MATHEMATICS (TURKISH, PHD) | |||||
PhD | TR-NQF-HE: Level 8 | QF-EHEA: Third Cycle | EQF-LLL: Level 8 |
Course Code | Course Name | Semester | Theoretical | Practical | Credit | ECTS |
MAT5004 | Differential Geometry | Fall | 3 | 0 | 3 | 12 |
The course opens with the approval of the Department at the beginning of each semester |
Language of instruction: | Tr |
Type of course: | Departmental Elective |
Course Level: | |
Mode of Delivery: | Face to face |
Course Coordinator : | Prof. Dr. ERTUĞRUL ÖZDAMAR |
Course Objectives: | The purpose of this course is to equip graduate students with the fundamental concepts of differential geometry. |
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. |
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. |
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. |
Course Notes: | Elementary Differential Geometry, Barret O'Neill |
References: | . |
Semester Requirements | Number of Activities | Level of Contribution |
Attendance | 14 | % 5 |
Laboratory | % 0 | |
Application | % 0 | |
Field Work | % 0 | |
Special Course Internship (Work Placement) | % 0 | |
Quizzes | % 0 | |
Homework Assignments | 2 | % 15 |
Presentation | % 0 | |
Project | % 0 | |
Seminar | % 0 | |
Midterms | 1 | % 30 |
Preliminary Jury | % 0 | |
Final | 1 | % 50 |
Paper Submission | % 0 | |
Jury | % 0 | |
Bütünleme | % 0 | |
Total | % 100 | |
PERCENTAGE OF SEMESTER WORK | % 50 | |
PERCENTAGE OF FINAL WORK | % 50 | |
Total | % 100 |
Activities | Number of Activities | Duration (Hours) | Workload |
Course Hours | 14 | 0 | 0 |
Laboratory | 0 | 0 | 0 |
Application | 0 | 0 | 0 |
Special Course Internship (Work Placement) | 0 | 0 | 0 |
Field Work | 0 | 0 | 0 |
Study Hours Out of Class | 0 | 0 | 0 |
Presentations / Seminar | 0 | 0 | 0 |
Project | 0 | 0 | 0 |
Homework Assignments | 2 | 50 | 100 |
Quizzes | 0 | 0 | 0 |
Preliminary Jury | 0 | ||
Midterms | 1 | 40 | 40 |
Paper Submission | 0 | ||
Jury | 0 | ||
Final | 1 | 50 | 50 |
Total Workload | 190 |
No Effect | 1 Lowest | 2 Low | 3 Average | 4 High | 5 Highest |
Program Outcomes | Level of Contribution |