Course Name Code Semester T+U Hours Credit ECTS
Semi Riemannian Geometry I MAT 606 0 3 + 0 3 6
Precondition Courses Students are assumed to be familiar with the courses of Analytic Geometry and Differential Geometry.
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
Course Category
Course Objective 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.
Course Content 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.
# Course Learning Outcomes Teaching Methods Assessment Methods
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,
Week Course Topics Preliminary Preparation
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
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
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