2024
Differential Geometry
Name: Differential Geometry
Code: MAT14223L
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese
Presentation
It is an optional curricular unit, which will be taught in the 6th semester and which intends that the student has mastery of the theory of differentiable varieties in its fundamentals and in the cut with the Euclidean vision of space.
Sustainable Development Goals
Learning Goals
The course consists in understanding topology, analysis and algebra applied to the study of differentiable manifolds. Consists of the mastery of the fundamentals of manifold theory, in a breakthrough from the euclidean viewpoint of space. The objectives are the understanding of the notions of manifold, tangent space, vector field, Lie bracket and the Lie algebra of vector fields. To meet with some fundamental applications, such as Lie groups and homogeneous spaces.
Contents
Review of topological spaces and metric spaces; completeness. Calculus on normed vector spaces. Notions of manifold, tangent space, tangent vector field, morphism between manifolds. Study of submanifolds; theorems on the construction of manifolds by inverse image; applications to Lie groups; introduction to the study of Riemannian manifolds; homogeneous manifolds.
Teaching Methods
Structured exposition, with worked exercises.
Evaluation:
The evaluation may be either continuous, done through between two and six partial tests and quizzes, done preferably during the classes, weighting 100% of the classification, the number of which is to be defined by the professor who is responsible for the course unit in each academic year, or by a final exam. Students who achieve a grade of 18 or above may have to do an extra oral exam. For these students, the final grade is the maximum between 17 and the simple average of the previous grade and the oral exam.
Formative evaluation is done in or by tasks to be done outside class, to improve the learning process; the elements of formative evaluation will have no weight on the final mark.
Evaluation:
The evaluation may be either continuous, done through between two and six partial tests and quizzes, done preferably during the classes, weighting 100% of the classification, the number of which is to be defined by the professor who is responsible for the course unit in each academic year, or by a final exam. Students who achieve a grade of 18 or above may have to do an extra oral exam. For these students, the final grade is the maximum between 17 and the simple average of the previous grade and the oral exam.
Formative evaluation is done in or by tasks to be done outside class, to improve the learning process; the elements of formative evaluation will have no weight on the final mark.
Certificações
