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Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS MAT 4016 RIEMANNIAN GEOMETRY ELECTIVE 4 0 0 7

Offered By

Mathematics

Level of Course Unit

First Cycle Programmes (Bachelor's Degree)

Course Coordinator

ASSOCIATE PROFESSOR ILHAN KARAKILIÇ

Offered to

Mathematics (Evening) Mathematics

Course Objective

This course aim to introduce students to the foundations of Riemannian geometry which is a natural generalization of the theory of surfaces in Euclidean spaces.

Learning Outcomes of the Course Unit

1   Will be able to give the definition of a Riemannian metric and a Riemannian manifold 2   will be able to explain what is a Riemannian geometry 3   will be able to calculate Riemannian metric, connection and curvature tensor of a Riemannian manifold 4   will be able to determine the properties of geodesics 5   will be able to apply global theorems of Riemannian geometry to deduce geometric and topological properties of a manifold

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description 1 Differentiable manifolds, Tangent space, Immersions and embeddings 2 Orientation, Vector fields, brackets, Topology of manifolds 3 Riemannian metrics 4 Affine connections, Riemannian connections 5 The geodesic flow, Minimizing properties of geodesics 6 Convex neighbourhoods 7 Curvature, Sectional curvature, Ricci curvature and scalar curvature 8 Problem solving Midterm 9 Tensors on Riemannian manifolds 10 The Jacobi equation and conjugate points 11 The second fundamental form, The fundamental equations 12 Complete manifolds, Hopf-Rinow and Hadamard theorems 13 Spaces of constant curvature, Variations of Energy 14 The Rouch Comparison and the Morse index theorems

Recomended or Required Reading

Textbooks: 1. Do Carmo, M. P., Riemannian Geometry, Birkhause, 1992, ISBN 978-0817634902 Supplementary Books: 2. Gallot, S., Hulin, D., Lafonteine, J., Riemannian Geometry, 3rd ed., Springer, 2004, ISBN 978-3540204930 3. Petersen, P., Riemannian Geometry, 2nd ed., Springer, 2006, ISBN 978-0387292465 References: 4. Berger, M., A Panaromic View of Riemannian Geometry, Springer, 2007, ISBN 978-3540653172

Planned Learning Activities and Teaching Methods

Lectures, Lecture notes and problem solving

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA 1 MTE MIDTERM EXAM 2 ASG ASSIGNMENT 3 FINS FINAL EXAM 4 FCG FINAL COURSE GRADE MTE * 0.30 + ASG * 0.20 + FIN * 0.50 5 RST RESIT 6 FCGR FINAL COURSE GRADE (RESIT) MTE * 0.30 + ASG * 0.20 + RST * 0.50

Further Notes About Assessment Methods

None

Assessment Criteria

To be announced.

Language of Instruction

English

Course Policies and Rules

To be announced.

Contact Details for the Lecturer(s)

E-mail: ilhan.karakilic@deu.edu.tr Office : 0 232 3018589

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours) Lectures 14 4 56 Preparations before/after weekly lectures 14 4 56 Preparation for midterm exam 1 25 25 Preparation for final exam 1 35 35 Final 1 2 2 Midterm 1 2 2 TOTAL WORKLOAD (hours) 176

Contribution of Learning Outcomes to Programme Outcomes

PO/LO PO.1 PO.2 PO.3 PO.4 PO.5 PO.6 PO.7 PO.8 PO.9 PO.10 PO.11 PO.12 PO.13 LO.1 5 3 4 3 3 LO.2 5 3 4 4 3 3 LO.3 5 4 3 4 3 3 LO.4 5 4 4 3 4 4 3 3 4 4 LO.5 5 4 4 3 4 4 3 3 4 4