# Manifolds MAT510

An introduction to smooth manifolds and related concepts in differential geometry.

*Course description for study year 2021-2022. Please note that changes may occur.*

Facts

**Emnekode**

MAT510

**Versjon**

1

**Vekting (SP)**

10

**Semester undervisningsstart**

Autumn

**Antall semestre**

1

**Vurderingsemester**

Autumn

**Undervisningsspråk**

English

**Timeplan**

**Learning outcome**

After completing this course, the student should understand how familiar concepts from vector calculus are subsumed by the framework of differential forms on a Euclidean space. The student should also have a decent grasp of the key notions in point-set topology, and be able to state definitions and give examples. Furthermore, the student should be able to perform simple calculations on smooth manifolds and work out detailed properties in examples.

**Content**

This course gives an introduction to smooth manifolds and related concepts in differential geometry. Smooth functions, directional derivatives, tangent vectors and differential forms on a Euclidean space will be reviewed. The basics of point-set topology will be introduced and the concept of a topological manifold will be defined. The notion of a smooth manifold will be introduced, with a plethora of familiar (and perhaps not so familiar) examples. Smooth maps between manifolds will be defined, together with the important concept of a diffeomorphism. Further concepts like tangent spaces, differentials, submanifolds, vector fields and integral curves will also be developed.

**Required prerequisite knowledge**

None

**Recommended prerequisites**

MAT100 Mathematical Methods 1, MAT110 Linear Algebra, MAT210 Real and Complex Calculus, MAT250 Abstract Algebra, MAT300 Vector Analysis, MAT320 Differential Equations

**Eksamen / vurdering**

Vurderingsform | Vekting | Varighet | Karakter | Hjelpemiddel |
---|---|---|---|---|

Written exam | 1/1 | 4 Hours | A - F | Basic calculator specified in general exam regulationsCompilation of mathematical formulae (Rottmann) |

**Course teacher(s)**

**Course coordinator:**Paul Francis de Medeiros

**Head of Department:**Bjørn Henrik Auestad

**Method of work**

5-6 hours lecturing and problem solving per week. Language of tuition: English.

**Open for**

**Course assessment**

Use of evaluation forms and/or conversation for students' evaluation of the course and teaching, according to current guidelines.

**Overlapping courses**

Course | Reduction (SP) |
---|---|

Mathematical Modelling (MAT500) | 10 |

**Literature**