COURSE UNIT TITLE

: DIFFERENTIAL GEOMETRY-II

DEGREE PROGRAMMES

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 (English)
Mathematics (English)

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.
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


Further Notes About Assessment Methods

None

Assessment Criteria

30% (Midterm examination) +30%(Homework assignment)+40% (Final examination)

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 SARIOĞLU
E-mail: celalcem.sarioglu@deu.edu.tr
Phone: +90 232 301 8585
Office: B212 (Mathematics Department)

Office Hours

To be announced later.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Lectures 14 3 42
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) 203

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

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