Universidade Nova de Lisboa

Universidade Nova de Lisboa

Faculdade de Ciências e Tecnologia

Differential Geometry

Code

10837

Academic unit

Faculdade de Ciências e Tecnologia

Department

Departamento de Matemática

Credits

6.0

Teacher in charge

Ana Cristina Malheiro Casimiro

Total hours

22

Teaching language

Português

Objectives

The main goal is to introduce to the students classic results of Differential Geometry of curves and surfaces on  ℝ3  and  to give them the methods of geometric vision in order to provide them enough tools to approach modern geometric theories.

Prerequisites

Linear Algebra I and II, Geometry, Calculus III and IV

Subject matter

  1. Curves in space: parametrisation by arc-length, curvature, torsion, Frenet trihedron.
  2. Surfaces in three dimensions: the first and second fundamental form, sectional curvature, principal curvature, mean and gaussian curvature, Weingarten linear map.
  3. Submanifolds in ℝn:: differential maps, tangent and cotangent spaces, differential of a differentiable map, immersions in ℝn, parametrised and closed submanifolds.
  4. Geodesics: definition, equations of geodesics, examples and applications.
  5. Gauss''''s Theorema Egregium:  isometries of surfaces, the Codazzi-Mainardi equations. (optional)
  6. The Gauss-Bonnet theorem. (optional)
  7. Tensor Calculus: fundaments of linear and multilinear algebra, differential forms, flux of a vector field, Lie bracket of two vector fields, geometric interpretation, Lie derivative. (optional)
  8. Integration on Manifolds: orientation of a manifold, integrationof differential forms, exterior derivative of a differential form, manifolds with boundary , Theorem of Stokes. (optional)

Bibliography

1.    M. P. Carmo, "Differential Geometry of curves and surfaces", Prentice Hall, 1976. 
2.    M. P. Carmo,  "Geometria Riemanniana", Projecto Euclides, IMPA, 1988.
3.    O’Neil, "Elementary differential geometry ", Academic Press, New York USA, 1966.
4.   Pressley, "Elementary differential geometry ", Springer Undergraduate Mathematics Series, 2001.
5.    Spivak, "Calculus on manifolds", Monograph Mathematics Series, 1965.

Teaching method

Tutorial orientation of the student, with indication of which matters to study, problems to solve resolution and clarification of doubts about theory and problems.

Evaluation method

The evaluation consist in the presentation, written or oral, by the student, of solved proposed exercises.

Courses