ECTS - Riemannian Geometry
Riemannian Geometry (MATH574) Course Detail
| Course Name | Course Code | Season | Lecture Hours | Application Hours | Lab Hours | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| Riemannian Geometry | MATH574 | Area Elective | 3 | 0 | 0 | 3 | 5 |
| Pre-requisite Course(s) |
|---|
| N/A |
| Course Language | English |
|---|---|
| Course Type | Elective Courses |
| Course Level | Ph.D. |
| Mode of Delivery | Face To Face |
| Learning and Teaching Strategies | Lecture, Discussion, Question and Answer. |
| Course Lecturer(s) |
|
| 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;
|
| 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 |
Sources
| 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 | X |
|---|---|
| Major Area Courses | |
| Supportive Courses | |
| 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 | Ability to carry out advanced research activities, both individual and as a member of a team | |||||
| 2 | Ability to evaluate research topics and comment with scientific reasoning | |||||
| 3 | Ability to initiate and create new methodologies, implement them on novel research areas and topics | |||||
| 4 | Ability to produce experimental and/or analytical data in systematic manner, discuss and evaluate data to lead scintific conclusions | |||||
| 5 | Ability to apply scientific philosophy on analysis, modelling and design of engineering systems | |||||
| 6 | Ability to synthesis available knowledge on his/her domain to initiate, to carry, complete and present novel research at international level | |||||
| 7 | Contribute scientific and technological advancements on engineering domain of his/her interest area | |||||
| 8 | Contribute industrial and scientific advancements to improve the society through research activities | |||||
ECTS/Workload Table
| Activities | Number | Duration (Hours) | Total Workload |
|---|---|---|---|
| Course Hours (Including Exam Week: 16 x Total Hours) | |||
| Laboratory | |||
| Application | |||
| Special Course Internship | |||
| Field Work | |||
| Study Hours Out of Class | 14 | 3 | 42 |
| Presentation/Seminar Prepration | |||
| Project | |||
| Report | |||
| 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 | ||