TÜRKÇE
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Information
| Code | MT0026 |
| Name | Riemann Geometry |
| Term | 2024-2025 Academic Year |
| Term | Spring |
| Duration (T+A) | 3-0 (T-A) (17 Week) |
| ECTS | 6 ECTS |
| National Credit | 3 National Credit |
| Teaching Language | Türkçe |
| Level | Yüksek Lisans Dersi |
| Type | Normal |
| Mode of study | Yüz Yüze Öğretim |
| Catalog Information Coordinator | Doç. Dr. NERGİZ POYRAZ |
| Course Instructor |
1 |
Course Goal / Objective
To teach the general properties of differantiable manifolds, tensors, immersion and imbeddindgs.
Course Content
Differentiable manifolds, Tangent Spaces, Vector Fields, Lie operator, Topology of Manifolds, Riemannian metrics, Riemannian Connections, Geodesics, Curvature tensor and Sectional Curvature, Tensor on Manifolds.
Course Precondition
None
Resources
Manfredo Perdigao do Carmo , Riemann Geometri , Birkhauser, 1992 W. M. Boothby, Diferansiyellenebilir manifoldlar ve Rimann Geometriye Giriş, Elsevier, 2003.
Notes
Manfredo Perdigao do Carmo , Riemann Geometri , Birkhauser, 1992
Course Learning Outcomes
| Order | Course Learning Outcomes |
|---|---|
| LO01 | Knows the concept of differentiable manifold, gives examples. |
| LO02 | Learns the concepts of Immersion and Imbedding. |
| LO03 | Learn vector fields, Lie bracket operator concepts. |
| LO04 | Learns the concepts of affine connection and Riemann connection. |
| LO05 | Learns the concepts of curvature tensor and cross-sectional curvature. |
| LO06 | Learns the concepts of Ricci curvature and scalar curvature. |
| LO07 | Knows the general properties of tensors. |
Relation with Program Learning Outcome
| Order | Type | Program Learning Outcomes | Level |
|---|---|---|---|
| PLO01 | Bilgi - Kuramsal, Olgusal | Knows in detail the relationship between the results in her area of expertise and other areas of mathematics. | 4 |
| PLO02 | Bilgi - Kuramsal, Olgusal | Knows in detail the relationship between the results in his area of expertise and other areas of mathematics. | 3 |
| PLO03 | Bilgi - Kuramsal, Olgusal | Establishes new mathematical models with the help of the knowledge gained in the field of specialization. | 2 |
| PLO04 | Bilgi - Kuramsal, Olgusal | Has basic knowledge in all areas of mathematics. | 5 |
| PLO05 | Bilgi - Kuramsal, Olgusal | It presents the knowledge gained in different fields of mathematics and their relations with each other in the simplest and most understandable way. | 4 |
| PLO06 | Bilgi - Kuramsal, Olgusal | Effectively uses the technical equipment needed to express mathematics. | 2 |
| PLO07 | Bilgi - Kuramsal, Olgusal | poses original problems related to field and presents different solution techniques. | |
| PLO08 | Bilgi - Kuramsal, Olgusal | carries out original and qualified scientific studies on the subject related to its field. | 3 |
| PLO09 | Bilgi - Kuramsal, Olgusal | Analyzes existing mathematical theories and develops new theories. | |
| PLO10 | Beceriler - Bilişsel, Uygulamalı | Knows the teaching-learning techniques in areas of mathematics that require expertise and uses these techniques effectively at every stage of education. | 2 |
| PLO11 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | To have knowledge of a foreign language at a level to be able to follow foreign sources related to the field and to communicate verbally and in writing with foreign stakeholders. | 5 |
| PLO12 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | presents and publishes its original works within the framework of scientific ethical rules for the benefit of its stakeholders. | |
| PLO13 | Yetkinlikler - Öğrenme Yetkinliği | Adheres to the ethical rules required by its scientific title | 5 |
Week Plan
| Week | Topic | Preparation | Methods |
|---|---|---|---|
| 1 | Differentiable manifolds | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 2 | Tangent spaces | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 3 | Immersion and Imbeddings and examples | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 4 | Orientation in manifolds | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 5 | Vector fields, Lie bracket operator | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 6 | Topology of manifolds | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 7 | Riemann metrics | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 8 | Mid-Term Exam | All topics covered up to week 7 in recommended resources | Ölçme Yöntemleri: Yazılı Sınav |
| 9 | Affine and Riemann connections | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 10 | geodesics | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 11 | Convex neighborhoods | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 12 | Curvature tensor and cross-sectional curvature | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 13 | Ricci curvature and scalar curvature | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 14 | Tensors on manifolds | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 15 | Tensors on manifolds 1 | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 16 | Term Exams | All topics covered in recommended resources | Ölçme Yöntemleri: Yazılı Sınav |
| 17 | Term Exams | All topics covered in recommended resources | Ölçme Yöntemleri: Yazılı Sınav |
Student Workload - ECTS
| Works | Number | Time (Hour) | Workload (Hour) |
|---|---|---|---|
| Course Related Works | |||
| Class Time (Exam weeks are excluded) | 13 | 3 | 39 |
| Out of Class Study (Preliminary Work, Practice) | 13 | 5 | 65 |
| Assesment Related Works | |||
| Homeworks, Projects, Others | 3 | 5 | 15 |
| Mid-term Exams (Written, Oral, etc.) | 0 | 0 | 0 |
| Final Exam | 1 | 25 | 25 |
| Total Workload (Hour) | 144 | ||
| Total Workload / 25 (h) | 5,76 | ||
| ECTS | 6 ECTS | ||