# Differentiable Manifolds (MAE728)

### General

School School of Science Department of Mathematics Undergraduate MAE728 7 Differentiable Manifolds Lectures, laboratory exercises (Weekly Teaching Hours: 3, Credits: 6) Special Background - Greek, English Yes See eCourse, the Learning Management System maintained by the University of Ioannina.

### 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. 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
Lectures 39
Autonomous Study 111
Course total 150
Student Performance Evaluation

Weakly homeworks and written final examination.

### 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. Books and other resources, not provided by Eudoxus:

• 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.