MATHEMATICS (TURKISH, PHD) | |||||
PhD | TR-NQF-HE: Level 8 | QF-EHEA: Third Cycle | EQF-LLL: Level 8 |
Course Code | Course Name | Semester | Theoretical | Practical | Credit | ECTS |
MAT6027 | Semi-Riemannian Geometry | Fall | 3 | 0 | 3 | 8 |
The course opens with the approval of the Department at the beginning of each semester |
Language of instruction: | Tr |
Type of course: | Departmental Elective |
Course Level: | |
Mode of Delivery: | |
Course Coordinator : | Prof. Dr. ERTUĞRUL ÖZDAMAR |
Course Objectives: | Yari-Riemann geometrisi dersi, Lorentz ve genelde de Yari-Riemann geometriye bir giriş dersidir. During the course, basic information aimed to meet the needs of researchers. |
The students who have succeeded in this course; upon succeeding this course the student will be able to 1) know the concepts of manifold and the curvature tensor, and understood very well the role of the inner-dot product, 2) calculate the distance, area and curvature on Semi-Riemannian manifolds. 3) explain the properties of geodesics on Semi-Riemannian manifolds, and also the concepts as chronology and time cone 4) understand paradoxes in relativity. 5) Explain all properties of Warped product and the metric matrices. |
Manifold theory, Tensors, Semi-Riemannian Manifolds; Isometries,Levi-Civita connection,paralel translation, geodesics, the exponential mapping, curvature, sectional curvature, Semi-Riemann surfaces, Semi-Riemannian Submanifolds; Ricci and scalar curvature, Semi-Riemann product manifolds, local isometries, Riemannian and Lorentz Geometry; Gauss lemma, convex open sets, arclength, Riemaniann distance, Lorentz causal character, time cone, Local Lorentz geometry, geodesics,completeness and extendibility, Constructions; Deck transformations, volume elements, vector bundles, local isometries, Warped products, Isometries; isometry groups, space forms, homogeneous spaces |
Week | Subject | Related Preparation | |
1) | Preliminiars and Manifold theory | ||
2) | Manifold theory | ||
3) | Tensors | ||
4) | Tensors | ||
5) | Semi-Riemannian Manifolds; Isometries,Levi-Civita connection,paralel translation, geodesics, the exponential mapping, curvature, sectional curvature, Semi-Riemann surfaces | ||
6) | Semi-Riemannian Submanifolds; Ricci and scalar curvature | ||
7) | Semi-Riemann product manifolds, local isometries | ||
8) | Riemannian and Lorentz Geometry; Gauss lemma, convex open sets, arclength, Riemaniann distance | ||
9) | Lorentz causal character, time cone | ||
10) | Local Lorentz geometry, geodesics,completeness and extendibility | ||
11) | Constructions; Deck transformations, volume elements, vector bundles, local isometries | ||
12) | Warped products | ||
13) | Isometries; isometry groups, space forms | ||
14) | homogeneous spaces |
Course Notes: | Semi-Riemannian Geometry With Applications to Relativity,Barrett O'Neill, 103 Academic Press, ISBN: 0125267401 |
References: | . |
Semester Requirements | Number of Activities | Level of Contribution |
Attendance | 14 | % 5 |
Laboratory | % 0 | |
Application | % 0 | |
Field Work | % 0 | |
Special Course Internship (Work Placement) | % 0 | |
Quizzes | % 0 | |
Homework Assignments | 3 | % 15 |
Presentation | % 0 | |
Project | % 0 | |
Seminar | % 0 | |
Midterms | 2 | % 35 |
Preliminary Jury | % 0 | |
Final | 1 | % 45 |
Paper Submission | % 0 | |
Jury | % 0 | |
Bütünleme | % 0 | |
Total | % 100 | |
PERCENTAGE OF SEMESTER WORK | % 55 | |
PERCENTAGE OF FINAL WORK | % 45 | |
Total | % 100 |
Activities | Number of Activities | Duration (Hours) | Workload |
Course Hours | 14 | 3 | 42 |
Laboratory | 0 | 0 | 0 |
Application | 0 | 0 | 0 |
Special Course Internship (Work Placement) | 0 | 0 | 0 |
Field Work | 0 | 0 | 0 |
Study Hours Out of Class | 0 | 0 | 0 |
Presentations / Seminar | 0 | 0 | 0 |
Project | 0 | 0 | 0 |
Homework Assignments | 3 | 25 | 75 |
Quizzes | 0 | 0 | 0 |
Preliminary Jury | 0 | ||
Midterms | 2 | 30 | 60 |
Paper Submission | 0 | ||
Jury | 0 | ||
Final | 1 | 25 | 25 |
Total Workload | 202 |
No Effect | 1 Lowest | 2 Low | 3 Average | 4 High | 5 Highest |
Program Outcomes | Level of Contribution | |
1) | Ability to assimilate mathematic related concepts and associate these concepts with each other. | |
2) | Ability to gain qualifications based on basic mathematical skills, problem solving, reasoning, association and generalization. | |
3) | Be able to organize events, for the development of critical and creative thinking and problem solving skills, by using appropriate methods and techniques. | |
4) | Ability to make individual and team work on issues related to working and social life. | |
5) | Ability to transfer ideas and suggestions, related to topics about his/her field of interest, written and verball. | |
6) | Ability to use mathematical knowledge in technology. | |
7) | To apply mathematical principles to real world problems. | |
8) | Ability to use the approaches and knowledge of other disciplines in Mathematics. | |
9) | Be able to set up and develope a solution method for a problem in mathematics independently, be able to solve and evaluate the results and to apply them if necessary. | |
10) | To be able to link abstract thought that one has to concrete events and to transfer the solutions and examine and interpret the results scientifically by forming experiments and collecting data. | |
11) | To be able to conduct a research either as an individual or as a team member, and to be effective in each related step of the project, to take role in the decision process, to plan and manage the project by using time effectively. | |
12) | To be able to acquire necessary information and to make modeling in any field that mathematics is used and to improve herself/himself. |