Riemannian Geometry (ΓΕ3)
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Graduate Courses Outlines - Department of Mathematics
General
| School | School of Science |
|---|---|
| Academic Unit | Department of Mathematics |
| Level of Studies | Graduate |
| Course Code | ΓΕ3 |
| Semester | 2 |
| Course Title | Riemannian Geometry |
| Independent Teaching Activities | Lectures (Weekly Teaching Hours: 3, Credits: 7.5) |
| Course Type | Special Background |
| Prerequisite Courses |
Differential Geometry (ΓΕ2) |
| Language of Instruction and Examinations |
Greek |
| Is the Course Offered to Erasmus Students | Yes (in English) |
| Course Website (URL) | See eCourse, the Learning Management System maintained by the University of Ioannina. |
Learning Outcomes
| Learning outcomes |
In this lecture we introduce basic notions of Riemannian Geometry. More precisely, we introduce among others the notions of Riemannian metric, Levi-Civita connection, holonomy, curvature operator, Ricci curvature, sectional curvature, scalar curvature and Jacobi field. |
|---|---|
| General Competences |
|
Syllabus
- Riemannian metrics, isometries, conformal maps.
- Geodesics and exponential maps.
- Parallel transport and holonomy.
- Hopf-Rinow’s Theorem.
- Curvature operator, Ricci curvature, scalar curvature.
- Riemannian submanifolds.
- Gauss-Codazzi-Ricci equations.
- 1st and 2nd variation of length.
- Jacobi fields.
- Comparison theorems.
- Homeomorphic sphere theorem.
Teaching and Learning Methods - Evaluation
| Delivery |
Face-to-face | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Use of Information and Communications Technology | - | ||||||||||
| Teaching Methods |
| ||||||||||
| Student Performance Evaluation |
Written final examination, presentations of HomeWorks. |
Attached Bibliography
- M. do Carmo, Riemannian Geometry, Birkhaüser Boston, Inc., Boston, MA, 1992.
- J.-Η. Eschenburg, Comparison theorems in Riemannian Geometry, Lecture Notes, Universität Augsburg, 1994.
- J. Jost, Riemannian Geometry and Geometric Analysis, Seventh edition, Universitext, Springer, 2017.
- J. Lee, Riemannian manifolds: An introduction to curvature, Graduate Texts in Mathematics, 176, Springer, 1997.
- P. Petersen, Riemannian Geometry, Third edition, Graduate Texts in Mathematics, 171, Springer, 2016.
- Δ. Κουτρουφιώτης, Διαφορική Γεωμετρία, Πανεπιστήμιο Ιωαννίνων, 1994.