Differential Geometry

Code:

M355

Acronym:

M355

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2013/2014 - 2S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Physics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:AST	1	Plano de Estudos a partir de 2008	3	-	7,5	-
L:F	0	Plano de estudos a partir de 2008	3	-	7,5	-
L:M	15	Plano de estudos a partir de 2009	1	-	7,5	-
			2
			3

Teaching language

Portuguese

Objectives

Students should acquire knowledge about the application of methods of differential and integral calculus to the study of geometry with emphasis on the differential geometry of surfaces. They should be able to apply this knowledge independently to analyze and solve mathematical problems in contexts where methods of differential geometry are relevant.

Learning outcomes and competences

Described in the Learning Outcomes.

Working method

Presencial

Program

Basic point set topology necessary for the development of the theory. Smooth surfaces and implicitly defined surfaces. Differentiable functions on surfaces and maps between surfaces. Tangent plane. Orientation. First fundamental form. Area. Geodesics and normal curvature. Geodesic curves. The second fundamental form. Curvature of surfaces. Gauss' Theorema Egregium. The Gauss-Bonnet theorem.

Additionally, one or more of the following optional topics will be treated:

- Global differential geometry of curves and surfaces;
- Classification of topological surfaces;
- Surfaces and abstract manifolds;
- Surfaces of constant curvature and the parallel axiom;
- Differential forms and applications;
- Variational problems and minimal surfaces;
- Applications of differential geometry to physics.

Mandatory literature

Manfredo P. do Carmo; Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976

Complementary Bibliography

Paulo Ventura Araújo; Geometria Diferencial, IMPA, 2004

Teaching methods and learning activities

Classes: Lectures, problem sessions, student presentations.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	50,00
Teste	50,00
Total:	100,00

Eligibility for exams

Attendance.

Calculation formula of final grade

The final mark is the average of the marks of the test(s) and of the final exam.

Special assessment (TE, DA, ...)

By special exam, oral or written.

Classification improvement

The final mark obtained by distributed assessment can be improved by taking the resit exam in one of the two exam periods immediately following the course, in accordance with the "Regulamento de Avaliação do Aproveitamento dos Estudantes" of FCUP.

Observations

Pre-requisites: Fundamentals of Linear Algebra, Calculus, Geometry and Analysis in R^n. (Vector Analysis / Multivariable Calculus)