One of the intrinsic methods in elasticity is to consider the Cauchy-Green tensor as the primary unknown, instead of the deformation realizing this tensor, as in the classical approach.
Then one can ask whether it is possible to recover the deformation from its Cauchy-Green tensor. From a differential geometry viewpoint, this amounts to finding an isometric immersion of a Riemannian manifold into the Euclidian space of the same dimension, say d. It is well known that this is possible, at least locally, if and only if the Riemann curvature tensor vanishes. However, the classical results assume at least a C2 regularity for the Cauchy-Green tensor (a.k.a. the metric tensor). From an elasticity theory viewpoint, weaker regularity assumptions on the data would be suitable.
We generalize this classical result under the hypothesis that the Cauchy-Green tensor is only of class W^{1,p} for some p>d.
The proof is based on a general result of PDE concerning the solvability and stability of a system of first order partial differential equations with L^p coefficients.