## Admission requirements

Topology (required)

Linear Algebra 1,2 (required)

Analysis 2,3 (required)

Algebra 1 (required)

Algebra 2 (recommended)

Complex Analysis (recommended)

Differentiable manifolds 1 (recommended, *NOT* required)

## Description

In this course, we introduce abstract smooth manifolds. This notion generalises that of a smooth surface in three dimensional space, which serves as a motivating example. We discuss the main properties of and structures on manifolds, such as vector bundles, vector fields and derivations, flows, differential forms, orientation and integrals. We then prove Stokes’ theorem. We introduce Riemannian metrics and prove the divergence theorem. This paves the way for a discussion of DeRham 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

The schedule for the course can be found on MyTimeTable.

## Mode of instruction

weekly lectures and question hour (optional)

## Assessment Method

Written exam and homework. During the course, there will be six homework assignments to be handed in at predetermined dates. Out of these, the five best count towards the final grade. For the final grade, the homework counts for 25% and the written exam for 75%, unless the grade for the written exam is higher than the aforementioned weighted average. In the latter case, the final grade is equal to the the grade for the written exam.

## Reading list

John. M. Lee, “Introduction to smooth manifolds”

## Registration

From the academic year 2022-2023 on every student has to register for courses with the new enrollment tool 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 see this page for more information.

Please note that it is compulsory to both preregister and confirm 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. Confirming your exam participation is possible until ten days before the exam.

Extensive FAQ's on MyStudymap can be found here.

## Contact

Bram Mesland, b[dot]mesland[at]math[dot]leidenuniv[dot]nl

## Remarks

Please note that the course "Differentiable Manifolds 1" is *NOT* a prequisite for Differentiable Manifolds 2. The courses treat related subject matter but are entirely independent of one another.