# Introduction to Riemannian Geometry

 Title ΕΙΣΑΓΩΓΗ ΣΤΗ ΓΕΩΜΕΤΡΙΑ RIEMANN / Introduction to Riemannian Geometry Code 0333Α Faculty Sciences School Mathematics Cycle / Level 1st / Undergraduate Teaching Period Spring Common No Status Active Course ID 600019940

### Programme of Study: UPS of School of Mathematics (2014-today)

Registered students: 32
OrientationAttendance TypeSemesterYearECTS
CoreElective Courses belonging to the selected specializationSpring-5.5

 Academic Year 2021 – 2022 Class Period Spring Instructors from Other Categories Kleanthis Polymerakis 39hrs Weekly Hours 3 Class ID 600187086
Course Type 2016-2020
• Scientific Area
Course Type 2011-2015
Knowledge Deepening / Consolidation
Mode of Delivery
• Face to face
Digital Course Content
Erasmus
The course is also offered to exchange programme students.
Language of Instruction
• Greek (Instruction, Examination)
Prerequisites
Required Courses
• 0110 ELEMENTS OF LINEAR ALGEBRA
• 0305 ELEMENTS OF ANALYTIC GEOMETRY
• 0306 INTRODUCTION TO GEOMETRY I
• 0307 INTRODUCTION TO GEOMETRY II
• 0108Α Linear Algebra
• 0303Α Classical Differential Geometry I
• 0304Α Differential Manifolds
Learning Outcomes
Deepening on introductory concepts of Riemannian geometry.
General Competences
• Apply knowledge in practice
• Work autonomously
• Work in teams
Course Content (Syllabus)
The notion of the metric and of the isometry map. Theory of connections. the notion of the covariant derivative. Levi-Civita connection. Geodesics. Curvature tensor. Sectional curvature, Ricci curvature and scalar curvature. Hypersurfaces of a Riemannian manifold. Riemannian manifolds of constant curvature. Elements from the theory of smooth manifolds. Riemannian manifolds. Linear connections. Geodesics and curvature. Sectional curvature. Jacobi fields and second fundamental form.
Educational Material Types
• Notes
• Book
Use of Information and Communication Technologies
Use of ICT
• Use of ICT in Course Teaching
• Use of ICT in Communication with Students
• Use of ICT in Student Assessment
Course Organization
Lectures1304.3
Seminars
Exams30.1
Total1655.5
Student Assessment
Description
Written Examination
Student Assessment methods
• Written Exam with Short Answer Questions (Formative, Summative)
• Written Exam with Extended Answer Questions (Formative, Summative)
• Written Exam with Problem Solving (Formative, Summative)
Bibliography
Course Bibliography (Eudoxus)
Δημητρίου Κουτρουφιώτη, Διαφορική Γεωμετρία, Πανεπιστήμιο Ιωαννίνων 1994. Ανδρέα Αρβανιτογεώργου, Γεωμετρία Πολλαπλοτήτων – Πολλαπλότητες Riemann και Ομάδες Lie, Ελληνικά Ακαδημαϊκά Ηλεκτρονικά Συγγράμματα και Βοηθήματα, www.kallipos.gr
Additional bibliography for study
M. P. do Carmo, Riemannian Geometry, Birkhäuser 1992. John M. Lee, Riemannian manifolds. An introduction to curvature, GTM 176, Springer-Verlag 1997. Andrew Mclnerney, First Steps in Differential Geometry, Riemannian, Contact, Symplectic, Springer 2013. Leonor Godinho, Jose Natario, An Introduction to Riemannian Geometry, With Applications to Mechanics and Relativity, Springer 2014. Loring W. Tu, An introduction to Manifolds, Universitext, Springer 2011.
Last Update
22-07-2021