Riemannian Geometry (MATH574) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Riemannian Geometry MATH574 3 0 0 3 5
Pre-requisite Course(s)
Consent of the Department
Course Language English
Course Type N/A
Course Level Natural & Applied Sciences Master's Degree
Mode of Delivery Face To Face
Learning and Teaching Strategies Lecture, Discussion, Question and Answer.
Course Coordinator
Course Lecturer(s)
Course Assistants
Course Objectives This course is designed to provide necessary background and further knowledge in Riemannian Geometry for graduate students of Mathematics. The content of the course serves as theory of modern geometries as well as the indispensable tool for mathematical modeling in classical physics and engineering applications.
Course Learning Outcomes The students who succeeded in this course;
  • understand the fundamental notions of connection and curvature, the geometry of submanifolds,
  • learn metric properties of geodesics and Jacobi fields,
  • learn sectional curvature, Ricci tensor and scalar curvature.
Course Content Review of differentiable manifolds and tensor fields, Riemannian metrics, the Levi-Civita connections, geodesics and exponential map, curvature tensor, sectional curvature, Ricci tensor, scalar curvature, Riemannian submanifolds, the Gauss and Codazzi equations.

Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Differentiable manifolds pp. 1-25
2 Vector fields, brackets. Topology of manifolds pp. 25-35
3 Riemannian metrics pp. 35-48
4 Affine connections, Riemannian connections pp. 48-60
5 Geodesics pp. 61-75
6 Convex neighborhoods pp. 75-88
7 Curvature, Sectional curvature pp. 88-97
8 Midterm
9 Ricci curvature, Scalar curvature pp. 97-100
10 Tensors on Riemannian manifolds pp. 100-110
11 Jacobi Fields pp. 110-124
12 Isometric immersions pp. 124-144
13 Complete manifolds, Hopf-Rinow and Hadamard Theorems pp .144-155
14 Spaces of constant curvature pp. 155-190
15 Variations of energy pp. 191-210
16 Final Exam


Course Book 1. M. P. Do Carmo, Riemannian Geometry, Birkhauser, 1992
Other Sources 2. T. J. Willmore, Riemannian Geometry, Oxford Science Publication, 2002
3. I. Chavel, Riemannian Geometry, Cambridge Univ. Press, 1993

Evaluation System

Requirements Number Percentage of Grade
Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments 6 30
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 1 30
Final Exam/Final Jury 1 40
Toplam 8 100
Percentage of Semester Work 60
Percentage of Final Work 40
Total 100

Course Category

Core Courses
Major Area Courses
Supportive Courses X
Media and Managment Skills Courses
Transferable Skill Courses

The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
1 2 3 4 5
1 Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area.
2 Has the ability to obtain, to evaluate, to interpret and to apply information by doing scientific research.
3 Can apply gained knowledge and problem solving abilities in inter-disciplinary research.
4 Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary.
5 Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study.
6 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework.
7 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility.
8 Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation.
9 Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields.
10 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge.
11 Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications.

ECTS/Workload Table

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)
Special Course Internship
Field Work
Study Hours Out of Class 14 3 42
Presentation/Seminar Prepration
Homework Assignments 6 3 18
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 1 7 7
Prepration of Final Exams/Final Jury 1 10 10
Total Workload 77