Differential Geometry

Code:

M3007

Acronym:

M3007

Level:

300

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2017/2018 - 2S

Active?	Yes
Web Page:	https://moodle.up.pt/course/view.php?id=2062
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	0	Official Study Plan	3	-	6	56	162
L:CC	0	Plano de estudos a partir de 2014	2	-	6	56	162
			3
L:F	4	Official Study Plan	2	-	6	56	162
			3
L:G	0	study plan from 2017/18	3	-	6	56	162
L:M	16	Official Study Plan	2	-	6	56	162
			3
L:Q	0	study plan from 2016/17	3	-	6	56	162

Teaching language

Suitable for English-speaking students

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.

keywords

Physical sciences > Mathematics > Geometry

Evaluation Type

Distributed evaluation with final exam

Assessment Components

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

Calculation formula of final grade

Test(s): 50%
Final exam: 50%

Special assessment (TE, DA, ...)

By oral and/or written exam. The marks of the tests cannot be used for this purpose.

Classification improvement

By exam. The marks of the tests cannot be used for this purpose.

Observations

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