Differentiable Manifolds

Code:

M505

Acronym:

M505

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2025/2026 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Doctoral Program in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
IUD-M	9	PE do Prog Inter-Univ Dout Mat	1	-	9	60	243

Teaching language

English

Objectives

To treat the basic theory of differential manifolds.

Learning outcomes and competences

The student should acquire a thorough knowledge of the theory of differential manifolds and be able to use its tools in mathematical problem solving and research.

Working method

Presencial

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

Elements of general topology. Calculus of functions of several variables. Basic algebra.

Program

Topological and differentiable manifolds, differentiable maps; partitions of unity. Tangent and cotangent bundles; the differential of a smooth map. Immersions, submersions, and submanifolds; inverse images of regular values, transversality, Sard's Theorem; Whitney embedding.

Vector fields and flows; the Lie bracket of vector fields; Foliations, distributions and the Frobenius theorem; Lie groups and Lie algebras; Lie group and Lie algebra actions and representations; quotients, principal bundles, and homogenous spaces. Fibre bundles and Vector bundles.

Differential forms, exterior derivative. Integration on manifolds. Stokes' theorem. Elements of homological algebra; de Rham cohomology. The Poincaré Lemma. Homotopy and homotopy invariance of de Rham cohomology. Euler characteristic. The Mayer-Vietoris sequence.

Some additional topics may be treated, such as: Degree of a map; the index of a vector field with isolated singularities and the Poincaré-Hopf Theorem. Constructions with vector and fiber bundles; vector valued forms; classification of bundles. Symplectic, Riemannian, complex, or other geometric structures.

Mandatory literature

Fernandes, Rui Loja; Lectures on Differential Geometry, World Scientific, 2025. ISBN: 978-981-12-5264-8
Jaques Lafontaine; An Introduction to Differential Manifolds, Springer, 2015

Complementary Bibliography

Ib Madsen; From calculus to cohomology. ISBN: 0-521-58956-8
Raoult Bott; Differential forms in algebraic topology. ISBN: 0-387-90613-4
Tu, L.W.; An Introduction to Manifolds, Springer, 2008
Fulton, W.; Algebraic Topology - A First Course, Springer, 1997
Sutherland, W.A. ; Introduction to Metric and Topological Spaces, Oxford University Press, 1975
Barden, D. and Thomas, C.; An introduction to differential manifolds, Imperial College Press, 2003

Teaching methods and learning activities

Lectures, problem sessions, student presentations.

keywords

Physical sciences > Mathematics > Geometry

Evaluation Type

Evaluation with final exam

Assessment Components

designation	Weight (%)
Apresentação/discussão de um trabalho científico	60,00
Trabalho escrito	40,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	187,00
Frequência das aulas	56,00
Total:	243,00

Eligibility for exams

Attendance is not compulsory.

Calculation formula of final grade

The final mark is the mark obtained in the exam.

Special assessment (TE, DA, ...)

By written and/or oral exam.

Classification improvement

By final exam.