Course Name Code Semester T+U Hours Credit ECTS
Tensor Geometry I MAT 621 0 3 + 0 3 6
Precondition Courses Students are assumed to be familiar with Differential Geometry I and Differential Geometry II
Recommended Optional Courses
Course Language Turkish
Course Level Doctorate Degree
Course Type Optional
Course Coordinator Prof.Dr. MEHMET ALİ GÜNGÖR
Course Lecturers
Course Assistants
Course Category
Course Objective The tensor geometry I course aims to give the fundamental knowledge for the studies of graduate students who study at geometry branch.
Course Content Tensors, covariant and contravariant tensors, mixed tensors, tensor product of two tensors, symmetric and anti-symmetric tensors, exterior product and exterior algebra, parallel vector fields, Levi-Civita parallelism, tensors on the Riemannnian manifolds, sectional curvature, Ricci curvature, scalar curvature, Ricci identity.
# Course Learning Outcomes Teaching Methods Assessment Methods
1 He/She knows the fundamental consepts of analysis and algebra of tensor. Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
2 He/She defines the fundamental calculations related to the parallelism on n- Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
3 He/She computes the metric, curvatures and connections in the sense of Riemannian Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
4 He/She computes the Laplacian, sectional curvatures, Ricci curvatures and scalar curvatures. Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
Week Course Topics Preliminary Preparation
1 Tensors
2 Covariant and contravariant tensors
3 Mixed tensors
4 Tensor product of two tensors
5 Symmetric and anti-symmetric tensors
6 Exterior product and exterior algebra
7 Parallel vector fields
8 Levi-Civita parallelism
9 Mid term exam
10 Tensors on the Riemannnian manifolds
11 Sectional curvature
12 Ricci curvature
13 Scalar curvature
14 Ricci identity
Course Notes 1. Hacısalihoğlu H. H. ve Ekmekçi N., Tensör Geometri, Ankara Üni., Fen Fakültesi,2003.
Course Resources 1. Kobayashi, S., and Nomizu, K., Foundations of differential geometry, Number 15, Volume II, New York, 1969.
2. ONeill B., Elementary Differential Geometry, Academic Press, New York, 1997.
3. D. C. Kay, , Schaums outline of theory and problems, McGraw-Hill, 1988.
4. C. T. J. Dodson, T. Poston, Tensor geometry, Graduate Texts in Mathematics, 130. Springer-Verlag, Berlin, 1991.
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 15 15
Assignment 1 10 10
Final examination 1 25 25
Total Workload 146
Total Workload / 25 (Hours) 5.84
dersAKTSKredisi 6