Differential Geometry (BSc)

6 EC

Semester 1, period 1, 2

51228DIF6Y

Owner Bachelor Wiskunde
Coordinator prof. dr. S. Shadrin
Part of Exchange Programme Faculty of Science, specialisation BSc Mathematics, year 1Minor Mathematical Themes, year 1Bachelor Mathematics, year 3Double bachelor Mathematics and Physics, year 3Double bachelor Mathematics and Computer Science, year 3
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Course manual 2024/2025

Course content

This course is an introduction to the theory of smooth manifolds and differential-geometric structures on them. We'll discuss smooth manifolds, smooth maps, tangent and cotangent bundles, submanifolds, Whitney's embedding and approximation theorems, more general tensors, differential forms, integration, de Rham cohomology, integral curves and manifolds, Lie derivatives, Lie groups, and Lie algebras. 

Study materials

Literature

  • John M. Lee. Introduction to smooth manifolds. Second Edition. Graduate Texts in Mathematics 218. Springer

Objectives

  • The student is able to work with differentiable manifolds in coordinates and in coordinate-free way.
  • The student is able to reproduce arguments and constructions in differential geometry.
  • The student is able to connect topological properties and differential-geometric structures.
  • The student is able to perform differential-geometric computation in particular examples.

Teaching methods

  • Lecture
  • Seminar

Learning activities

Activiteit

Aantal uur

Hoorcollege

28

Werkcollege

28

Tussentoets

2

Tentamen

3

Zelfstudie

107

Total

168

Attendance

Programme's requirements concerning attendance (OER-B):

  • Each student is expected to actively participate in the course for which he/she is registered.
  • If a student cannot be present due to personal circumstances with a compulsory part of the programme, he / she must report this as quickly as possible in writing to the relevant lecturer and study advisor.
  • It is not allowed to miss obligatory parts of the programme's component if there is no case of circumstances beyond one's control.
  • In case of participating qualitatively or quantitatively insufficiently, the examiner can expel a student from further participation in the programme's component or a part of that component. Conditions for sufficient participation are stated in advance in the course manual and on Canvas.
  • In the first and second year, a student should be present in at least 80% of the seminars and tutor groups. Moreover, participation to midterm tests and obligatory homework is required. If the student does not comply with these obligations, it can be decided to deny the positive effects of homework to be counted towards the final grade of the course. In case of personal circumstances, as described in OER-A Article A-6.4, a different arrangement will be proposed in consultation with the study advisor.

Assessment

Item and weight Details

Final grade

0.6 (60%)

Tentamen

Must be ≥ 5.5

0.2 (20%)

Homework (average)

0.2 (20%)

Deeltoets

The final grade is 0.6*Exam+0.2*Midterm+0.2*Homework. It is required that the grade for the exam at least 5.5; otherwise the final grade is at most 5.0.

The grade after the retake is 1.0*retake.

The grade for the homework is the average of 5 assignments. 

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

Course structure

Weeknummer Onderwerpen Studiestof
1 Definitions of topological manifold, smooth manifold, examples (Chapter 1 of the book).  
2

Regular coordinate balls, smooth manifold chart lemma, more examples, manifolds with boundary (Chapter 1 of the book.)

  
3 Smooth functions, partition of unity, applications of the partition of unity (Chapter 2).  
4 Applications of partition of unity (Chapter 2). Tangent vectors and bundle. (Chapters 3).  
5 Tangent vectors and bundle (Chapters 3). Smooth maps of constant rank (Chapter 4).  
6

Smooth maps of constant rank, constant rank theorem, smooth coverings (Chapter 4).

  
7 Further details on smooth coverings (Chapter 4). Submanifolds, Whitney theorem (parts of Chapters 5, 6).  
8

Embedding in a Euclidian space. Smooth vector fields and related algebraic structures (parts of Chapter 8).

  
9 Smooth vector fields and related algebraic structures (parts of Chapter 8). Integral curves and flows of vector fields (Sections "Integral curves" and "Flows" of Chapter 9).  
10 Flows of vector fields. Their usage for differentiation of functions and vector fields (Parts of Chapter 9).  
11 Lie derivative. Canonical form of a vector field near a regular point. (Parts of Chapter 9). An introduction to multilinear algebra (Chapter 12, sections "Multilinear algebra" and "Symmetric and alternating tensors").  
12 Alternating tensors (Chapter 14, section "The Algebra of Alternating Tensors"). Vector bundles, sections (very briefly; Chapter 10, sections "Vector bundles" and "Local and Global Sections of Vector Bundles"). Differential forms (Chapter 14, section "Differential Forms on Manifolds").   
13 Differential form as differential graded commutative associative algebra, de Rham differential (Chapter 14). Orientation (Chapter 15).   
14 Induced orientation on the boundary. Integration on manifolds, Stokes theorem. (Chapter 16).  

Additional information

Analysis on R^n, Topology

Contact information

Coordinator

  • prof. dr. S. Shadrin

Staff

  • Jort de Groot