the proof of the theorem, it remains to adequately choose the initial values F 0j at x0 in step (ii). (iii) Let Ω be a connected and simply-connected open subset of R3 and let (gij ) ∈ C 2 (Ω; S3> ) be a matrix ﬁeld satisfying ∂j Γikq − ∂k Γijq + Γpij Γkqp − Γpik Γjqp = 0 in Ω, the functions Γijq , Γpij , and g pq being deﬁned by Γijq := 1 (∂j giq + ∂i gjq − ∂q gij ), 2 Γpij := g pq Γijq , (g pq ) := (gij )−1 .

8] An immersion as a function of its metric tensor 49 that the mappings Θn uniformly converge on every compact subset of Ω toward a limit Θ ∈ C 1 (Ω; E3 ) that satisﬁes ∇Θ(x) = lim ∇Θn (x) = I for all x ∈ Ω. n→∞ This shows that (Θ − id) is a constant mapping since Ω is connected. Consequently, Θ = id since in particular Θ(x0 ) = limn→∞ Θn (x0 ) = x0 . We have thus established that lim |Θn − id|0,K = 0 for all K n→∞ Ω. 8-1. We next establish the sequential continuity of the mapping F at those matrix ﬁelds C ∈ C 2 (Ω; S3> ) that can be written as C = ∇ΘT ∇Θ with an injective mapping Θ ∈ C 3 (Ω; E3 ).

A1 ∧ a2 | The vector ﬁelds ai : ω → R3 deﬁned in this fashion constitute the covariant bases along the surface ω, while the vector ﬁelds ai : ω → R3 deﬁned by the relations ai · aj = δji constitute the contravariant bases along ω. 5.