MATH546 DIFFERENTIAL GEOMETRY II
| Course Code: | 2360546 |
| METU Credit (Theoretical-Laboratory hours/week): | 3 (3.00 - 0.00) |
| ECTS Credit: | 8.0 |
| Department: | Mathematics |
| Language of Instruction: | English |
| Level of Study: | Graduate |
| Course Coordinator: | Assoc.Prof.Dr. İBRAHİM ÜNAL |
| Offered Semester: | Spring Semesters. |
Course Objectives
By the end of this course, students should be able to:
- Understand Lie groups and Lie algebras and actions of Lie groups on manifolds.
- Comprehend the theory of principal fiber bundles and learn connections and curvature on principal bundles.
- Define almost complex structures and integrability conditions and study both almost complex and complex manifolds.
- Learn Hermitian metrics and understand their compatibility with complex structures.
- Learn Kähler manifolds and analyze their properties.
- Understand the role of the Kähler form and its implications in geometry.
- Define symmetric spaces and investigate their classification.
- Analyze geometric and algebraic properties of symmetric spaces and study examples of compact and noncompact symmetric space.
Course Content
Lie groups; principle fibre bundles; almost complex and complex manifolds; Hermitian and Kaehlerian geometry; symmetric spaces.
Course Learning Outcomes
By the end of this course, students will be able to:
- Define and explain the structure and properties of Lie groups and Lie algebras.
- Demonstrate understanding of principal fiber bundles, including their local trivializations, transition functions, and structure groups.
- Construct and analyze connections and curvature on principal bundles, and understand their geometric significance.
- Identify and distinguish almost complex and complex manifolds, and determine integrability conditions using tools such as the Nijenhuis tensor.
- Define and work with Hermitian metrics and Kähler structures, and compute associated geometric quantities such as the Kähler form.
- Apply differential and complex geometric concepts to study the local and global structure of complex manifolds.
- Describe and analyze the geometry of symmetric spaces, including their classification and curvature properties.
Program Outcomes Matrix
| Level of Contribution | |||||
| # | Program Outcomes | 0 | 1 | 2 | 3 |
| 1 | Acquires mathematical thinking skills (problem solving, generating ways of thinking, forming correspondence, generalizing etc.) and can use them in related fields. | ✔ | |||
| 2 | Gains academic maturity through self-study. | ✔ | |||
| 3 | Can design mathematics related problems, devise solution methods and apply them when appropriate. | ✔ | |||
| 4 | Carries out parts of a mathematical research program independently. | ✔ | |||
| 5 | Has a command of Turkish and English languages so that he/she can actively communicate (read, write, listen and speak). | ✔ | |||
| 6 | Contributes to solving global, environmental and social problems either individually or as being part of a social group. | ✔ | |||
| 7 | Respects ethical values and rules; applies them in professional and social issues. | ✔ | |||
| 8 | Can work cooperatively in a team and also individually. | ✔ | |||
| 9 | Gets exposed to academic culture through interaction with others. | ✔ | |||
| 10 | Comprehends necessity of knowledge, can define it and acquires it; uses knowledge effectively and shares it with others. | ✔ | |||
0: No Contribution 1: Little Contribution 2: Partial Contribution 3: Full Contribution