Differential Geometry

Code:

M3007

Acronym:

M3007

Level:

300

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2024/2025 - 2S

Active?	Yes
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	48	162
L:CC	4	study plan from 2021/22	2	-	6	48	162
			3
L:F	8	Official Study Plan	3	-	6	48	162
L:G	5	study plan from 2017/18	2	-	6	48	162
			3
L:M	19	Official Study Plan	3	-	6	48	162
L:MA	1	Official Study Plan	3	-	6	48	162
L:Q	0	study plan from 2016/17	3	-	6	48	162

Teaching language

Portuguese

Objectives

Study of smooth surfaces in 3-dimensional Euclidean space.

Learning outcomes and competences

As described in the above section.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Linear Algebra; Real Analysis I, II, III

Program

1) Smooth curves in Euclidean space; smooth plane curves.

2) Regular smooth surfaces in R^3: parametrizations; differentiable functions on surfaces; tangent spaces; orientation of surfaces; the first fundamental form (how to measure the area of regions, the length of curves and angles on surfaces).


3) The geometry of the Gauss mapping: the second fundamental form; curvature of surfaces.

4) Intrinsic geometry of surfaces: conformal mappings and isometries; covariant derivative and parallel transport; geodesic curvature; Gauss-Bonnet's theorem; geodesic curves.

5) Intrinsic distance on connected surfaces. Complete surfaces. Hopf-Rinow's theorem.

Mandatory literature

Araújo Paulo Ventura; Geometria diferencial. ISBN: 8-52440-136-2

Complementary Bibliography

Klingenberg Wilhelm; A course in differential geometry. ISBN: 0-387-90255-4
Carmo Manfredo Perdigão do; Differential geometry of curves and surfaces. ISBN: 0-13-212589-7

Teaching methods and learning activities

Formal lectures, complemented by exercises for the students to solve.

keywords

Physical sciences > Mathematics > Geometry

Evaluation Type

Evaluation with final exam

Assessment Components

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

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	114,00
Frequência das aulas	48,00
Total:	162,00

Eligibility for exams

Class attendance is not compulsory.

Calculation formula of final grade

Final examination only.

Special assessment (TE, DA, ...)

In some special circumstances, assessment of the students may consist of an oral examination only.