Riemannian Geometry (ΓΕ3)

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	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	Written final examination, presentations of HomeWorks.