We show that surfaces with assigned director field immersed in the three-dimensional Euclidean space can be defined intrinsically by four tensor fields, of which two are of order two, one of order one, and one of order zero. We then show how a surface and its assigned director field can be reconstructed from these four tensors and prove that the reconstruction operator is continuous between ad hoc functional spaces with as little regularity as possible. These results have applications in the Cosserat theory of nonlinearly elastic shells, the strain energy of which are defined precisely in terms of the four tensor fields defined in this paper.
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