Engelska

Varje hösttermin

The overall goal of the course is to provide a classic introduction to differential geometry, important for further studies in the subject and in relevant areas of physics and technology. The purpose is further to develop the PhD students' ability to solve problems and communicate mathematical reasoning.

*The geometry of curves in Euclidean space, their curvature and torsion and how these determine the curves.

*The geometry of surfaces in Euclidean space, their first and second fundamental forms, the Gauss map, principal curvatures, Gaussian curvature and mean curvature.

*Theorema Egregium and a deep analysis of geodesics and their behaviour both locally and globally.

*Gauss-Bonnet's Theorem: two different local versions and the famous global version

be able to give an account of the concepts and methods within classic differential geometry that are treated in the course,

be able to identify the most important results in the course and give an account of their proofs,

give a detailed account of the theory behind the methods used in differential geometry within the framework of the course.

be able to integrate knowledge from the different parts of the course in connection with problem solving,

be able to describe the solution to a mathematical problem within the course framework in speech and writing, logically coherent and with adequate terminology,

be able to plan and carry out relevant assignments for the course using appropriate methods within a given time-frame.

be able to argue for the importance of differential geometry as a tool in other areas, e.g. physics.

Föreläsningar

Seminarier

Skriftlig tentamen

Muntlig tentamen

övrigt

Oral presentation of a group assignment during the course

Underkänd, godkänd

FMAB22 Linear Algebra and FMAB35 Calculus in Several Variables.

Pressley, A.N.: Elementary Differential Geometry. Springer, 2010. ISBN 9781848828902.
Gudmundsson, S.: An Introduction to Gaussian Geometry. 2024.
Carmo, Manfredo P. do: Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition. Courier Dover Publications, 2016. ISBN 9780486806990.
Woodward, L. & Bolton, J.: A First Course in Differential Geometry: Surfaces in Euclidean Space. Cambridge University Press, 2018. ISBN 9781108441025.

The lecture notes by Gudmundsson are available as
http://www.matematik.lu.se/matematiklu/personal/sigma/Gauss.pdf .
The other titles are recommended supplementary reading.

FMAP20F

2025-01-14

Maria Sandsten