# Curves and Surfaces

Vak
2024-2025

Linear Algebra 1,2, Analysis 2,3, Complex Analysis, Algebra 1.

## Description

In this course, we will focus on curves and surfaces embedded into three-dimensional space. We begin the course with the basics of curves: curvature and torsion. Then, the basics of surfaces in three-dimensional space will be discussed: the First and Second Fundamental Form, the Gauss map, and the principal curvatures. We continue the course with the intrinsic geometry of surfaces proving Gauss-Bonnet Theorem. We end the course with the definition of topological manifolds and with the classification of orientable compact topological surfaces.

## Course objectives

Multivariate differentiation: implicit function theorem, inverse function theorem
Curves: Parametrized curves, curvature, canonical form and global properties of plane curves
Surfaces: Regular surfaces, tangent planes, fundamental forms, orientation, Gauss-map, parallel-transport, geodesics, Gauss-Bonnet theorem
Introduction to Topological Manifolds: Classification of surfaces, Whitney's embedding theorem

## Timetable

The schedule for the course can be found on MyTimeTable.

In MyTimetable, you can find all course and programme schedules, allowing you to create your personal timetable. Activities for which you have enrolled via MyStudyMap will automatically appear in your timetable.

Questions? Watch the video, read the instructions, or contact the ISSC helpdesk.

Note: Joint Degree students from Leiden/Delft need to combine information from both the Leiden and Delft MyTimetables to see a complete schedule. This video explains how to do it.

## Mode of instruction

Lectures (maybe online), and discussion sessions

## Assessment method

Weekly lectures and problem sessions. Written exam plus weekly homework. The final grade is the weighted average of the written exam (80%) and the homework grade (20%).

Manfredo P. Do Carmo: Differential Geometry of Curves and Surfaces

## Registration

As a student, you are responsible for enrolling on time through MyStudyMap.

In this short video, you can see step-by-step how to enrol for courses in MyStudyMap.
Extensive information about the operation of MyStudyMap can be found here.

There are two enrolment periods per year:

• Enrolment for the fall opens in July

• Enrolment for the spring opens in December

Note:

• It is mandatory to enrol for all activities of a course that you are going to follow.

• Your enrolment is only complete when you submit your course planning in the ‘Ready for enrolment’ tab by clicking ‘Send’.

• Not being enrolled for an exam/resit means that you are not allowed to participate in the exam/resit.

## Contact

hablicsekhm@math.leidenuniv.nl

## Remarks

Software
Starting from the 2024/2025 academic year, the Faculty of Science will use the software distribution platform Academic Software. Through this platform, you can access the software needed for specific courses in your studies. For some software, your laptop must meet certain system requirements, which will be specified with the software. It is important to install the software before the start of the course. More information about the laptop requirements can be found on the student website.