| Course Name | Code | Semester | T+U Hours | Credit | ECTS |
|---|---|---|---|---|---|
| Tensor Geometry II | MAT 622 | 0 | 3 + 0 | 3 | 6 |
| Precondition Courses | Students are assumed to be familiar with Differential Geometry I and Differential Geometry II |
| Recommended Optional Courses | |
| Course Language | Turkish |
| Course Level | Doctorate Degree |
| Course Type | Optional |
| Course Coordinator | Prof.Dr. MEHMET ALİ GÜNGÖR |
| Course Lecturers | |
| Course Assistants | |
| Course Category | |
| Course Objective | The tensor geometry II course aims to give the fundamental knowledge for the studies of graduate students who study at geometry branch. |
| Course Content | Transformations between manifolds, differential and tensors, connections and tensors, affine connections, Cartan structure equations, curvatures, Semi Euclidean space, semi- Riemannian surfaces, semi symmetric Riemannian manifolds, total absolute curvature, Lipschitz Killing curvature, total mean curvature |
| # | Course Learning Outcomes | Teaching Methods | Assessment Methods |
|---|---|---|---|
| 1 | He/She defines the transformations between manifolds | Lecture, Question-Answer, Discussion, Group Study, Problem Solving, | Testing, Homework, |
| 2 | He/She defines the fundamental operations on transformations | Lecture, Question-Answer, Discussion, Group Study, Problem Solving, | Testing, Homework, |
| 3 | He/She knows the concepts of connections, tensors, affine connections and the Cartan structure equations | Lecture, Question-Answer, Discussion, Group Study, Problem Solving, | Testing, Homework, |
| 4 | He/She calculates total curvature, Lipschitz-Killing curvature and total absolute curvature | Lecture, Question-Answer, Discussion, Group Study, Problem Solving, | Testing, Homework, |
| Week | Course Topics | Preliminary Preparation |
|---|---|---|
| 1 | Transformations between manifolds | |
| 2 | Differential and tensors | |
| 3 | Connections and tensors | |
| 4 | Affine connections | |
| 5 | Cartan structure equations | |
| 6 | Curvatures | |
| 7 | Semi Euclidean space | |
| 8 | Semi- Riemannian surfaces | |
| 9 | Mid term exam | |
| 10 | Semi symmetric Riemannian manifolds | |
| 11 | Total absolute curvature | |
| 12 | Lipschitz Killing curvature | |
| 13 | Lipschitz Killing curvature | |
| 14 | Total mean curvature |
| Resources | |
|---|---|
| Course Notes | 1. Hacısalihoğlu H. H. ve Ekmekçi N., Tensör Geometri, Ankara Üni., Fen Fakültesi,2003. |
| Course Resources | 1. Kobayashi, S., and Nomizu, K., Foundations of differential geometry, Number 15, Volume II, New York, 1969. 2. ONeill B., Elementary Differential Geometry, Academic Press, New York, 1997. 3. D. C. Kay, , Schaums outline of theory and problems, McGraw-Hill, 1988. 4. C. T. J. Dodson, T. Poston, Tensor geometry, Graduate Texts in Mathematics, 130. Springer-Verlag, Berlin, 1991. |
| Order | Program Outcomes | Level of Contribution | |||||
|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | |||
| 0 | Develop strategic, political and practice plans and evaluate the results by taking into account the quality process in his/her area of expertise | ||||||
| 1 | At a master´s degree level, student reaches new knowledge via scientific researches, the use of knowledge of the same field as him/her or of different field from him/her, and the use of knowledge based on the competence in his/her field; s/he interprets the knowledge and prospects for the fields of application. | X | |||||
| 2 | Student completes the missing or limited knowledge by using the scientific methods. | X | |||||
| 3 | Student freely poses a problem of his/her field, develops a solution method, solves the problem, and evaluates the result. | X | |||||
| 4 | Student conveys, orally or in writing, his/her studies or the current developments in his/her field to the people in or out of his/her field. | X | |||||
| 5 | Student finds a solution to the unforeseen complex problems in his/her studies by developing new approaches. | X | |||||
| 6 | At a doctorate degree level, student prepares at least one scientific article of his/her field to be published in an international indexed journal, and s/he extends its popularity. | X | |||||
| 7 | Student analyzes the works that have been published before, approaches the same subjects with different proof methods, or determines the open problems about the current subject matters. | X | |||||
| 8 | Student looks for the scientists studying on the same field as him/her, and s/he gets in touch with them for a collaborative work. | X | |||||
| 9 | Student knows enough foreign language to do a collaborative work with the scientists studying on the same field as him/her abroad. | X | |||||
| 10 | Student follows the necessary technological developments in his/her field, and s/he uses them. | X | |||||
| 11 | Student looks out for the scientific and ethic values while gathering, interpreting and publishing data. | X | |||||
| Evaluation System | |
|---|---|
| Semester Studies | Contribution Rate |
| 1. Ara Sınav | 70 |
| 1. Ödev | 30 |
| Total | 100 |
| 1. Yıl İçinin Başarıya | 50 |
| 1. Final | 50 |
| Total | 100 |
| ECTS - Workload Activity | Quantity | Time (Hours) | Total Workload (Hours) |
|---|---|---|---|
| Course Duration (Including the exam week: 16x Total course hours) | 16 | 3 | 48 |
| Hours for off-the-classroom study (Pre-study, practice) | 16 | 3 | 48 |
| Mid-terms | 1 | 15 | 15 |
| Assignment | 1 | 10 | 10 |
| Final examination | 1 | 25 | 25 |
| Total Workload | 146 | ||
| Total Workload / 25 (Hours) | 5.84 | ||
| dersAKTSKredisi | 6 | ||