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 | Natural & Applied Sciences Master's Degree |
| 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 | |
|---|---|
| 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. | X | ||||
| 2 | Has the ability to obtain, to evaluate, to interpret and to apply information by doing scientific research. | X | ||||
| 3 | Can apply gained knowledge and problem solving abilities in inter-disciplinary research. | X | ||||
| 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. | X | ||||
| 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. | X | ||||
| 6 | Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework. | X | ||||
| 7 | Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility. | X | ||||
| 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. | X | ||||
| 9 | Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields. | X | ||||
| 10 | Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge. | X | ||||
| 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. | X | ||||
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 | ||