MATH545 DIFFERENTIAL GEOMETRY I
| Course Code: | 2360545 |
| 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: | Fall Semesters. |
Course Objectives
By the end of this course, students will be able to:
- Understand the fundamentals of differentiable manifolds and apply them.
- Define and analyze Riemannian metrics.
- Comprehend the Levi-Civita Connection and compute covariant derivatives.
- Formulate geodesics and understand the exponential map.
- Understand the Riemann curvature tensor and its properties.
- Compute sectional curvature, Ricci curvature, and scalar curvature and interpret them geometrically.
- Analyze submanifolds of Riemannian manifolds and understand the Gauss and Codazzi equations.
Course Content
Review of differentiable manifolds and tensor fields. Riemannian metrics, the Levi-civita connections. Geodesics and exponential map. Curvature tensor, sectional curvature. Ricci tensor, scalar curvature. Riemannian submanifolds. Gauss and Codazzi equations.
Course Learning Outcomes
By the end of this course, students will be able to:
- Define and explain the basic concepts of differentiable manifolds, including charts, atlases, tangent spaces, and tensor fields.
- Construct and analyze Riemannian metrics on differentiable manifolds and use them to compute lengths, angles.
- Demonstrate understanding of the Levi-Civita connection, and compute covariant derivatives and parallel transport on Riemannian manifolds.
- Derive and solve the geodesic equations on Riemannian manifolds.
- Compute and interpret the Riemann curvature tensor, sectional curvature, Ricci tensor, and scalar curvature.
- Analyze the geometric implications of curvature in Riemannian geometry.
- Study Riemannian submanifolds and use the Gauss and Codazzi equations to understand their intrinsic and extrinsic geometry.
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