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 |
MAT6025 | Differentiable Manifolds | Fall | 3 | 0 | 3 | 8 |
This catalog is for information purposes. Course status is determined by the relevant department at the beginning of semester. |
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 differentiable manifolds course aims to give the fundamental knowledge as differentiable structures, analysis on manifolds , action groups and lie groups for the studies of graduate students who intends to study at geometry. |
The students who have succeeded in this course; upon succeeding this course the student 1) be able to know, basic information on the differentiable structures on a set use induced topology, special structures, especially the Grassmann manifolds, sub and quotient manifolds effectively. 2) be able to know the necessary and sufficient conditions of existence and uniqueness of the solution of differential equations on a manifold, and interpret the structures of solution sets. 3) be able to know the concepts of linear connections, curvature, torsion and relate them with horizontal distributions. 4) be able to associate Integral manifolds and distributions with linear connections. 5) be able to know the basic properties of Lie groups and associate them with Lie algebras and matrices. |
Some classical theory of differentiable functions Atlas, differentiable structures on a set Examples of differentiable structures on a set Differentiable manifolds Differentiable functions The induced topology on a manifold Differentiable varieties, Grassmann manifolds Topological restrictions on a manifold, Partitions of unity Manifold structure on a topological space, properties of the induced topology Partial differentiation, tangent vectors, derived linear functions, The invers function Theorem, Leibniz's rule. İmmersions, submanifolds, regular submanifolds, some topological properties of submanifolds. Submersions, The fibres of submersions, Quotient manifolds Transformation groups, Examples of quotient manifolds. Vector fields, tangent bundle, orientable manifolds, ○-related vector fields Differential equations of first order Linear connections, curvature, torsion, horizontal distribution, Riemann connections Differential equations of second order, sprays Distributions Lie groups exponential function Lie transformation groups. |
Week | Subject | Related Preparation |
1) | Preliminaires | |
2) | Some classical theory of differentiable functions Atlas, differentiable structures on a set Examples of differentiable structures on a set Differentiable manifolds | |
3) | Differentiable functions The induced topology on a manifold Differentiable varieties, Grassmann manifolds Topological restrictions on a manifold, Partitions of unity | |
4) | Manifold structure on a topological space, properties of the induced topology Partial differentiation, tangent vectors, derived linear functions, The invers function Theorem, Leibniz's rule. | |
5) | İmmersions, submanifolds, regular submanifolds, some topological properties of submanifolds. Submersions, The fibres of submersions, Quotient manifolds | |
6) | Transformation groups, Examples of quotient manifolds. | |
7) | Vector fields, tangent bundle, orientable manifolds, ○-related vector fields | |
8) | Differential equations of first order | |
9) | Linear connections, curvature, torsion, horizontal distribution, Riemann connections | |
10) | Differential equations of second order, sprays | |
11) | Distributions | |
12) | Lie groups | |
13) | exponential function | |
14) | Lie transformation groups |
Course Notes / Textbooks: | Differentiable Manifolds an Introduction ,F Brickell, R. S. Clark. |
References: | . |
Semester Requirements | Number of Activities | Level of Contribution |
Attendance | 1 | % 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 |
Activities | Number of Activities | Duration (Hours) | Workload |
Course Hours | 14 | 3 | 42 |
Homework Assignments | 3 | 20 | 60 |
Midterms | 2 | 30 | 60 |
Final | 1 | 35 | 35 |
Total Workload | 197 |
No Effect | 1 Lowest | 2 Low | 3 Average | 4 High | 5 Highest |
Program Outcomes | Level of Contribution |