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Curves and Surfaces


Admission requirements

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


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

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


The schedule for the course can be found on MyTimeTable.

You will find the timetables for all courses and degree programmes of Leiden University in the tool MyTimetable (login). Any teaching activities that you have sucessfully registered for in MyStudyMap will automatically be displayed in MyTimeTable. Any timetables that you add manually, will be saved and automatically displayed the next time you sign in.

MyTimetable allows you to integrate your timetable with your calendar apps such as Outlook, Google Calendar, Apple Calendar and other calendar apps on your smartphone. Any timetable changes will be automatically synced with your calendar. If you wish, you can also receive an email notification of the change. You can turn notifications on in ‘Settings’ (after login).

For more information, watch the video or go the the 'help-page' in MyTimetable. Please note: Joint Degree students Leiden/Delft have to merge their two different

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%).

Reading list

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


From the academic year 2022-2023 on every student has to register for courses with the new enrollment tool MyStudyMap. There are two registration periods per year: registration for the fall semester opens in July and registration for the spring semester opens in December. Please see this page for more information.

Please note that it is compulsory to both preregister and confirm your participation for every exam and retake. Not being registered for a course means that you are not allowed to participate in the final exam of the course. Confirming your exam participation is possible until ten days before the exam.
Extensive FAQ's on MyStudymap can be found here.