Differentiable Manifolds (MAE622)
Undergraduate Courses Outlines - Department of Mathematics
General
| School |
School of Science |
|---|---|
| Academic Unit |
Department of Mathematics |
| Level of Studies |
Undergraduate |
| Course Code |
MAE622 |
| Semester |
6 |
| Course Title |
Differentiable Manifolds |
| 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 |
In this lecture, the fundamental concept of a differentiable manifold will be developed. In particular, this lecture is a basic prerequisite for the upcoming class of Riemannian geometry. After a quick review of basic facts from general topology we will introduce the notions of a smooth manifold, tangent bundle, vector field, submanifold, connection, geodesic curve, parallel transport and Riemannian metric. On the completion of this course we expect that the students fully understand these important concepts and the main theorems that will be presented in the lectures. |
|---|---|
| General Competences |
|
Syllabus
Review of basic facts from general topology, smooth manifolds, tangent bundle, vector fields, immersions and embeddings, Lie bracket, Frobenius' theorem, Whitney's embedding theorem, connections and parallel transport, Riemannian metrics.
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.
- V. Guillemin & A. Pollack, Differentiable Topology, Prentice-Hall, Inc, Englewood Cliffs, 1974.
- J. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, 218, 2013.
- J. Milnor, Topology From the Differentiable Viewpoint, Princeton University Press, NJ, 1997.
- L. Tu, An Introduction to Manifolds, Universitext. Springer, New York, 2011.
- Δ. Κουτρουφιώτης, Διαφορική Γεωμετρία, Πανεπιστήμιο Ιωαννίνων, 1994.