Differentiable Manifolds

Code:

M4135

Acronym:

M4135

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2024/2025 - 2S

Active?	Yes
Web Page:	https://moodle2324.up.pt/course/view.php?id=1715
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Master in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
M:M	14	Official study plan since 2024/2025	1	-	9	63	243

Teaching language

Portuguese

Objectives

Introduction to the theory of differentiable manifolds.

Learning outcomes and competences

Students should grasp the basic concepts of the theory of differentiable manifolds. They should also be able to deepen their knowledge of the theory by self-study, using the vast available literature.

Working method

Presencial

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

First degree with a strong mathematical content, including at least two semesters of linear algebra and three semesters of real analysis (comprising differential equations).

Program

Differentiable manifolds. Derivative of differentiable maps between manifolds, submersions, immersions and embeddings. Tangent and cotangent buncle of a manifold. 
Transversality, homotopy and stability, Sard's Theorem and Morse functions, Whitney's Theorem, partitions of unity, tubular neighbourhoods, genericity and its relation to transversality.
Vector fields on a manifold: flow, derivations and one-parameter group of diffeomorphisms, Lie groups. 
Exterior algebra, differential forms , integration of differential forms, Stokes theorem.
De Rham cohomology (if time allows).

Mandatory literature

Victor Guillemin; Differential topology. ISBN: 0-13-212605-2
John M. Lee; Introduction to smooth manifolds. ISBN: 0-387-95448-1
Dennis Barden; An introduction to differential manifolds. ISBN: 1-86094-355-1

Complementary Bibliography

Serge Lang; Differential manifolds. ISBN: 0-387-96113-5

Teaching methods and learning activities

Lectures will include time for theoretical exposition of concepts, examples of application and also for resolution of exercises by the students.

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	180,00
Frequência das aulas	63,00
Total:	243,00

Eligibility for exams

Class atendance is not compulsory.

Calculation formula of final grade

Final examination only.