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Riemannian Geometry
Provider: Faculty of Science

Activity no.: 5548-23-07-31 
Enrollment deadline: 24/04/2023
PlaceDepartment of Mathematical Sciences
Universitetsparken 5, 2100 København Ø
Date and time24.04.2023, at: 08:00 - 23.06.2023, at: 16:00
Regular seats50
ECTS credits7.50
Contact personNina Weisse    E-mail address: weisse@math.ku.dk
Enrolment Handling/Course OrganiserNiels Martin Møller    E-mail address: nmoller@math.ku.dk
Written languageEnglish
Teaching languageEnglish
Semester/BlockBlock 4
Scheme groupA (Tues 8-12 + Thurs 8-17)
Exam requirementsThe student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course
Exam formContinuous assessment
Exam details7 written assignments during the course of which the 5 best count equally. In addition, one 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 aidsAll aids allowed
Grading scalePassed / Not passed
Course workload
Course workload categoryHours
Preparation106.00
Lectures40.00
Theory exercises32.00
Exam28.00
Sum206.00
Content
1. Differentiable manifolds and vector bundles.

2. Linear connections and curvature tensor

3. Riemannian metric, the Levi-Civita connection

4. Curvature

5. Geodesics and the exponential map

6. Extremal properties of geodesics

Learning outcome
At the end of the course the students are expected to have acquired the following knowledge and associated tool box:
- the mathematical framework of Riemannian geometry, including the basic theory of vector bundles
- the Levi-Civita connection
- the Riemann curvature tensor and its basic properties including the Bianchi identities
- immersed submanifolds and the second fundamental form, including examples
- geodesics and the exponential map and extremal properties

Skills:
- be able to work rigorously with problems from Riemannian geometry
- be able to treat a class of variational problems by rigorous methods
- be able to use extremal properties of geodesics to analyse global properties of manifolds

Competences:
The course aims at training the students in representing, modelling and handling geometric problems by using advanced mathematical concepts and techniques from Riemannian geometry.

Teaching and learning methods
3+2 lectures (including seminars by students) and 2+2 tutorials per week during 8 weeks.

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