COURSE UNIT TITLE

: DIFFERENTIAL GEOMETRY-I

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 5021 DIFFERENTIAL GEOMETRY-I 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 differential geometry in an advance way and make the students familiar with them. (It is supposed that, the students took an undergraduate differential geometry course or they are familiar with the local theory of curves of surfaces.)

Learning Outcomes of the Course Unit

1   will be able to describe what is a manifold.
2   will be able to use linear tangent mapping.
3   will be able to compute curvature and torsion of a connection.
4   will be able to describe geodesics on a manifold.
5   will be able to use differential forms.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Introduction, Topology review, Differential Calculus review.
2 Topological manifold, differentiable manifold, Tangent space, Partitions of unity.
3 Differentiable mappings, immersions, Submanifolds, embeddings.
4 The Whitney theorem, The Sard theorem.
5 Tangent space, Tangent vector, Linear tangent mapping, Cotangent mapping.
6 Vector bundles, The bracket [X,Y], Exterior differential.
7 Orientable manifolds, Manifolds with boundary.
8 Manifolds with boundary. + Midterm.
9 Integration of vector fields, Lie derivative, The Frobenious theorem.
10 Linear connections, Chrsitophel Symbols, Torsion and curvature of a connection.
11 Parallel Transport, Geodesics, Covariant derivative.
12 Riemannian manifolds, riemannian connection, Exponential mapping.
13 Some operators on differential forms.
14 Spectrum of a manifold.

Recomended or Required Reading

Textbook:
1. Thierry Aubin, A Course in Differential Geometry, American Mathematical Society, 2001.
Supplementary Books:
2. S. Kumaresan, A Course in Differential Geometry and Lie Groups, Hindustan Book Agency, 2002.
3. Manfredo Perdigao do Carmo, Riemannian Geometry, Birkhauser, 1992.
4. Loring W. Tu, Differential Geometry: Connections, Curvature, and Characteristic Classes, Springer 2017.
References:
5. Sigurdur Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, American Mathematical Society, 2001.
6. Shoshichi Kobayashi, Katsumi Nomizu, Foundations of Differential Geometry, Volumes 1 and 2, Wiley-Interscience, 1996.
7. S.S. Chern, W.H. Chen, K.S. Lam, Lectures on Differential Geometry, World Scientific, 2000.
8. Michael Spivak, A Compherensive Introduction to Differential Geometry, Volumes 1,2,3,4 and 5, Publish or Perish, 1999.
9. Wolfgang Kühnel, Differential Geometry, Curves, Surfaces, Manifolds, 3rd edition, American Mathematical Society, 2015.
10. Barrett O Neill, Elementary Differential Geometry, Second revised Edition, Academic Press, 2016.
11. Manfredo Perdigao do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, 1976.

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/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11
LO.1334434343
LO.2334434343
LO.3334434343
LO.4334434343
LO.5334434343

CONTACT INFORMATION
Dokuz Eylül Üniversitesi Cumhuriyet Bulvarı No: 144 35210 Alsancak / İZMİR
Phone: +90(232) 412 12 12 - Fax: +90 (232) 464 81 35

Copyright 2015 DEÜ. BİD

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