International audienceWe establish that the linearized strains in curvilinear coordinates associated with a given displacement ﬁeld necessarily satisfy “Saint Venant equations in curvilinear coordinates”. Furthermore, we show that these equations are also sufficient, in the following sense: If a symmetric matrix ﬁeld deﬁned over a simply- connected open set satisﬁes the Saint Venant equations in curvilinear coordinates, then its coefficients are the linearized strains associated with a displacement ﬁeld. In addition, our proof provides an explicit algorithm for recovering such a displacement ﬁeld from its linear strains in curvilinear coordinates. This algorithm may be viewed as the linear counterpart of the reconstruction of an immersion from a given ﬂat Riemannian metric
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