# Differential Geometry, 6 credits (TATA74)

Differentialgeometri, 6 hp

### Main field of study

Mathematics Applied Mathematics

First cycle

Programme course

### Examiner

Milagros Izquirdo Barrios

### Director of studies or equivalent

Jesper Thorén
Course offered for Semester Period Timetable module Language Campus VOF
6MMAT Mathematics, Master's Programme 1 (Autumn 2017) 1, 2 3, 3 Swedish/English Linköping v
6MMAT Mathematics, Master's Programme (Mathematics) 1 (Autumn 2017) 1, 2 3, 3 Swedish/English Linköping v
6KMAT Mathematics 5 (Autumn 2017) 1, 2 3, 3 Swedish/English Linköping v

### Main field of study

Mathematics, Applied Mathematics

First cycle

G2X

### Course offered for

• Mathematics, Master's Programme
• Mathematics

### Specific information

The course is only offered every second year. It will be offered during 2015

### Entry requirements

Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshold requirements for progression within the programme, or corresponding.

### Prerequisites

Linear algebra, Calculus in several variables, Vector calculus.

### Intended learning outcomes

The aim of the course is to provide knowledge of the geometry of curves and surfaces. The course integrates concepts from different parts of mathematics, such as linear algebra, calculus and differential equations. It also provides intuitive examples for many concepts in linear algebra, calculus and differential equations. These examples are fundamental to physics and mechanics: they play a role in our understanding of the movements of particles and the theory of relativity. After completing this course, students should be able to:

• Determine and calculate curvature of curves in different coordinate systems.
• Parametrise surfaces and use the metric tensor. Calculate isometries.
• Treat geodesic curves and parallel translation
• Calculate and analyse curvature of surfaces in different settings.
• Know the concept of tensor and recognise tensors that are used in mechanics, image processing and theory of relativity.
• Apply geometry of curves and surfaces to computer aided graphics

### Course content

Curves: tangents, curvature and torsion. Contact. Different types of curves. Regular surfaces: tangent plane. The first fundamental form: normal and geodesic curvature. Geodesics and parallel transport. Gauss' formulae.
The second fundamental form: Weingarten's equation, principal, Gauss and mean curvature. Minimal and developable surfaces. Riemann's and Ricci's tensors, Codazzi-Mainardi's equations. Gauss' ''Theorema Egregium''. Isometrical and conformal mappings. Gauss-Bonnet theorem.

### Teaching and working methods

Lectures and tutorials.
The course runs over the entire autumn semester.

### Examination

 UPG1 Assignments U, 3, 4, 5 6 credits TEN1 Written Examination U, 3, 4, 5 6 credits

A written exam will be offered in August for those who did not pass the assignments. Examination is either by hand-in assignments or TEN1. Ten 1 can only give the grade 3.

Four-grade scale, LiU, U, 3, 4, 5

### Department

Matematiska institutionen

Jesper Thorén

### Examiner

Milagros Izquirdo Barrios

### Course website and other links

http://www.mai.liu.se/und/kurser/index-amne-tm.html

### Education components

Preliminary scheduled hours: 48 h
Recommended self-study hours: 112 h

### Course literature

##### Books
A. Pressley, (2001) Elementary Differential Geometry Springer-Verlag

### Books

A. Pressley, (2001) Elementary Differential Geometry Springer-Verlag

 UPG1 Assignments U, 3, 4, 5 6 credits TEN1 Written Examination U, 3, 4, 5 6 credits

A written exam will be offered in August for those who did not pass the assignments. Examination is either by hand-in assignments or TEN1. Ten 1 can only give the grade 3.

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