Riemannian Geometry (MAE825)
Undergraduate Courses Outlines - Department of Mathematics
General
| School |
School of Science |
|---|---|
| Academic Unit |
Department of Mathematics |
| Level of Studies |
Undergraduate |
| Course Code |
MAE722 |
| Semester |
7 |
| Course Title |
Riemannian Geometry |
| Independent Teaching Activities |
Lectures, laboratory exercises (Weekly Teaching Hours: 3, Credits: 6) |
| Course Type |
Special Background |
| Prerequisite Courses | - |
| Language of Instruction and Examinations |
Greek, English |
| Is the Course Offered to Erasmus Students |
Yes |
| Course Website (URL) | - |
Learning Outcomes
| Learning outcomes |
The main task is to present the fundamental concepts of Riemannian geometry, i.e., the concepts of curvatures and differential form on manifolds with boundary. Moreover, we will introduce the notions of Riemannian submanifold and will investigate the Gauss-Codazzi-Ricci equations. The lecture will be completed with the presentation of the sphere theorem, a deep and important result that connects geometry with topology. On the completion of the course we expect that the student fully understand the main theorems that were presented during the lectures. |
|---|---|
| General Competences |
|
Syllabus
Riemannian metrics, curvature operator, Schur's theorem, differential forms, integration on manifolds, Stokes' theorem, Riemannian submanifolds, sphere theorem.
Teaching and Learning Methods - Evaluation
| Delivery |
Classroom (face-to-face) | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Use of Information and Communications Technology | - | ||||||||||
| Teaching Methods |
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| Student Performance Evaluation |
Weekly exercises and homeworks, presentations, final written exam in Greek (in case of Erasmus students in English) which includes resolving application problems. |
Attached Bibliography
See Eudoxus. Additionally:
- M. do Carmo, Riemannian Geometry, Birkhaüser Boston, Inc., Boston, MA, 1992.
- J. Eschenburg, Comparison Theorems in Riemannian Geometry, Universität Augsburg, 1994.
- J. Jost, Riemannian Geometry and Geometric Analysis, Universitext, Springer, 2017.
- J. Lee, Riemannian manifolds: An Introduction to Curvature, Vol. 176, Springer, 1997.
- P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, 171, Springer, 2016.
- Δ. Κουτρουφιώτης, Διαφορική Γεωμετρία, Πανεπιστήμιο Ιωαννίνων, 1994.