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- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Hattori Toshiaki
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Mon3-4(H115)
- Group
- -
- Course number
- MTH.B408
- Credits
- 1
- Academic year
- 2017
- Offered quarter
- 4Q
- Syllabus updated
- 2017/3/17
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
The aim of this lecture is to familiarize the students with the basic language of and some fundamental theorems in Riemannian geometry. This course is a continuation of [MTH.B407 : Advanced topics in Geometry C1].
Student learning outcomes
Students are expected to
・understand the definition of geodesic and the theorem on completeness.
・understand that Einstein equation is a second order non-linear partial differential equation for Riemannian metrics.
Keywords
Parallel translation, geodesic, exponential map, normal cocordinate neighborhood, Einstein equation, Hopf-Rinow Theorem
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
Standard lecture course
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Parallel translation | Details will be provided during in class. |
| Class 2 | The definition of geodesics and the equations of geodesics | Details will be provided in class. |
| Class 3 | Exponential map | Details will be provided in class. |
| Class 4 | Normal coordinate neighborhood, Gauss' lemma | Details will be provided in class. |
| Class 5 | Geodesics are locally minimizing curves | Details will be provided in class. |
| Class 6 | Einstein equation, Hopf-Rinow theorem | Details will be provided in class. |
| Class 7 | Proof of Hopf-Rinow theorem | Details will be provided in class. |
| Class 8 | Jacobi field | Details will be provided in class. |
Textbook(s)
None required
Reference books, course materials, etc.
M.do Carmo, Riemannian Geometry, Birkhauser
Assessment criteria and methods
Exams and reports. Details will be provided in class.
Related courses
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
- ZUA.B301 ： Geometry I
- MTH.B331 ： Geometry III
- MTH.B407 ： Advanced topics in Geometry C1
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Students are expected to have passed [Advanced topics in Geometry C1].
