Equivalence Between Riemann-Christoffel and Gauss-Codazzi-Mainardi Conditions for a Shell

Authors

  • D. Léonard-Fortuné 21, rue du Hameau du Cherpe, 86280 Saint Benoit, France
  • B. Miara Université Paris-Est, 92 300 Marne la Vallée
  • C. Vallée Université de Poitiers, Institut Pprime, UPR CNRS 3346, SP2MI, bd Marie et Pierre Curie, Téléport 2, BP 30179, 86962, Futuroscope Chasseneuil cedex, France

Keywords:

Surfaces, 3D manifolds, Pfaffian systems, Frobenius integrability conditions, Riemann-Christoffel curvature tensor, moving frames, Cartan differential geometry, Tensorial calculus.

Abstract

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.

Downloads

Published

2016-09-03

Issue

Vol. 13 No. 5 (2016)

Section

Articles