Program Overview

Program Overview

The University of Copenhagen offers a Master's program in Mathematics, with a course on Riemannian Geometry (NMAK20006U).

Course Description

The course covers the mathematical framework of Riemannian geometry, including the basic theory of vector bundles, the Levi-Civita connection, and the Riemann curvature tensor. It also explores immersed submanifolds, geodesics, and the exponential map.

Learning Outcomes

Upon completing the course, students are expected to have acquired the following knowledge and skills:

• The mathematical framework of Riemannian geometry
• The Levi-Civita connection and the Riemann curvature tensor
• Immersed submanifolds and the second fundamental form
• Geodesics and the exponential map
• Extremal properties of geodesics

Skills and Competences

The course aims to train students in representing, modeling, and handling geometric problems using advanced mathematical concepts and techniques from Riemannian geometry. Students will be able to:

• Work rigorously with problems from Riemannian geometry
• Treat a class of variational problems by rigorous methods
• Use extremal properties of geodesics to analyze global properties of manifolds

Literature and Academic Qualifications

The course uses lecture notes and/or a textbook. Recommended academic qualifications include Geometri 2 or corresponding knowledge of differentiable manifolds, as well as academic qualifications equivalent to a BSc degree.

Teaching and Learning Methods

The course consists of lectures and tutorials, with 3+2 lectures (including seminars by students) and 2+2 tutorials per week during 8 weeks.

Workload

The workload for the course is as follows:

• Lectures: 40 hours
• Preparation: 106 hours
• Theory exercises: 32 hours
• Exam: 28 hours
• Total: 206 hours

Assessment

The course is assessed through continuous assessment, with 7 written assignments during the course, of which the 5 best count equally. Additionally, students must give a seminar talk of 45 minutes about a topic to be specified during the course. The written work and seminar talk count with equal weights in the final grade.

Exam Details

• Type of assessment: Continuous assessment
• Type of assessment details: 7 written assignments and a seminar talk
• Aid: All aids allowed
• Marking scale: 7-point grading scale
• Censorship form: No external censorship, one internal examiner

Re-exam

The re-exam consists of resubmission of all 7 assignments from the continuous assessment, of which the 5 best count equally, and a half an hour oral re-exam without any preparation time. The written work and oral re-exam count with equal weights in the final grade.

Course Information

• Language: English
• Course code: NMAK20006U
• Credit: 7.5 ECTS
• Level: Full Degree Master
• Duration: 1 block
• Placement: Block 4
• Schedule: A
• Course capacity: No limitation, unless registering in the late-registration period or as a credit or single subject student.

Study Board and Contracting Department

• Study Board: Study Board of Mathematics and Computer Science
• Contracting department: Department of Mathematical Sciences
• Contracting faculty: Faculty of Science

Course Coordinators

• Niels Martin Møller
• Artemis Aikaterini Vogiatzi

Additional Information

The course is also available as continuing and professional education. The University of Copenhagen offers various services, including libraries, museums, and attractions, as well as a job portal and alumni network.