MT0017 Differentiable Manifolds

6 ECTS - 3-0 Duration (T+A)- . Semester- 3 National Credit

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

Update Time: 09.05.2024 11:37