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| MATL5140 | Riemannian Manifolds | 3+0+0 | ECTS:7.5 | | Year / Semester | Fall Semester | | Level of Course | Second Cycle | | Status | Elective | | Department | DEPARTMENT of MATHEMATICS | | Prerequisites and co-requisites | None | | Mode of Delivery | | | Contact Hours | 14 weeks - 3 hours of lectures per week | | Lecturer | Prof. Dr. Yasemin SAĞIROĞLU | | Co-Lecturer | | | Language of instruction | Turkish | | Professional practise ( internship ) | None | | | | The aim of the course: | | To examine the concept of the Riemann manifold and the differential structures on it. |
| Programme Outcomes | CTPO | TOA | | Upon successful completion of the course, the students will be able to : | | | | PO - 1 : | He can recognize and process the Riemannian metric. | 1 - 4 - 5 - 6 | | | PO - 2 : | Know the concept of Riemannian manifold. | 1 - 4 - 5 - 6 | | | PO - 3 : | Know the curvatures and form calculations on the Riemannian manifold. | 1 - 4 - 5 - 6 | | | PO - 4 : | It has knowledge about Riemannian submanifolds. | 1 - 4 - 5 - 6 | | | CTPO : Contribution to programme outcomes, TOA :Type of assessment (1: written exam, 2: Oral exam, 3: Homework assignment, 4: Laboratory exercise/exam, 5: Seminar / presentation, 6: Term paper), PO : Learning Outcome | | |
| Riemannian metric, Riemannian manifold, covariant derivative, parallel translation, geodesics and normal coordinates, curvature tensors, sectional curvature, Ricci curvature and scalar curvature. Space forms, conformal changes of Riemannian metric, Riemannian submanifolds, induced connection, second fundamental form. |
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| Course Syllabus | | Week | Subject | Related Notes / Files | | Week 1 | Basic structures on manifolds | | | Week 2 | Basic structures on manifolds | | | Week 3 | Riemannian metric | | | Week 4 | Riemannian manifold | | | Week 5 | Covariant derivative | | | Week 6 | Parallel translation | | | Week 7 | Godesics, exponential function | | | Week 8 | Normal coordinates | | | Week 9 | Midterm | | | Week 10 | Curvature tensors, sectional curvature | | | Week 11 | Ricci curvature and scalar curvature. | | | Week 12 | Space forms | | | Week 13 | Conformal changes of Riemannian metric, short exam | | | Week 14 | Riemannian submanifolds, induced connection | | | Week 15 | Second fundamental form | | | Week 16 | Final exam | | | |
| 1 | Manifoldların Diferansiyel Geometrisi, Bayram Şahin, Nobel Yayın Dağıtım, 2012. | | | 2 | Riemannian Manifolds: An Introduction to Curvature, John M. Lee, Springer, 1997. | | | |
| Method of Assessment | | Type of assessment | Week No | Date | Duration (hours) | Weight (%) | | Mid-term exam | 9 | 17/11/2024 | 2 | 30 | | Quiz | 13 | 15/12/2024 | 1 | 20 | | End-of-term exam | 16 | 05/01/2025 | 2 | 50 | | |
| Student Work Load and its Distribution | | Type of work | Duration (hours pw) | No of weeks / Number of activity | Hours in total per term | | Yüz yüze eğitim | 14 | 3 | 42 | | Sınıf dışı çalışma | 10 | 15 | 150 | | Arasınav için hazırlık | 8 | 1 | 8 | | Arasınav | 2 | 1 | 2 | | Kısa sınav | 1 | 1 | 1 | | Dönem sonu sınavı için hazırlık | 10 | 1 | 10 | | Dönem sonu sınavı | 2 | 1 | 2 | | Total work load | | | 215 |
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