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
You will find the timetables for all courses and degree programmes of Leiden University in the tool MyTimetable (login). Any teaching activities that you have sucessfully registered for in MyStudyMap will automatically be displayed in MyTimeTable. Any timetables that you add manually, will be saved and automatically displayed the next time you sign in.
MyTimetable allows you to integrate your timetable with your calendar apps such as Outlook, Google Calendar, Apple Calendar and other calendar apps on your smartphone. Any timetable changes will be automatically synced with your calendar. If you wish, you can also receive an email notification of the change. You can turn notifications on in ‘Settings’ (after login).
For more information, watch the video or go the the 'help-page' in MyTimetable. Please note: Joint Degree students Leiden/Delft have to merge their two different timetables into one. This video explains how to do this.
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
Please register for the course in MyStudyMap. There are two registration periods per year: registration for the fall semester opens in July and registration for the spring semester opens in December.
Please note that it is compulsory to register your participation for every exam and retake. Not being registered for a course means that you are not allowed to participate in the final exam of the course. Not being registered for an exam means your grade will not be processed.
Contact
Lecturer:
Dr. Emre Can Sertöz
Course assistant:
Amira Tlemsani
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.