Classical Differential Geometry (ΓΕ1)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΓΕ1
Semester	1
Course Title	Classical Differential Geometry
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	Topology, Calculus of Several Variables, Complex Analysis.
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 Classical Differential Geometry. More precisely, we introduce among others the notions of a manifold as a subset of the Euclidean space. Then, we present various local and global theorems concerning minimal submanifolds.
General Competences	Work autonomously. Work in teams. Develop critical thinking skills.

Syllabus

• Manifolds of the Euclidean space.
• Tangent and normal bundles.
• 1st and 2nd fundamental forms.
• Weingarten operator and Gauss map.
• Convex hypersurfaces.
• Hadamard’s Theorem.
• 1st and 2nd variation of area.
• Minimal submanifolds.
• Weierstrass representation.
• Bernstein’s Τheorem.

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, presentations of the HomeWorks in the blackboard, 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.