Volume 13, Issue 5
Equivalence between Riemann-Christoffel and Gauss-Codazzi-Mainardi Conditions for a Shell

D. L&#233onard-Fortun&#233, B. Miara & C. Vall&#233e

Int. J. Numer. Anal. Mod., 13 (2016), pp. 820-830

Published online: 2016-09

Preview Purchase PDF 84 2075
Export citation
  • Abstract

We establish the equivalence between the vanishing three-dimensionnal Riemann- Christoffel curvature tensor of a shell and the two-dimensionnal 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.

  • Keywords

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

  • AMS Subject Headings

58A15 58A17 58B21

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{IJNAM-13-820, author = {}, title = {Equivalence between Riemann-Christoffel and Gauss-Codazzi-Mainardi Conditions for a Shell}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2016}, volume = {13}, number = {5}, pages = {820--830}, abstract = {We establish the equivalence between the vanishing three-dimensionnal Riemann- Christoffel curvature tensor of a shell and the two-dimensionnal 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.}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/467.html} }
TY - JOUR T1 - Equivalence between Riemann-Christoffel and Gauss-Codazzi-Mainardi Conditions for a Shell 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 KW - 3D manifolds KW - Pfaffian systems KW - Frobenius integrability conditions KW - Riemann-Christoffel curvature tensor KW - moving frames KW - Cartan differential geometry KW - Tensorial calculus AB - We establish the equivalence between the vanishing three-dimensionnal Riemann- Christoffel curvature tensor of a shell and the two-dimensionnal 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.
D. Léonard-Fortuné, B. Miara & C. Vallée. (1970). Equivalence between Riemann-Christoffel and Gauss-Codazzi-Mainardi Conditions for a Shell. International Journal of Numerical Analysis and Modeling. 13 (5). 820-830. doi:
Copy to clipboard
The citation has been copied to your clipboard