If the Riemann–Christoffel tensor associated with a field of class of positive definite symmetric matrices of order three vanishes in a connected and simply connected open subset , then this field is the metric tensor field associated with a deformation of class of the set , uniquely determined up to isometries of . We establish here that the mapping defined in this fashion is continuous, for ad hoc metrizable topologies.
Si le tenseur de Riemann–Christoffel associé à un champ de classe de matrices symétriques définies positives d'ordre trois s'annule sur un ouvert connexe et simplement connexe , alors ce champ est celui du tenseur métrique associé à une déformation de classe de l'ensemble , déterminée de façon unique à une isométrie de près. On établit ici la continuité de l'application ainsi définie, pour des topologies métrisables convenables.
Published online:
Philippe G. Ciarlet 1, 2; Florian Laurent 3
@article{CRMATH_2002__335_5_489_0, author = {Philippe G. Ciarlet and Florian Laurent}, title = {Up to isometries, a deformation is a continuous function of its metric tensor}, journal = {Comptes Rendus. Math\'ematique}, pages = {489--493}, publisher = {Elsevier}, volume = {335}, number = {5}, year = {2002}, doi = {10.1016/S1631-073X(02)02504-9}, language = {en}, }
TY - JOUR AU - Philippe G. Ciarlet AU - Florian Laurent TI - Up to isometries, a deformation is a continuous function of its metric tensor JO - Comptes Rendus. Mathématique PY - 2002 SP - 489 EP - 493 VL - 335 IS - 5 PB - Elsevier DO - 10.1016/S1631-073X(02)02504-9 LA - en ID - CRMATH_2002__335_5_489_0 ER -
Philippe G. Ciarlet; Florian Laurent. Up to isometries, a deformation is a continuous function of its metric tensor. Comptes Rendus. Mathématique, Volume 335 (2002) no. 5, pp. 489-493. doi : 10.1016/S1631-073X(02)02504-9. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02504-9/
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[3] Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity, North-Holland, Amsterdam, 1988
[4] On the recovery of a surface with prescribed first and second fundamental forms, J. Math. Pure Appl., Volume 81 (2002), pp. 167-185
[5] P.G. Ciarlet, F. Laurent, On the continuity of a deformation as a function of its Cauchy–Green tensor, 2002, to appear
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