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Information
| Code | MT0017 |
| Name | Differentiable Manifolds |
| Term | 2024-2025 Academic Year |
| Semester | . Semester |
| 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 Goal / Objective
The aim of this course is to provide knowledge about differantiable structure and differantiable manifolds.
Course Content
Diffeomorphism, Topological Manifold, Differentiable Manifold, Tangent Vector, Vector field, 1-Forms, Tensor Field, Various Derivatives on Manifold: İnterior, Exterior and Covariant Derivative, Immersions, Submersions, Distributions, Curvature Tensor Field, Torsion Tensor Field, Structure Equations, Vector Bundles, Integration on Manifolds, Riemannan Manifolds, Riemannian Connection,Sectional Curvature, Ricci Tensor, Scalar Curvature, Constant Sectional Curvature Riemannian Manifolds, Integration on Riemannian Manifolds, Stokes Theorem, Riemannian Submanifolds, Gauss-Codazzi Equations, Totally Geodesic, Umbilical and Minimal Submanifolds, Submanifolds of Space Form.
Course Precondition
None
Resources
1. Bayram Şahin, Manifoldların Diferensiyel Geometrisi, Nobel Yayıncılık, 2012. 2. Boothby, William M. An introduction to differentiable manifolds and Riemannian geometry. Vol. 120. Academic press, 1986.
Notes
1. Bayram Şahin, Manifoldların Diferensiyel Geometrisi, Nobel Yayıncılık, 2012. 2. Boothby, William M. An introduction to differentiable manifolds and Riemannian geometry. Vol. 120. Academic press, 1986.
Course Learning Outcomes
| Order | Course Learning Outcomes |
|---|---|
| LO01 | Understand differentiable manifold and differentiable concepts on it. |
| LO02 | Examines the concepts of curvature tensor field and torsion tensor field. |
| LO03 | Understands Riemann manifolds and submanifolds. |
| LO04 | Learns integration on Riemann manifolds. |
| LO05 | Fully Comprehends Geodetic, Umbilical and Minimal Submanifolds, Submanifolds of Space Form. |
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. | 3 |
| PLO02 | Bilgi - Kuramsal, Olgusal | Knows in detail the relationship between the results in his area of expertise and other areas of mathematics. | 2 |
| PLO03 | Bilgi - Kuramsal, Olgusal | Establishes new mathematical models with the help of the knowledge gained in the field of specialization. | |
| PLO04 | Bilgi - Kuramsal, Olgusal | Has basic knowledge in all areas of mathematics. | 4 |
| 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. | 5 |
| 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. | 4 |
| 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. | 4 |
| 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 | 4 |
Week Plan
| Week | Topic | Preparation | Methods |
|---|---|---|---|
| 1 | Algebraic Concepts, Diffeomorphism, Topological Manifold | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 2 | Differentiable Manifold and Differentiable Concepts On It: Tangent Vector, Vector Field, 1-Form, Tensor Field | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 3 | Various Derivatives on Manifold: Inner, Outer and Covariant Derivative | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 4 | Immersions and Submersions, Distributions | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 5 | Curvature Tensor Field, Torsion Tensor Field | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 6 | Structure Equations | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 7 | Vector Bundles, Integration on Manifolds | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 8 | Mid-Term Exam | Pages related to the subject in the source books | Ölçme Yöntemleri: Yazılı Sınav |
| 9 | Riemann Manifolds and Riemann Connection | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 10 | Curvature of Section, Ricci Tensor, Scalar Curvature | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 11 | Riemann Manifolds with Fixed Section Curvature | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 12 | Integration on Riemann Manifolds, Stokes Theorem | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 13 | Riemann Submanifolds, Gauss-Codazzi Equations | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 14 | Fully Geodetic, Umbilical and Minimal Submanifolds, Submanifolds of Space Form | Pages related to the subject in the source books | Öğretim Yöntemleri: Anlatım, Tartışma |
| 15 | An overview | Pages related to the subject in the source books | Ölçme Yöntemleri: Yazılı Sınav |
| 16 | Term Exams | An overview | Ölçme Yöntemleri: Yazılı Sınav |
| 17 | Term Exams | An overview | 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 | ||