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| MATL7912 | Semi-Riemannian Geometry | 3+0+0 | ECTS:7.5 | | Year / Semester | Spring Semester | | Level of Course | Third 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 | Doç. Dr. Gül TUĞ | | Co-Lecturer | Prof. Dr. Yasemin Sağıroğlu | | Language of instruction | Turkish | | Professional practise ( internship ) | None | | | | The aim of the course: | | The aim of the course is to introduce the Semi-Riemann metric and to analyze geometric concepts such as covariant derivative, Levi-Civita connection and curvature on Semi-Riemannian manifolds with the help of this metric. |
| Programme Outcomes | CTPO | TOA | | Upon successful completion of the course, the students will be able to : | | | | PO - 1 : | Learns the concepts of covariant derivative and connection on Semi-Riemannian manifolds. | 1 | 1,3, | | PO - 2 : | Calculates the sectional curvature, mean curvature, and Ricci curvature. | 1 | 1,3, | | PO - 3 : | Learns the casual characters of vectors in 3-dimensional Lorentz-Minkowski spacetime. | 1 | 1,3, | | PO - 4 : | Characterizes curves and surfaces in 3-dimensional Lorentz-Minkowski spacetime. | 1 | 1,3, | | 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 | | |
| Semi-Riemannian manifolds, isometries, Levi-Civita connection, parallel transport, geodesic curves, curvature tensor, sectional curvature, semi-Riemannian hypersurfaces, Ricci curvature, scalar curvature, tangent and normal spaces, induced metric, submanifolds, Total geodesic hypersurfaces, Codazzi equations, Total umbilical hypersurfaces, normal connection, curves and surfaces in the 3-dimensional Lorentz-Minkowski space. |
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| Course Syllabus | | Week | Subject | Related Notes / Files | | Week 1 | Semi-Riemannian metric and Levi-Civita connection | | | Week 2 | Parallel transport | | | Week 3 | Geodesics | | | Week 4 | Riemannian curvature tensor, first and second Bianchi idendities | | | Week 5 | Sectional curvature | | | Week 6 | Ricci and Scalar curvatures | | | Week 7 | Semi-Riemannian submanifolds | | | Week 8 | Totally geodesic submanifolds | | | Week 9 | Midterm | | | Week 10 | Codazzi equations and semi-Riemannian hypersurfaces | | | Week 11 | Totally umbilical hypersurfaces, Normal connection | | | Week 12 | 3-dimensional Lorentz Minkowski space-time | | | Week 13 | Characterizations of curves in 3-dimensional Lorentz Minkowski space | | | Week 14 | Characterizations of surfaces in 3-dimensional Lorentz Minkowski space | | | Week 15 | Relations with the theory of Special Relativity | | | Week 16 | Final | | | |
| 1 | O'Neill, 1983; Semi-Riemannian Geometry with Applications to Relativity | | | |
| 1 | Lopez, 2014; Differential Geometry of Curves and Surfaces in Lorentz-Minkowski space | | | 2 | Sternberg, 2003; Semi-Riemann Geometry and General Relativity | | | |
| Method of Assessment | | Type of assessment | Week No | Date | Duration (hours) | Weight (%) | | Mid-term exam | 9 | - | 2 | 50 | | In-term studies (second mid-term exam) | - | | | | | Quiz | - | | | | | Laboratory exam | - | | | | | Project | - | | | | | Practice | - | | | | | Clinic Practice | - | | | | | Oral exam | - | | | | | Presentation | - | | | | | Homework/Assignment/Term-paper | - | | | | | End-of-term exam | 16 | - | 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 | 3 | 14 | 42 | | Sınıf dışı çalışma | 7 | 14 | 98 | | Arasınav için hazırlık | 8 | 2 | 16 | | Arasınav | 3 | 2 | 6 | | Ödev | 5 | 5 | 25 | | Dönem sonu sınavı için hazırlık | 10 | 2 | 20 | | Dönem sonu sınavı | 3 | 1 | 3 | | Total work load | | | 210 |
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