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 | Is independently able to build a problem in the area of study, solve the problem by developing solution techniques and assess the solutions. | X | ||||
| 2 | Is capable of creating a groundwork in the fundamental branches of mathematics as well as in his/her research area | X | ||||
| 3 | follows the latest national and international literature in Mathematics and in his/her area of research; and uses them in his/her related studies | X | ||||
| 4 | observes and adopts the scientific ethical values in his/her professional and social life | X | ||||
| 5 | presents in Turkish and English in academic/scientific events the results of his/her research or the latest studies and findings on a special topic and participates in discussions | X | ||||
| 6 | Develops skills to work independently or as a member of a team | X | ||||
| 7 | Develops competences in the areas of creative and critical thinking, problem solving and producing original studies. Follows recent scientific studies, is capable of making an analysis, synthesis and assessment of the knowledge acquired | X | ||||
| 8 | Is open to lifelong improvement of his/her acquired knowledge, skills and competences. | X | ||||
| 9 | Is able to apply the acquired knowledge and problem-solving skills to interdisciplinary studies, proposes different solution methods to problems in terms of mathematical models and from a mathematical point of view | X | ||||
| 10 | Uses the mathematical based softwares, informatics and communication technologies for scientific purposes | 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 | ||