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

D. Léonard-Fortuné, B. Miara & C. Vallée

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

Published online: 2016-09

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  • 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.

  • 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

danielle.fortune123@orange.fr (D. Léonard-Fortuné)

bernadette.miara@gmail.com (B. Miara)

claude.vallee@lms.univ-poitiers.fr (C. Vallée)

  • BibTex
  • RIS
  • TXT
@Article{IJNAM-13-820, author = {Léonard-Fortuné , D.Miara , B. and Vallée , C.}, 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-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.

}, 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 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.

Léonard-Fortuné , D.Miara , B. and Vallée , C.. (2016). Equivalence Between Riemann-Christoffel and Gauss-Codazzi-Mainardi Conditions for a Shell. International Journal of Numerical Analysis and Modeling. 13 (5). 820-830. doi:
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