Dokuz Eylül University Information Package / Courses Catalog

Description of Individual Course Units

Course Unit Code	Course Unit Title	Type Of Course	D	U	L	ECTS
MAT 5026	DIFFERENTIAL GEOMETRY-II	ELECTIVE	3	0	0	8

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

ASSISTANT PROFESSOR CELAL CEM SARIOĞLU

Offered to

Mathematics
Mathematics

Course Objective

The aim of this course is to introduce concepts of Riemannian geometry make the students familiar with them.

Learning Outcomes of the Course Unit

1	will be able to describe what is a Riemannian manifold.
2	will be able to describe Riemannian metric.
3	will be able to describe curvature and the sectional curvature.
4	will be able to compute Ricci and scalar curvature.
5	will be able to minimize geodesic and energy.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week	Subject	Description
1	Differentiable manifolds, tangent space, immersions and embeddings.
2	Vector fields, brackets, topology of manifolds.
3	Riemannian metrics.
4	Affine connections, Riemannian connections.
5	Geodesic, the geodesic flow, minimizing properties of geodesics.
6	Convex neighborhoods.
7	Curvature, Sectional curvature.
8	Sectional curvature. + Midterm.
9	Ricci curvature and scalar curvature, Tensors on Riemannian manifolds.
10	The Jacobi fields and equations, Isometric immersions.
11	Complete manifolds, Hopf-Rinow theorems, the theorem of Hadamard.
12	Spaces of constant curvature.
13	Variations of Energy.
14	The Rauch comparison theorem, The Morse index theorem.

Recomended or Required Reading

Textbook:
1. Manfredo Perdigao do Carmo, Riemannian Geometry, Birkhauser, 1992.
Supplementary Books:
2. S. Kumaresan, A Course in Differential Geometry and Lie Groups, Hindustan Book Agency, 2002.
3. Loring W. Tu, Differential Geometry: Connections, Curvature, and Characteristic Classes, Springer 2017.
4. Thierry Aubin, A Course in Differential Geometry, American Mathematical Society, 2001.
References:
5. Peter Petersen, Riemannian Geometry, Springer 1998.
6. Sigurdur Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, American Mathematical Society, 2001.
7. Shoshichi Kobayashi, Katsumi Nomizu, Foundations of Differential Geometry, Volumes 1 and 2, Wiley-Interscience, 1996.
8. S.S. Chern, W.H. Chen, K.S. Lam, Lectures on Differential Geometry, World Scientific, 2000.
9. Michael Spivak, A Compherensive Introduction to Differential Geometry, Volumes 1,2,3,4 and 5, Publish or Perish, 1999.

Planned Learning Activities and Teaching Methods

Lecture notes, Presentation, Problem solving, Homework Assignments.

Assessment Methods

SORTING NUMBER	SHORT CODE	LONG CODE	FORMULA
1	ASG	ASSIGNMENT
2	MTE	MIDTERM EXAM
3	FIN	FINAL EXAM
4	FCG	FINAL COURSE GRADE	ASG * 0.30 + MTE * 0.30 + FIN * 0.40
5	RST	RESIT
6	FCGR	FINAL COURSE GRADE (RESIT)	ASG * 0.30 + MTE * 0.30 + RST * 0.40

*** Resit Exam is Not Administered in Institutions Where Resit is not Applicable.

Further Notes About Assessment Methods

None

Assessment Criteria

To be succesful, at the end of the term, the relative grade must be 75 or greater.

Language of Instruction

English

Course Policies and Rules

Attending at least 70 percent of lectures is mandatory.

Contact Details for the Lecturer(s)

Asst. Prof. Dr. Celal Cem Sarıoğlu
E-mail: celalcem.sarioglu@deu.edu.tr
Office: +90 232 301 8607

Office Hours

TBA

Work Placement(s)

None

Workload Calculation

Activities	Number	Time (hours)	Total Work Load (hours)
Lectures	13	3	39
Preparations before/after weekly lectures	13	4	52
Preparation for midterm exam	1	25	25
Preparation for final exam	1	30	30
Preparing assignments	6	8	48
Midterm	1	3	3
Final	1	3	3
TOTAL WORKLOAD (hours)			200

Contribution of Learning Outcomes to Programme Outcomes

PO/LO	PO.1	PO.2	PO.3	PO.4	PO.6	PO.8	PO.9	PO.10	PO.11
LO.1	3	3	4	4	3	4	3	4	3
LO.2	3	3	4	4	3	4	3	4	3
LO.3	3	3	4	4	3	4	3	4	3
LO.4	3	3	4	4	3	4	3	4	3
LO.5	3	3	4	4	3	4	3	4	3

COURSE UNIT TITLE

DEGREE PROGRAMMES

Description of Individual Course Units

Offered By

Level of Course Unit

Course Coordinator

Offered to

Course Objective

Learning Outcomes of the Course Unit

Mode of Delivery

Prerequisites and Co-requisites

Recomended Optional Programme Components

Course Contents

Recomended or Required Reading

Planned Learning Activities and Teaching Methods

Assessment Methods

Further Notes About Assessment Methods

Assessment Criteria

Language of Instruction

Course Policies and Rules

Contact Details for the Lecturer(s)

Office Hours

Work Placement(s)

Workload Calculation

Contribution of Learning Outcomes to Programme Outcomes

CONTACT INFORMATION