MATHEMATICS (TURKISH, PHD)
PhD	TR-NQF-HE: Level 8	QF-EHEA: Third Cycle	EQF-LLL: Level 8

Course Introduction and Application Information

Course Code	Course Name	Semester	Theoretical	Practical	Credit	ECTS
MAT6027	Semi-Riemannian Geometry	Fall Spring	3	0	3	8

This catalog is for information purposes. Course status is determined by the relevant department at the beginning of semester.

Basic information

Language of instruction:	Turkish
Type of course:	Departmental Elective
Course Level:
Mode of Delivery:
Course Coordinator :	Prof. Dr. ERTUĞRUL ÖZDAMAR
Recommended Optional Program Components:	None
Course Objectives:	Yari-Riemann geometrisi dersi, Lorentz ve genelde de Yari-Riemann geometriye bir giriş dersidir. During the course, basic information aimed to meet the needs of researchers.

Learning Outcomes

The students who have succeeded in this course;
upon succeeding this course the student will be able to
1) know the concepts of manifold and the curvature tensor, and understood very well
the role of the inner-dot product,
2) calculate the distance, area and curvature on Semi-Riemannian manifolds.
3) explain the properties of geodesics on Semi-Riemannian manifolds, and also the concepts as chronology and time cone
4) understand paradoxes in relativity.
5) Explain all properties of Warped product and the metric

matrices.

Course Content

Manifold theory, Tensors, Semi-Riemannian Manifolds; Isometries,Levi-Civita connection,paralel translation, geodesics, the exponential mapping, curvature, sectional curvature, Semi-Riemann surfaces, Semi-Riemannian Submanifolds; Ricci and scalar curvature, Semi-Riemann product manifolds, local isometries, Riemannian and Lorentz Geometry; Gauss lemma, convex open sets, arclength, Riemaniann distance, Lorentz causal character, time cone, Local Lorentz geometry, geodesics,completeness and extendibility, Constructions; Deck transformations, volume elements, vector bundles, local isometries, Warped products, Isometries; isometry groups, space forms, homogeneous spaces

Weekly Detailed Course Contents

Week	Subject	Related Preparation
1)	Preliminiars and Manifold theory
2)	Manifold theory
3)	Tensors
4)	Tensors
5)	Semi-Riemannian Manifolds; Isometries,Levi-Civita connection,paralel translation, geodesics, the exponential mapping, curvature, sectional curvature, Semi-Riemann surfaces
6)	Semi-Riemannian Submanifolds; Ricci and scalar curvature
7)	Semi-Riemann product manifolds, local isometries
8)	Riemannian and Lorentz Geometry; Gauss lemma, convex open sets, arclength, Riemaniann distance
9)	Lorentz causal character, time cone
10)	Local Lorentz geometry, geodesics,completeness and extendibility
11)	Constructions; Deck transformations, volume elements, vector bundles, local isometries
12)	Warped products
13)	Isometries; isometry groups, space forms
14)	homogeneous spaces

Sources

Course Notes / Textbooks:	Semi-Riemannian Geometry With Applications to Relativity,Barrett O'Neill, 103 Academic Press, ISBN: 0125267401
References:	.

Evaluation System

Semester Requirements	Number of Activities	Level of Contribution
Attendance	14	% 5
Homework Assignments	3	% 15
Midterms	2	% 35
Final	1	% 45
Total		% 100
PERCENTAGE OF SEMESTER WORK		% 55
PERCENTAGE OF FINAL WORK		% 45
Total		% 100

ECTS / Workload Table

Activities	Number of Activities	Duration (Hours)	Workload
Course Hours	14	3	42
Homework Assignments	3	25	75
Midterms	2	30	60
Final	1	25	25
Total Workload			202

Contribution of Learning Outcomes to Programme Outcomes

No Effect	1 Lowest	2 Low	3 Average	4 High	5 Highest

	Program Outcomes	Level of Contribution
1)	Ability to assimilate mathematic related concepts and associate these concepts with each other.	5
2)	Ability to gain qualifications based on basic mathematical skills, problem solving, reasoning, association and generalization.
3)	Be able to organize events, for the development of critical and creative thinking and problem solving skills, by using appropriate methods and techniques.	5
4)	Ability to make individual and team work on issues related to working and social life.
5)	Ability to transfer ideas and suggestions, related to topics about his/her field of interest, written and verball.	4
6)	Ability to use mathematical knowledge in technology.	3
7)	To apply mathematical principles to real world problems.	2
8)	Ability to use the approaches and knowledge of other disciplines in Mathematics.	4
9)	Be able to set up and develope a solution method for a problem in mathematics independently, be able to solve and evaluate the results and to apply them if necessary.	5
10)	To be able to link abstract thought that one has to concrete events and to transfer the solutions and examine and interpret the results scientifically by forming experiments and collecting data.	3
11)	To be able to conduct a research either as an individual or as a team member, and to be effective in each related step of the project, to take role in the decision process, to plan and manage the project by using time effectively.	2
12)	To be able to acquire necessary information and to make modeling in any field that mathematics is used and to improve herself/himself.	4