Ders Adı | Kodu | Yarıyıl | T+U Saat | Kredi | AKTS |
---|---|---|---|---|---|
Semı Rıemannıan Geometry I | MAT 606 | 0 | 3 + 0 | 3 | 6 |
Ön Koşul Dersleri | Students are assumed to be familiar with the courses of Analytic Geometry and Differential Geometry. |
Önerilen Seçmeli Dersler | |
Dersin Dili | Türkçe |
Dersin Seviyesi | Doktora |
Dersin Türü | Seçmeli |
Dersin Koordinatörü | Prof.Dr. MURAT TOSUN |
Dersi Verenler | Prof.Dr. MEHMET ALİ GÜNGÖR, |
Dersin Yardımcıları | |
Dersin Kategorisi | Diğer |
Dersin Amacı | The aim of the course Semi-Riemannian Geometry I is to give some fundamental acknowledges which are base for studies of graduate students studying on geometry. |
Dersin İçeriği | Differentiable manifolds, differentiable maps between manifolds, tangent vectors, differential maps, curves, one-forms, submanifolds, immersions and subimmersions, topology of manifolds, some special manifolds, integral curves, definition of tensor, tensor fields, contractions, covariant tensors, tensor derivation, symmetric bilinear forms, scalar products, Semi-Riemannian manifolds, isometries, Levi-Civita connection, parallel translation, geodesics, the exponential map, curvature tensor, sectional curvature, semi-Riemannian surfaces, metric contraction, Ricci and scalar curvature, local isometries. |
# | Ders Öğrenme Çıktıları | Öğretim Yöntemleri | Ölçme Yöntemleri |
---|---|---|---|
1 | He/She defines differentiable manifolds, | Lecture, Question-Answer, Discussion, Group Study, Problem Solving, | Testing, Homework, |
2 | He/She defines Semi-Riemannian manifolds, | Lecture, Question-Answer, Discussion, Group Study, Problem Solving, | Testing, Homework, |
3 | He/She illustrates manifolds, | Lecture, Question-Answer, Discussion, Group Study, Problem Solving, | Testing, Homework, |
4 | He/She defines concepts of curvature tensor, sectional curvature, | Lecture, Question-Answer, Discussion, Group Study, Problem Solving, | Testing, Homework, |
5 | He/She computes sectional curvature Semi-Riemannian surfaces, | Lecture, Question-Answer, Discussion, Group Study, Problem Solving, | Testing, Homework, |
6 | He/She computes Ricci curvature Semi-Riemannian surfaces, | Lecture, Question-Answer, Discussion, Group Study, Problem Solving, | Testing, Homework, |
7 | He/She computes scalar curvature Semi-Riemannian surfaces, | Lecture, Question-Answer, Discussion, Group Study, Problem Solving, | Testing, Homework, |
8 | He/She develops the geometry by concepts of Semi-Riemannian geometry. | Lecture, Question-Answer, Discussion, Group Study, Problem Solving, | Testing, Homework, |
Hafta | Ders Konuları | Ön Hazırlık |
---|---|---|
1 | Differentiable manifolds, differentiable maps between manifolds, tangent vectors, differential maps | Page 1-10 |
2 | Curves, one-forms, submanifolds, immersions and subimmersions | Page 10-21 |
3 | Topology of manifolds, some special manifolds | Page 21-34 |
4 | Definition of tensor, tensor fields, contractions. | Page 34-40 |
5 | Covariant tensors, tensor derivation | Page 40-46 |
6 | Symmetric bilinear forms, scalar products | Page 46-58 |
7 | Semi-Riemannian manifolds, isometries, Levi-Civita connection | Page 58-65 |
8 | Parallel translation, geodesics | Page 65-70 |
9 | applications and Midterm exam | |
10 | The exponential map | Page 70-74 |
11 | Curvature tensor, sectional curvature | Page 74-87 |
12 | Semi-Riemannian surfaces, metric contraction | Page 87-89 |
13 | Ricci and scalar curvature | Page 89-90 |
14 | Local isometries | Page 90-97 |
Kaynaklar | |
---|---|
Ders Notu | [1] Barrett O´Neill, Semi-riemannian Geometry: With Applications to Relativity (Pure & Applied Mathematics S.), June ,1983 |
Ders Kaynakları | [2] Ramon Vazquez-Lorenzo, Demir N. Kupeli, Eduardo Garcia-Rio, Osserman Manifolds in Semi-Riemannian Geometry (Lecture Notes in Mathematics, 1777) [3] Hacısalihoğlu H. H. , Diferensiyel Geometri, Ankara Üni., Fen Fakültesi,1983 |
Sıra | Program Çıktıları | Katkı Düzeyi | |||||
---|---|---|---|---|---|---|---|
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 |
Değerlendirme Sistemi | |
---|---|
Yarıyıl Çalışmaları | Katkı Oranı |
1. Ara Sınav | 70 |
1. Ödev | 30 |
Toplam | 100 |
1. Yıl İçinin Başarıya | 50 |
1. Final | 50 |
Toplam | 100 |
AKTS - İş Yükü Etkinlik | Sayı | Süre (Saat) | Toplam İş Yükü (Saat) |
---|---|---|---|
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 | 20 | 20 |
Assignment | 1 | 15 | 15 |
Final examination | 1 | 25 | 25 |
Toplam İş Yükü | 156 | ||
Toplam İş Yükü / 25 (Saat) | 6,24 | ||
Dersin AKTS Kredisi | 6 |