Riemannian Geometry (MAE825)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE722
Semester	7
Course Title	Riemannian Geometry
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)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The main task is to present the fundamental concepts of Riemannian geometry, i.e., the concepts of curvatures and differential form on manifolds with boundary. Moreover, we will introduce the notions of Riemannian submanifold and will investigate the Gauss-Codazzi-Ricci equations. The lecture will be completed with the presentation of the sphere theorem, a deep and important result that connects geometry with topology. On the completion of the course we expect that the student fully understand the main theorems that were presented during the lectures.

General Competences

Working independently Decision-making Production of free, creative and inductive thinking Criticism and self-criticism

Syllabus

Riemannian metrics, curvature operator, Schur's theorem, differential forms, integration on manifolds, Stokes' theorem, Riemannian submanifolds, sphere theorem.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures (13X3) 39 Working independently 78 Exercises-Homeworks 33 Course total 150
Student Performance Evaluation	Weekly exercises and homeworks, presentations, final written exam in Greek (in case of Erasmus students in English) which includes resolving application problems.

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.
• J. Eschenburg, Comparison Theorems in Riemannian Geometry, Universität Augsburg, 1994.
• J. Jost, Riemannian Geometry and Geometric Analysis, Universitext, Springer, 2017.
• J. Lee, Riemannian manifolds: An Introduction to Curvature, Vol. 176, Springer, 1997.
• P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, 171, Springer, 2016.
• Δ. Κουτρουφιώτης, Διαφορική Γεωμετρία, Πανεπιστήμιο Ιωαννίνων, 1994.