TY - JOUR T1 - Equivalence Between Riemann-Christoffel and Gauss-Codazzi-Mainardi Conditions for a Shell AU - Léonard-Fortuné , D. AU - Miara , B. AU - Vallée , C. JO - International Journal of Numerical Analysis and Modeling VL - 5 SP - 820 EP - 830 PY - 2016 DA - 2016/09 SN - 13 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/467.html KW - Surfaces, 3D manifolds, Pfaffian systems, Frobenius integrability conditions, Riemann-Christoffel curvature tensor, moving frames, Cartan differential geometry, Tensorial calculus. AB -

We establish the equivalence between the vanishing three-dimensional Riemann- Christoffel curvature tensor of a shell and the two-dimensional Gauss-Codazzi-Mainardi compatibility conditions on its middle surface. Additionally, we produce a new proof of Gauss theorema egregium and Bonnet theorem (reconstructing a surface from its two fundamental forms). This is performed in the very elegant framework of Cartan's moving frames.