RIEMANNIAN GEOMETRY
| Title | ΓΕΩΜΕΤΡΙΑ RIEMANN / RIEMANNIAN GEOMETRY |
| Code | Θ033 |
| Faculty | Sciences |
| School | Mathematics |
| Cycle / Level | 2nd / Postgraduate |
| Teaching Period | Winter/Spring |
| Coordinator | Manousos Maridakis |
| Common | Yes |
| Status | Active |
| Course ID | 600025922 |
Programme of Study: PMS Tmīmatos Mathīmatikṓn (2025-2030)
Registered students: 0
| Orientation | Attendance Type | Semester | Year | ECTS |
|---|---|---|---|---|
| THEŌRĪTIKA MATHĪMATIKA | Elective Courses belonging to the selected specialization | Spring | - | 10 |
| Academic Year | 2025 – 2026 |
| Class Period | Spring |
| Faculty Instructors | |
| Weekly Hours | 3 |
| Total Hours | 39 |
| Class ID | 600268428
|
Course Type 2021
Specialization / Direction
Mode of Delivery
- Face to face
- Distance learning
Digital Course Content
- e-Study Guide https://qa.auth.gr/en/class/1/600268428
- eLearning:
Erasmus
The course is also offered to exchange programme students.
Language of Instruction
- Greek (Instruction, Examination)
- English (Instruction, Examination)
Prerequisites
General Prerequisites
General background on Differential Manifolds at graduate level.
Learning Outcomes
Upon successful completion of the course, students will be able:
1) to have a basic knowledge of fundamental examples of Riemannian manifolds,
2) to study basic properties of geodesics as the geodesic completeness and learn how to calculate using normal
coordinates,
3) to calculate Riemannian curvature in basic examples using the symmetries of the curvature tensor,
4) to extract local geometric properties by using the 2nd variation of the energy and the use of Jacobi fields,
5) to use foundational theorems of global nature about Riemannian manifolds.
General Competences
- Apply knowledge in practice
- Work autonomously
- Work in teams
- Advance free, creative and causative thinking
Course Content (Syllabus)
Tensor bundles on manifolds, Riemannian metrics, Levi Civita connection, parallel transport, theory of geodesics, geodesic completeness, Riemann, sectional, Ricci and scalar curvatures, Energy and length variations of geodesics, Jacobi fields and theorems of global Riemannian geometry.
Keywords
Riemannian metrics, Levi-Civita connection, geodesics, Riemann's curvature tensor, Ricci curvature, sectional curvature, energy-length of a path, Jacobi fields
Educational Material Types
- Notes
- Multimedia
- Book
Use of Information and Communication Technologies
Use of ICT
- Use of ICT in Course Teaching
- Use of ICT in Communication with Students
- Use of ICT in Student Assessment
Description
Online exercise hours.
Course Organization
| Activities | Workload | ECTS | Individual | Teamwork | Erasmus |
|---|---|---|---|---|---|
| Lectures | 39 | ||||
| Reading Assigment | 200 | ||||
| Tutorial | 20 | ||||
| Written assigments | 25 | ||||
| Exams | 16 | ||||
| Total | 300 |
Student Assessment
Description
Course grades will be based mostly on homework assignments and a presentation of a special topic.
Student Assessment methods
- Written Exam with Extended Answer Questions (Formative, Summative)
- Written Assignment (Formative, Summative)
- Oral Exams (Formative, Summative)
- Performance / Staging (Formative, Summative)
- Written Exam with Problem Solving (Formative, Summative)
Bibliography
Course Bibliography (Eudoxus)
Riemannian Geometry, by M. do Carmo.
Comparison Theorems in Riemannian Geometry, by J.H. Eschenburg.
Additional bibliography for study
Comparison Theorems in Riemannian Geometry, by J. Cheeger and D. Ebin.
Riemannian Geometry, by P. Petersen.
Last Update
12-05-2025
