Rotation fields and the fundamental theorem of Riemannian geometry in ℝ3

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

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Original languageEnglish
Pages (from-to)415-421
Journal / PublicationComptes Rendus Mathematique
Volume343
Issue number6
Publication statusPublished - 15 Sept 2006

Abstract

Let Ω be a simply-connected open subset of ℝ3. We show in this Note that, if a smooth enough field U of symmetric and positive-definite matrices of order three satisfies the compatibility relation (due to C. Vallée)

CURL Λ + COF Λ = 0 in Ω

where the matrix field Λ is defined in terms of the field U by

Λ = (1/det U){U(CURL U)TU - (1/2)(tr[U(CURL U)T])U}

then there exists, typically in spaces such as Wloc2,∞(Ω; ℝ3) or C2(Ω; ℝ3), an immersion Θ : Ω → ℝ3 such that U2 = ΘTΘ in Ω. In this approach, one directly seeks the polar factorization Θ = RU of the gradient of the unknown immersion Θ in terms of a rotation R and a pure stretch U.