Riemannian Geometry (MAE825)

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Undergraduate Courses Outlines - Department of Mathematics

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) -

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 Eudoxus. Additionally:

  • 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.