Subject: Differential Geometry

Scientific Area:

Mathematics

80 Hours

Number of ECTS:

7,5 ECTS

Language:

Portuguese

# Overall objectives:

1 - To develop skills in order to recognize curves and surfaces embedded in euclidean spaces. To define and compute the curvature and torsion of a regular curve as well as the gaussian curvature and mean curvature of a surface.
2 - Recognize differentiable manifolds and tangent bundles as the natural place for the differential calculus. To develop skills to use local maps in order to study the local properties of differentiable manifolds and differentiable functions.
3 - To know the more basic aspects of Riemannian geometry.

# Syllabus:

1 - Differentiable manifolds. Parametrizable sets and their tangent structures (tensor fields). Maps between parametrizable sets. Study of the classical cases: curves and surfaces in a vector space. Differentiable manifolds: maps, atlases, tangent structures.
2 - Classification problems - the classical cases where the action is that of a the group of affine transformations and the group of rigid motions: Curves (arc length, curvatures). Surfaces in 3 dimensional space (fundamental forms; curvatures)
3 - Riemannian geometry: metric, parallel transport; fundamental tensors. Examples.

# Literature/Sources:

M. do Carmo , Differential Geometry of Curves And Surfaces , Prentice Hall
M. Berger, B. Gostiaux , Differential Geometry: Manifolds, Curves and Surfaces , Springer
B. O'Neil , Elementary Differential Geometry , Academic Press
Jean-Pierre Serre , Lie algebras and Lie groups , Springer

# Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
The lectures are expository. Several examples are presented to facilitate the apprehension of concepts and serve as motivation to the students. The students are provided with a list of problems and exercises that they should try to solve out of the classroom. In practical classes these problems and exercises will be discussed and solved in order to overcome any difficulties that students may have The evaluation consists of 2 tests, each one counting for 50% of the final mark.