Course Name Code Semester T+U Hours Credit ECTS
Semi-Riemannian Geometry II MAT 607 0 3 + 0 3 6
Precondition Courses Students are assumed to be familiar with the course Differential Geometry I and Differential Geometry II.
Recommended Optional Courses
Course Language Turkish
Course Level Doctorate Degree
Course Type Optional
Course Coordinator Prof.Dr. MURAT TOSUN
Course Lecturers Prof.Dr. MEHMET ALİ GÜNGÖR,
Course Assistants Research assistants of geometry
Course Category
Course Objective 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.
Course Content 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.
# Course Learning Outcomes Teaching Methods Assessment Methods
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,
Week Course Topics Preliminary Preparation
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
Resources
Course Notes [1] Barrett O´Neill, Semi-Riemannian Geometry: With Applications to Relativity (Pure & Applied Mathematics S.), June, 1983.
Course Resources [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
Order Program Outcomes Level of Contribution
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
Evaluation System
Semester Studies Contribution Rate
1. Ara Sınav 70
1. Ödev 30
Total 100
1. Yıl İçinin Başarıya 50
1. Final 50
Total 100
ECTS - Workload Activity Quantity Time (Hours) Total Workload (Hours)
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
Total Workload 156
Total Workload / 25 (Hours) 6.24
dersAKTSKredisi 6