Differentiable Manifolds (MAE728)

Undergraduate Courses Outlines - Department of Mathematics

General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE728
Semester	7
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)	http://users.uoi.gr/ansavas/lectures/id-5.html

Learning Outcomes

Learning outcomes	In this lecture we introduce basic notions of modern Differential Geometry. More precisely, we introduce among others the notions of manifold, tangent bundle, connection, parallel transport and Riemannian metric.
General Competences	work autonomously work in teams develop critical thinking skills.

Syllabus

• Smooth manifolds.
• Smooth maps.
• Tangent vectors.
• Vector fields.
• Regular values and Sard's Theorem.
• Homotopy and Isotopy.
• Lie bracket.
• Frobenius' 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 39 Autonomous Study 111 Course total 150
Student Performance Evaluation	Weakly homeworks and written final examination.

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.