TY - JOUR
T1 - Another approach to the fundamental theorem of Riemannian geometry in R3, by way of rotation fields
AU - Ciarlet, Philippe G.
AU - Gratie, Liliana
AU - Iosifescu, Oana
AU - Mardare, Cristinel
AU - Vallée, Claude
PY - 2007/3
Y1 - 2007/3
N2 - In 1992, C. Vallée showed that the metric tensor field C = ∇ΘT∇Θ associated with a smooth enough immersion Θ : Ω → R3 defined over an open set Ω ⊂ R3 necessarily satisfies the compatibility relation CURL Λ + COF Λ = 0 in Ω, where the matrix field Λ is defined in terms of the field U = C1/2 by Λ = 1/det U {U (CURL U)TU - ½ (tr[U(CURL U)T])U}. The main objective of this paper is to establish the following converse: If a smooth enough field C of symmetric and positive-definite matrices of order three satisfies the above compatibility relation over a simply-connected open set Ω ⊂ R3, then there exists, typically in spaces such as W2, ∞loc (Ω ; R3) or C2 (Ω ; R3), an immersion Θ : Ω → R3 such that C = ∇ΘT∇Θ in Ω. This global existence theorem thus provides an alternative to the fundamental theorem of Riemannian geometry for an open set in R3, where the compatibility relation classically expresses that the Riemann curvature tensor associated with the field C vanishes in Ω. The proof consists in first determining an orthogonal matrix field R defined over Ω, then in determining an immersion Θ such that ∇Θ = RC1/2 in Ω, by successively solving two Pfaff systems. In addition to its novelty, this approach thus also possesses a more "geometrical" flavor than the classical one, as it directly seeks the polar factorization ∇Θ = RU of the immersion gradient in terms of a rotation R and a pure stretch U = C1/2. This approach also constitutes a first step towards the analysis of models in nonlinear three-dimensional elasticity where the rotation field is considered as one of the primary unknowns.
AB - In 1992, C. Vallée showed that the metric tensor field C = ∇ΘT∇Θ associated with a smooth enough immersion Θ : Ω → R3 defined over an open set Ω ⊂ R3 necessarily satisfies the compatibility relation CURL Λ + COF Λ = 0 in Ω, where the matrix field Λ is defined in terms of the field U = C1/2 by Λ = 1/det U {U (CURL U)TU - ½ (tr[U(CURL U)T])U}. The main objective of this paper is to establish the following converse: If a smooth enough field C of symmetric and positive-definite matrices of order three satisfies the above compatibility relation over a simply-connected open set Ω ⊂ R3, then there exists, typically in spaces such as W2, ∞loc (Ω ; R3) or C2 (Ω ; R3), an immersion Θ : Ω → R3 such that C = ∇ΘT∇Θ in Ω. This global existence theorem thus provides an alternative to the fundamental theorem of Riemannian geometry for an open set in R3, where the compatibility relation classically expresses that the Riemann curvature tensor associated with the field C vanishes in Ω. The proof consists in first determining an orthogonal matrix field R defined over Ω, then in determining an immersion Θ such that ∇Θ = RC1/2 in Ω, by successively solving two Pfaff systems. In addition to its novelty, this approach thus also possesses a more "geometrical" flavor than the classical one, as it directly seeks the polar factorization ∇Θ = RU of the immersion gradient in terms of a rotation R and a pure stretch U = C1/2. This approach also constitutes a first step towards the analysis of models in nonlinear three-dimensional elasticity where the rotation field is considered as one of the primary unknowns.
KW - Classical differential geometry
KW - Fundamental theorem of Riemannian geometry
KW - Nonlinear three-dimensional elasticity
KW - Pfaff systems
KW - Polar factorization
UR - http://www.scopus.com/inward/record.url?scp=33947172442&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-33947172442&origin=recordpage
U2 - 10.1016/j.matpur.2006.10.009
DO - 10.1016/j.matpur.2006.10.009
M3 - RGC 21 - Publication in refereed journal
VL - 87
SP - 237
EP - 252
JO - Journal des Mathematiques Pures et Appliquees
JF - Journal des Mathematiques Pures et Appliquees
SN - 0021-7824
IS - 3
ER -