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| MATL5120 | Differentiable Manifolds | 3+0+0 | ECTS:7.5 | | Year / Semester | Spring Semester | | Level of Course | Second Cycle | | Status | Elective | | Department | DEPARTMENT of MATHEMATICS | | Prerequisites and co-requisites | None | | Mode of Delivery | Face to face | | Contact Hours | 14 weeks - 3 hours of lectures per week | | Lecturer | -- | | Co-Lecturer | - | | Language of instruction | Turkish | | Professional practise ( internship ) | None | | | | The aim of the course: | | To introduce differentiable manifold concept, tangent and cotangent spaces. Recognize the geometric structure of hypersurfaces of Euclidean space and investigate their geometric properties. To introduce tensor and differential forms. |
| Programme Outcomes | CTPO | TOA | | Upon successful completion of the course, the students will be able to : | | | | PO - 1 : | Recognize some basic definitions and theorems of topology. | 1,2,6,8 | | | PO - 2 : | Learn the basic definitions and theorems of differentiable manifold. | 1,2,6,8 | | | PO - 3 : | Learn hypersurfaces of Euclidean space, presentation of Gauss and Weingarten of these, Gauss and Codazzi equations and their using. | 1,2,6,8 | | | PO - 4 : | Recognize tensors on manifolds, differential forms and their features. | 1,2,6,8 | | | 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 | | |
| Topological spaces, differentiable manifolds, tangent space, vector fields, Lie bracket, diffeomorphism, the inverse function theorem, submanifolds, hypersurfaces, standart connection of Euclidean spaces, Weingarten and Gauss maps, tensors and differential forms, Lie derivative, Riemannian connection, Riemannian manifolds. |
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| Course Syllabus | | Week | Subject | Related Notes / Files | | Week 1 | Topological spaces, product topology, metric topology | | | Week 2 | Quotient topology, connectedness, compactness | | | Week 3 | Differentiable manifolds and examples | | | Week 4 | Tangent spaces, vector fields, Lie bracket | | | Week 5 | The inverse function theorem, hypersurfaces of Euclidean spaces | | | Week 6 | Standard connection of Euclidean spaces | | | Week 7 | Weingarten and Gauss map | | | Week 8 | Gauss and Codazzi equations | | | Week 9 | Mid-term exam | | | Week 10 | Tensors | | | Week 11 | Differential forms | | | Week 12 | Lie derivative | | | Week 13 | Riemannian manifolds, Short Exam | | | Week 14 | Riemannian connection | | | Week 15 | Riemannian curvature tensor | | | Week 16 | Final exam | | | |
| 1 | Boothby, W.M., An Introduction to Differential Manifolds and Riemannian Geometry, Academic Press Inc. 1975. | | | |
| 1 | Do Carmo, M.P., Riemannian Geometry, Birkehauser, 1990. | | | |
| Method of Assessment | | Type of assessment | Week No | Date | Duration (hours) | Weight (%) | | Mid-term exam | 9 | 12/04/2017 | 2 | 30 | | Quiz | 13 | 10/05/2017 | 1 | 30 | | End-of-term exam | 16 | 29/05/2017 | 2 | 40 | | |
| 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 | 3 | 14 | 42 | | Sınıf dışı çalışma | 5 | 14 | 70 | | Arasınav için hazırlık | 6 | 7 | 42 | | Arasınav | 2 | 1 | 2 | | Kısa sınav | 1 | 1 | 1 | | Dönem sonu sınavı için hazırlık | 8 | 5 | 40 | | Dönem sonu sınavı | 2 | 1 | 2 | | Total work load | | | 199 |
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