# Differential Geometry (ΓΕ2)

### General

School School of Science Department of Mathematics Graduate ΓΕ2 1 Differential Geometry Lectures (Weekly Teaching Hours: 3, Credits: 7.5) Special Background Linear Algebra, Topology, Calculus of Several Variables. Greek Yes (in English) See eCourse, the Learning Management System maintained by the University of Ioannina.

### Learning Outcomes

Learning outcomes In this lecture we introduce basic notions of Differential Geometry. More precisely, we introduce among others the notions of manifold, manifold with boundary, vector bundle, connection, parallel transport, submanifold, differential form and de Rham cohomology. Work autonomously Work in teams Develop critical thinking skills.

### Syllabus

• Topological and smooth manifolds.
• Tangent and cotangent bundles.
• Vector fields and their flows.
• Submanifolds and Frobenius’ Theorem.
• Vector bundles.
• Connection and parallel transport.
• Differential forms.
• De Rham cohomology.
• Integration.
• Stokes’ Theorem.

### Teaching and Learning Methods - Evaluation

Delivery Face-to-face.
Use of Information and Communications Technology -
Teaching Methods
Lectures 39
Autonomous Study 78
Solution of Exercises - Homeworks 70.5
Course total 187.5
Student Performance Evaluation

Weakly HomeWorks, presentation in the blackboard of the HomeWorks, written final examination.

### Attached Bibliography

• M. do Carmo, Riemannian Geometry, Birkhaüser Boston, Inc., Boston, MA, 1992.
• J. Jost, Riemannian Geometry and Geometric Analysis, Universitext, Springer, 2017.
• J. Lee, Introduction to smooth manifolds, Second edition, Graduate Texts in Mathematics, 218, Springer, 2013.
• Δ. Κουτρουφιώτης, Διαφορική Γεωμετρία, Πανεπιστήμιο Ιωαννίνων, 1994.