Admission requirements
Topology (required)
Linear Algebra 1,2 (required)
Analysis 2,3 (required)
Algebra 1 (required)
Algebra 2 (recommended)
Complex Analysis (recommended)
Curves and Surfaces (recommended, NOT required)
Description
In this course, we introduce the concept of abstract smooth manifolds, generalizing the notion of a smooth surface in three-dimensional space, which serves as a motivating example. We will discuss the main properties of manifolds and structures on them, including vector bundles, vector fields and derivations, flows, differential forms, orientation, and integrals. In particular, we will introduce Riemannian metrics and prove the divergence theorem. Towards the end, we will prove Stokes’ theorem. This will pave the way for a discussion on De Rham cohomology, its fundamental properties, and its relation to topology.
Course objectives
Obtain a thorough understanding of the geometry and topology of manifolds. Develop the ability to carry out explicit calculations of differential and integral calculus on manifolds in a concrete setting. Apply these techiques in examples from geometry and dynamical systems.
Timetable
In MyTimetable, you can find all course and programme schedules, allowing you to create your personal timetable. Activities for which you have enrolled via MyStudyMap will automatically appear in your timetable.
Additionally, you can easily link MyTimetable to a calendar app on your phone, and schedule changes will be automatically updated in your calendar. You can also choose to receive email notifications about schedule changes. You can enable notifications in Settings after logging in.
Questions? Watch the video, read the instructions, or contact the ISSC helpdesk.
Note: Joint Degree students from Leiden/Delft need to combine information from both the Leiden and Delft MyTimetables to see a complete schedule. This video explains how to do it.
Mode of instruction
Lectures.
Optional office hours with the course assistant.
Assessment Method
Choose your own adventure:
75% Final exam + 25% Homework, OR
100% Final exam + 0% Homework.
There will be six homework assignments and the top five will count towards your homework grade. Homework is not mandatory; but if you do them, it can only improve your grade.
Reading list
John. M. Lee, “Introduction to smooth manifolds”
The book can be downloaded on Springerlink and a hardcopy can be ordered via the Mycopy option. https://link.springer.com/book/10.1007/978-1-4419-9982-5
Registration
As a student, you are responsible for enrolling on time through MyStudyMap.
In this short video, you can see step-by-step how to enrol for courses in MyStudyMap.
Extensive information about the operation of MyStudyMap can be found here.
There are two enrolment periods per year:
Enrolment for the fall opens in July
Enrolment for the spring opens in December
See this page for more information about deadlines and enrolling for courses and exams.
Note:
It is mandatory to enrol for all activities of a course that you are going to follow.
Your enrolment is only complete when you submit your course planning in the ‘Ready for enrolment’ tab by clicking ‘Send’.
Not being enrolled for an exam/resit means that you are not allowed to participate in the exam/resit.
Contact
Lecturer:
Dr. Emre Can Sertöz <e.c.sertoz@math.leidenuniv.nl>
Course assistant:
Amira Tlemsani <a.tlemsani@math.leidenuniv.nl>
Remarks
Please note that the course "Curves and Surfaces" used to be called "Differentiable Manifolds 1," whereas our current course was called "Differentiable Manifolds 2." Nevertheless, "Curves and Surfaces" is NOT a prequisite for Differentiable Manifolds. The courses treat related subject matter but are entirely independent of one another.
Software
Starting from the 2024/2025 academic year, the Faculty of Science will use the software distribution platform Academic Software. Through this platform, you can access the software needed for specific courses in your studies. For some software, your laptop must meet certain system requirements, which will be specified with the software. It is important to install the software before the start of the course. More information about the laptop requirements can be found on the student website.