APPLIED MATHEMATICS (TURKISH, THESIS)
Master TR-NQF-HE: Level 7 QF-EHEA: Second Cycle EQF-LLL: Level 7

Course Introduction and Application Information

Course Code Course Name Semester Theoretical Practical Credit ECTS
MAT5004 Differential Geometry Fall 3 0 3 12
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: Face to face
Course Coordinator : Prof. Dr. ERTUĞRUL ÖZDAMAR
Recommended Optional Program Components: None
Course Objectives: The purpose of this course is to equip graduate students with the fundamental concepts of differential geometry.

Learning Outcomes

The students who have succeeded in this course;
The students who succeeded in this course;
o will be able to know the concepts of curve and surface and do basic calculations related to curves and surfaces
o will be able to know special curves and surfaces and use them as special demonstrations.
o will be able to apply concepts of vector spaces to tangent spaces and differentiability of mappings between surfaces.
o will be able to formulate the fundamental equations for both extrinsic and intrinsic geometry of surfaces .
o will be able to distinguish the geometrical ideas between the euclidean and non-euclidean geometries by using curvature properties of surfaces.

Course Content

1. Calculus on Euclidean Space,
2. Frame Fields, the dot product, the natural inner product on Euclidean space,
3. the geometry of curves in R3 , shape of a curve in R3 , curvature and torsion functions,
4. Frenet formulas, method of moving frames,
5. Euclidean Geometry, Rigid motion of the plane , Rigid motions (isometries) of Euclidean space,
6. Calculus on a Surface, definition of a surface in R3 and with some standard ways to construct surfaces,
7. Vector fields, differential forms, mappings,
8. Shape Operators, the shape of a surface M in R3 , shape operators,Gaussian curvature ,
9. Geometry of Surfaces in R3 , Shape of a surface related to its other properties, Shape of M if it is compact, or flat, or both?
10. Riemannian Geometry, the fundamentals of the Riemannian Geometry ,
11. Global Structure, global structure of geometric surfaces, The influence of Gaussian curvature on geodesics,
12. geodesics on connected surfaces
13. Gaussian curvature and geodesics
14. surfaces with constant curvature, surfaces whose curvature K obeys either K 0.

Weekly Detailed Course Contents

Week Subject Related Preparation
1) Calculus on Euclidean Space,
2) Frame Fields, the dot product, the natural inner product on Euclidean space,
3) The geometry of curves in R3 , shape of a curve in R3 , curvature and torsion functions,
4) Frenet formulas, method of moving frames,
5) Euclidean Geometry, Rigid motion of the plane , Rigid motions (isometries) of Euclidean space,
6) Calculus on a Surface, definition of a surface in R3 and with some standard ways to construct surfaces,
7) Vector fields, differential forms, mappings,
8) Shape Operators, the shape of a surface M in R3 , shape operators,Gaussian curvature ,
9) Geometry of Surfaces in R3 , Shape of a surface related to its other properties, Shape of M if it is compact, or flat, or both?
10) Riemannian Geometry, the fundamentals of the Riemannian Geometry,
11) Global Structure, global structure of geometric surfaces, The influence of Gaussian curvature on geodesics,
12) Geodesics on connected surfaces
13) Gaussian curvature and geodesics,
14) surfaces with constant curvature, surfaces whose curvature K obeys either K <0 or K > 0.

Sources

Course Notes / Textbooks: Elementary Differential Geometry,
Barret O'Neill
References: .

Evaluation System

Semester Requirements Number of Activities Level of Contribution
Attendance 14 % 5
Homework Assignments 2 % 15
Midterms 1 % 30
Final 1 % 50
Total % 100
PERCENTAGE OF SEMESTER WORK % 50
PERCENTAGE OF FINAL WORK % 50
Total % 100

ECTS / Workload Table

Activities Number of Activities Duration (Hours) Workload
Course Hours 14 0 0
Homework Assignments 2 50 100
Midterms 1 40 40
Final 1 50 50
Total Workload 190

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. 5
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. 3
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. 3
8) Ability to use the approaches and knowledge of other disciplines in Mathematics. 3
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. 4
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. 3
12) To be able to acquire necessary information and to make modeling in any field that mathematics is used and to improve herself/himself, 3