Riemannian Geometry (ΓΕ3)

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Graduate Courses Outlines - Department of Mathematics

General

School School of Science
Academic Unit Department of Mathematics
Level of Studies Graduate
Course Code ΓΕ3
Semester 2
Course Title Riemannian Geometry
Independent Teaching Activities Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type Special Background
Prerequisite Courses

Differential Geometry (ΓΕ2)

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 Riemannian Geometry. More precisely, we introduce among others the notions of Riemannian metric, Levi-Civita connection, holonomy, curvature operator, Ricci curvature, sectional curvature, scalar curvature and Jacobi field.

General Competences
  • Work autonomously
  • Work in teams
  • Develop critical thinking skills.

Syllabus

  • Riemannian metrics, isometries, conformal maps.
  • Geodesics and exponential maps.
  • Parallel transport and holonomy.
  • Hopf-Rinow’s Theorem.
  • Curvature operator, Ricci curvature, scalar curvature.
  • Riemannian submanifolds.
  • Gauss-Codazzi-Ricci equations.
  • 1st and 2nd variation of length.
  • Jacobi fields.
  • Comparison theorems.
  • Homeomorphic sphere theorem.

Teaching and Learning Methods - Evaluation

Delivery

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Use of Information and Communications Technology

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Teaching Methods
Activity Semester Workload
Lectures 39
ΧΧΧ 000
ΧΧΧ 000
Course total 187.5
Student Performance Evaluation

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Attached Bibliography

Πρότυπο:MAM199-Biblio