MAT6025 Differentiable ManifoldsBahçeşehir UniversityDegree Programs MATHEMATICS (TURKISH, PHD)General Information For StudentsDiploma SupplementErasmus Policy StatementNational QualificationsBologna Commission
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
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.

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 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.

Learning Outcomes

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.

Course Content

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.

Weekly Detailed Course Contents

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

Sources

Course Notes / Textbooks: Differentiable Manifolds an Introduction ,F Brickell, R. S. Clark.
References: .

Evaluation System

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

ECTS / Workload Table

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

Contribution of Learning Outcomes to Programme Outcomes

No Effect 1 Lowest 2 Low 3 Average 4 High 5 Highest
           
Program Outcomes Level of Contribution