Ders Adı Kodu Yarıyıl T+U Saat Kredi AKTS
Semı-Rıemannıan Geometry II MAT 607 0 3 + 0 3 6
Ön Koşul Dersleri Students are assumed to be familiar with the course Differential Geometry I and Differential Geometry II.
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 Research assistants of geometry
Dersin Kategorisi Diğer
Dersin Amacı The aim of the course Semi-Riemannian Geometry II is to give some fundamental acknowledges which are base for studies of graduate students studying on geometry.
Dersin İçeriği Semi-Riemannian submanifolds, tangents and normals, the induced connections, geodesics in submanifolds, totally geodesic submanifolds, semi-Riemannian hypersurfaces, hyperquadrics, Codazzi equation, totally umbilic hypersurfaces, normal connection, isometric immersions, two parameter maps, the Gauss lemma, convex open sets, arc length, Riemannian distance, Riemannian completeness, Lorentz causal character, time cones, local Lorentz geometry, geodesics in hyperquadrics, geodesics in surfaces, orientability, semi-Riemannian coverings, Lorentz time orientability, volume elements, Jakobi fields, locally symmetric manifolds, semi-ortogonal groups, some isometry groups.
# Ders Öğrenme Çıktıları Öğretim Yöntemleri Ölçme Yöntemleri
1 He/She defines Semi-Riemannian hypersurfaces, Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
2 He/She analyses the geodesics in submanifolds and hyperquadrics, Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
3 He/She adapts Codazzi equation to hypersurfaces, Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
4 He/She defines fundamental concepts of Lorentz geometry, Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
5 He/She compares Lorentz geometry with Euclidean geometry, Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
6 He/She defines time cones and orientability, Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
7 He/She adapts well known concepts of Differential geometry to Semi-Riemannian manifolds. Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
Hafta Ders Konuları Ön Hazırlık
1 Semi-Riemannian submanifolds, tangents and normals, the induced connections, Page 97-102
2 Geodesics in submanifolds, totally geodesic submanifolds, Page 102-106
3 Semi-Riemannian hypersurfaces, hyperquadrics, Page 106-114
4 Codazzi equation, totally umbilic hypersurfaces, Page 114-118
5 Normal connection, isometric immersions, Page 118-122
6 Two parameter maps, Page 122-126
7 The Gauss lemma, Page 126-131
8 Arc length, Riemannian distance, Riemannian completeness, Page 131-138
9 Applications and Midterm Exam
10 Lorentz causal character, time cones, local Lorentz geometry, Page 138-149
11 Geodesics in hyperquadrics, Geodesics in surfaces, orientability, Page 149-154
12 Semi-Riemannian coverings, Lorentz time orientability, volume elements, Jakobi fields, Page 191-215
13 Locally symmetric manifolds, Page 215-233
14 Semi-ortogonal groups, some isometry groups, Page 233-239
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