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Bachelor	TR-NQF-HE: Level 6	QF-EHEA: First Cycle	EQF-LLL: Level 6

Course Introduction and Application Information

Course Code	Course Name	Semester	Theoretical	Practical	Credit	ECTS
MAT4053	Differentiable Manifolds	Spring	3	0	3	6

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

Basic information

Language of instruction:	English
Type of course:	Non-Departmental Elective
Course Level:	Bachelor’s Degree (First Cycle)
Mode of Delivery:	Face to face
Course Coordinator :
Recommended Optional Program Components:	None
Course Objectives:	The differentiable manifolds course aims to give the fundamental knowledge 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
1)be able to test a differentiable structure given on a set
2)be able to give examples of Differentiable structures on a set
3) be able to check differentiablity of a function
4) be able to solve problems involving the derived map of a transformation between two manifolds
5) be able to use the properties of induced topology on a manifold,
6) be able to coordinatize Grassmann manifolds and can evaluate their dimensions,
7) be able to understand the existence problems by using the unity of partition
8)be able to explain the derived function of a function by using the Leibniz rule,
9) be able to explain submanifolds as images under Immersions
10) be able to coordinatize quotient manifolds and calculate their dimensions,
11) be able to construct Klein bottle and Mobius strip as an example of a quotient manifold

Course Content

Differentiable (diff.able) functions, Atlas, diff.able structures on a set, Examples of diff.able structures, diff.able manifolds, diff.able functions, The induced topology on a manifold, diff.able varieties, Grassmann manifolds, Manifold structure on a topological space, properties of the induced topology, Topological restrictions on a manifold, Partitions of unity, Partial differentiation, tangent vectors, 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.

Weekly Detailed Course Contents

Week	Subject	Related Preparation
1)	Preliminaires
2)	Some classical theory of differentiable functions
3)	Atlas, differentiable structures on a set
4)	Examples of differentiable structures on a set
5)	Differentiable manifolds
6)	Differentiable functions
7)	The induced topology on a manifold
8)	Differentiable varieties, Grassmann manifolds
9)	Topological restrictions on a manifold, Partitions of unity
10)	Manifold structure on a topological space, properties of the induced topology
11)	Partial differentiation, tangent vectors, derived linear functions, The invers function Theorem, Leibniz's rule.
12)	İmmersions, submanifolds, regular submanifolds, some topological properties of submanifolds.
13)	Submersions, The fibres of submersions, Quotient manifolds
14)	Transformation groups, Examples of quotient manifolds.

Sources

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

Evaluation System

Semester Requirements	Number of Activities	Level of Contribution
Midterms	2	% 45
Final	1	% 55
Total		% 100
PERCENTAGE OF SEMESTER WORK		% 45
PERCENTAGE OF FINAL WORK		% 55
Total		% 100

ECTS / Workload Table

Activities	Number of Activities	Duration (Hours)	Workload
Course Hours	14	3	42
Study Hours Out of Class	7	2	14
Midterms	2	20	40
Final	1	30	30
Total Workload			126

Contribution of Learning Outcomes to Programme Outcomes

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

	Program Outcomes	Level of Contribution
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