Algebraic topology - old and new: M.M.Postnikov memorial by Golasinski M., et al. (eds.)

By Golasinski M., et al. (eds.)

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Since each function ζj is continuously difdγ i dζj (t) = Γpij (γ(t)) (t)ζp (t) for all 0 ≤ t ≤ 1, ferentiable in [0, 1] and satisfies dt dt we have dζj (τ ) + o(t − τ ) dt = ζj (τ ) + (t − τ )Γpij (γ(τ ))ζp (τ ) + o(t − τ ) ζj (t) = ζj (τ ) + (t − τ ) for all t ∈ I. Equivalently, F j (x + (t − τ )ei ) = F j (x) + (t − τ )Γpij (x)F p (x) + o(t − x). This relation shows that each function F the set Ω, given at each x ∈ Ω by j possesses partial derivatives in ∂i F p (x) = Γpij (x)F p (x). Sect. 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.

Mardare [2004] has shown that the existence 2,∞ (Ω), with a resulting mapping Θ in the space theorem still holds if gij ∈ Wloc 2,∞ d Wloc (Ω; E ). , as Ω {−Γikq ∂j ϕ + Γijq ∂k ϕ + Γpij Γkqp ϕ − Γpik Γjqp ϕ} dx = 0 for all ϕ ∈ D(Ω). The existence result has also been extended “up to the boundary of the set Ω” by Ciarlet & C. Mardare [2004a]. More specifically, assume that the set Ω 36 Three-dimensional differential geometry [Ch. 1 satisfies the “geodesic property” (in effect, a mild smoothness assumption on the boundary ∂Ω, satisfied in particular if ∂Ω is Lipschitz-continuous) and that the functions gij and their partial derivatives of order ≤ 2 can be extended by continuity to the closure Ω, the symmetric matrix field extended in this fashion remaining positive-definite over the set Ω.

8-5. Let Ω be a connected and simply-connected open subset of R3 . Let C02 (Ω; S3> ) := {(gij ) ∈ C 2 (Ω; S3> ); Rqijk = 0 in Ω}, and, given any matrix field C = (gij ) ∈ C02 (Ω; S3> ), let F (C) ∈ C˙3 (Ω; E3 ) denote the equivalence class modulo R of any Θ ∈ C 3 (Ω; E3 ) that satisfies ∇ΘT ∇Θ = C in Ω. Then the mapping F : {C02 (Ω; S3> ); d2 } −→ {C˙ 3 (Ω; E3 ); d˙3 } defined in this fashion is continuous. Proof. Since {C02 (Ω; S3> ); d2 } and {C˙3 (Ω; E3 ); d˙3 } are both metric spaces, it suffices to show that convergent sequences are mapped through F into convergent sequences.

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