UE Differential and dynamic geometry

Degrees incorporating this pedagocial element :

Description

Introduction to curves and surfaces.

Curves: Frénet references, covariant derivatives, reference fields, connection forms, structural equations.

Surfaces: surfaces of R^3, plane tangents, differential forms, differentiable applications between surfaces.

Abstract Variety: Whitney's Theorem

Curvature: Normal curvature, Gauss curvature, Geodesics. Case of surfaces of revolution.

Geometry of the surfaces of R^3: Egregium theorem, Gauss Bonnet's theorem

 Possibility of complement: Transition to sub-varieties of Rn and abstract varieties

 Possibility of complement: Introduction to dynamic systems. Vector fields and dynamic systems on varieties

Prerequisites

Differential calculus of L3

Bibliography

  • Edmond Ramis, Claude Deschamps, Jacques Odoux, Cours de mathématiques spéciales. 5. Applications de l’analyse à la géométrie, Masson, 1981
  • Marcel Berger, Bernard Gostiaux, Géométrie différentielle : variétés, courbes et surfaces, seconde édition, Presses Universitaires de France, 1992
  • Manfredo Do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, 1976
  • Jacques Lafontaine, Introduction aux variétés différentielles, Grenoble Sciences, 2010