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School of Environment and Society
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- Common courses
- Liberal arts and basic science courses
- 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.B403
- Credits
- 1
- Academic year
- 2018
- Offered quarter
- 3Q
- Syllabus updated
- 2018/3/20
- 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 will be succeeded by [MTH.B404 : Advanced topics in Geometry D].
Student learning outcomes
Students are expexted to
・understand the definitions of Riemannian metric, sectional curvature, Ricci curvature, and scalar curvature.
・be familiar with the method of expressing them by using local coordinates of the underlying manifold.
Keywords
Riemannian metric, connection, covariant derivative, curvature tensor, Levi-Civita connection, sectional curvature, Ricci curvature, scalar curvature, Laplacian
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 | the definition and examples of Riemannian metrics | Details will be provided during each class session |
| Class 2 | the length of a curve, the distance and the volume form on a Riemannian manifold | Details will be provided during each class session |
| Class 3 | connection and covariant derivative, torsion tensor and curvature tensor | Details will be provided during each class session |
| Class 4 | Bianchi's identity, extending a covariant derivative to diffenrential of (general) tensor fields | Details will be provided during each class session |
| Class 5 | Levi-Civita connection and sectional curvature | Details will be provided during each class session |
| Class 6 | relationship between Gaussian curvature and sectional curvature, Ricci curvature and scalar curvature | Details will be provided during each class session |
| Class 7 | local expression of covariant derivative | Details will be provided during each class session |
| Class 8 | divergence and Green's theoem, Laplacian | Details will be provided during each class session |
Textbook(s)
Non required
Reference books, course materials, etc.
M. do Carmo, Riemannian Geometry, Birkhauser
S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, Springer
Assessment criteria and methods
Exams and reports. Details will be provided during class sessions.
Related courses
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
- MTH.B331 ： Geometry III
- MTH.B404 ： Advanced topics in Geometry D
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Students are expected to have passed [Geometry I], [Geometry II] and [Geometry III].
Office hours
to be determined
