Ders Bilgileri

#### Ders Tanımı

Ders Kodu Yarıyıl T+U Saat Kredi AKTS
SEMI-RIEMANNIAN 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.
 Dersin Dili Türkçe Dersin Seviyesi Doktora Dersin Türü SECMELI Dersin Koordinatörü Prof.Dr. MURAT TOSUN Dersi Verenler Dersin Yardımcıları Research assistants of geometry Dersin Kategorisi 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.
 Dersin Öğrenme Çıktıları Öğretim Yöntemleri Ölçme Yöntemleri 1 - He/She defines Semi-Riemannian hypersurfaces, 1 - 2 - 3 - 8 - 15 - A - C - 2 - He/She analyses the geodesics in submanifolds and hyperquadrics, 1 - 2 - 3 - 8 - 15 - A - C - 3 - He/She adapts Codazzi equation to hypersurfaces, 1 - 2 - 3 - 8 - 15 - A - C - 4 - He/She defines fundamental concepts of Lorentz geometry, 1 - 2 - 3 - 8 - 15 - A - C - 5 - He/She compares Lorentz geometry with Euclidean geometry, 1 - 2 - 3 - 8 - 15 - A - C - 6 - He/She defines time cones and orientability, 1 - 2 - 3 - 8 - 15 - A - C - 7 - He/She adapts well known concepts of Differential geometry to Semi-Riemannian manifolds. 1 - 2 - 3 - 8 - 15 - A - C -
 Öğretim Yöntemleri: 1:Lecture 2:Question-Answer 3:Discussion 8:Group Study 15:Problem Solving Ölçme Yöntemleri: A:Testing C:Homework

#### Ders Akışı

Hafta Konular ÖnHazırlık
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

#### Dersin Program Çıktılarına Katkısı

No Program Öğrenme Çıktıları KatkıDüzeyi
1 2 3 4 5
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

YARIYIL İÇİ ÇALIŞMALARI SIRA KATKI YÜZDESİ
AraSinav 1 70
Odev 1 30
Toplam 100
Yıliçinin Başarıya Oranı 50
Finalin Başarıya Oranı 50
Toplam 100

#### AKTS - İş Yükü

Etkinlik Sayısı Süresi(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(s) 6.24
Dersin AKTS Kredisi 6.24
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