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

Working independently Decision-making Production of free, creative and inductive thinking Criticism and self-criticism

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	Activity Semester Workload Lectures (13X3) 39 Working independently 78 Exercises-Homeworks 33 Course total 150
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 the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. 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.