Differential Geometry (ΓΕ2)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΓΕ2
Semester	1
Course Title	Differential Geometry
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	Linear Algebra, Topology, Calculus of Several Variables.
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	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.
General Competences	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	Activity Semester Workload 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.