## Module Handbook

• Dynamischer Default-Fachbereich geändert auf MAT

# Module MAT-40-21-M-6

## Module Identification

Module Number Module Name CP (Effort)
MAT-40-21-M-6 Manifolds 9.0 CP (270 h)

## Basedata

CP, Effort 9.0 CP = 270 h 1 Sem. irreg. [6] Master (General) [EN] English Gathmann, Andreas, Prof. Dr. (PROF | DEPT: MAT) [MAT-AGCA] Algebra, Geometry and Computer Algebra [MAT-88.105-SG] M.Sc. Mathematics [NORM] Active

## Notice

Without a proof of successful participation in the exercise classes, only 6 credit points will be awarded for the module.

## Courses

Type/SWS Course Number Title Choice in
Module-Part
Presence-Time /
Self-Study
SL SL is
required for exa.
PL CP Sem.
4V+2U MAT-40-21-K-6
Manifolds
P 84 h 186 h
U-Schein
- PL1 9.0 irreg.
• About [MAT-40-21-K-6]: Title: "Manifolds"; Presence-Time: 84 h; Self-Study: 186 h
• About [MAT-40-21-K-6]: The study achievement "[U-Schein] proof of successful participation in the exercise classes (ungraded)" must be obtained.

## Examination achievement PL1

• Form of examination: oral examination (20-30 Min.)
• Examination Frequency: irregular (by arrangement)
• Examination number: 84155 ("Manifolds")

## Contents

• Concept of manifold, in particular differentiable manifold,
• tangential and other vector bundles to manifolds,
• partition of unity,
• orientation
• differential equations on manifolds, Ehresmann's fibration theorem,
• differential forms and integration, general theorem of Stokes,
• special structures on manifolds (e.g. basic concepts of complex, Riemannian and symplectic manifolds).

## Competencies / intended learning achievements

Upon successful completion of this module, the students have studied and understand the concept of a differentiable manifold and the basic techniques required to deal with it. They have learnt how the familiar concepts about coordinate analysis and geometry are globalized through the concept of manifolds. Thus, the students have attained a deeper understanding of differential and integral calculus.They understand the proofs presented in the lecture and are able to comprehend and explain them.

By completing the given exercises, the students will have developed a skilled, precise and independent handling of the terms, propositions and techniques taught in the lecture. In addition, they have learnt how to apply these techniques to new problems, analyze them and develop solution strategies independently or by team work.

## Literature

• T. Bröcker, K. Jänich: Introduction to Differential Topology,
• K. Jänich: Vector Analysis,
• S. Lang: Introduction to Differentiable Manifolds,
• J.M. Lee: Introduction to Smooth Manifolds,
• M. Spivak: Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus.

## Registration

Registration for the exercise classes via the online administration system URM (https://urm.mathematik.uni-kl.de)

## Requirements for attendance of the module (informal)

Additional knowledge from the courses [MAT-12-25-K-3] and [MAT-12-27-K-3] is useful but not necessarily required.

None

## References to Module / Module Number [MAT-40-21-M-6]

Course of Study Section Choice/Obligation
[MAT-88.105-SG] M.Sc. Mathematics [Core Modules (non specialised)] Pure Mathematics [WP] Compulsory Elective
[MAT-88.706-SG] M.Sc. Mathematics International [Core Modules (non specialised)] Pure Mathematics [WP] Compulsory Elective
Module-Pool Name
[MAT-GM-MPOOL-5] General Mathematics (Introductory Modules M.Sc.)