@article{c355b8c5186f4334bc42184c2088cc12, title = "Explicit reconstruction of a displacement field on a surface by means of its linearized change of metric and change of curvature tensors", abstract = "Let ω be a simply-connected open subset in R2 and let θ : ω → R3 be a smooth immersion. If two symmetric matrix fields (γα β) and (ρα β) of order two satisfy appropriate compatibility relations in ω, then (γα β) and (ρα β) are the linearized change of metric and change of curvature tensor fields corresponding to a displacement vector field η of the surface θ (ω). We show here that, when the fields (γα β) and (ρα β) are smooth, the displacement vector η (y) at any point θ (y), y ∈ ω, of the surface θ (ω) can be explicitly computed by means of a {"}Ces{\`a}ro-Volterra path integral formula on a surface{"}, i.e., a path integral inside ω with endpoint y, and whose integrand is an explicit function of the functions γα β and ρα β and their covariant derivatives. To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008). {\textcopyright} 2008 Acad{\'e}mie des sciences.", author = "Ciarlet, {Philippe G.} and Liliana Gratie and Michele Serpilli", year = "2008", month = oct, doi = "10.1016/j.crma.2008.09.001", language = "English", volume = "346", pages = "1113--1117", journal = "Comptes Rendus Mathematique", issn = "1631-073X", publisher = "ELSEVIER FRANCE-EDITIONS SCIENTIFIQUES MEDICALES ELSEVIER", number = "19-20", }