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- Academic unit or major
- Mathematics
- Instructor(s)
- Yamada Kotaro
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Tue3-4(H104)
- Group
- -
- Course number
- ZUA.B332
- Credits
- 1
- Academic year
- 2019
- Offered quarter
- 2Q
- Syllabus updated
- 2019/3/18
- Lecture notes updated
- -
- Language used
- English
- Access Index
-
Course description and aims
Definition and meanings of the "curvature" of Riemannian manifolds, especially those obtained as submanifolds of (pseudo) Euclidean space, are introduced.
Student learning outcomes
Students are expected to know
- the integrability condition of linear system of partial differential equations,
- the sectional curvature of a Riemannian manifolds,
- the curvature as an integrability condition,
- and the local uniqueness of Riemannian manifolds of constant sectional curvature.
Keywords
Riemannian manifolds, curvature, integrability conditions
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
A standard lecture course.
Homeworks will be assigned for each lesson.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Linear ordinary differential equations | Details will be provided during each class session |
| Class 2 | The fundamental theorem for linear ordinary differential equations | Details will be provided during each class session |
| Class 3 | The second fundamental forms of hypersurfaces and the sectional curvature | Details will be provided during each class session |
| Class 4 | Spheres and hyperbolic spaces | Details will be provided during each class session |
| Class 5 | The curvature tensor and the sectional curvature | Details will be provided during each class session |
| Class 6 | Local uniqueness of Riemannian manifolds of constant sectoinal curvature | Details will be provided during each class session |
| Class 7 | Models of hyperbolic spaces | Details will be provided during each class session |
Textbook(s)
No textbook is set.
Lecture note will be provided.
Reference books, course materials, etc.
S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, Third Edition, Springer-Verlag, 2013.
M. P. do Carmo (transl. F. Flaherty), Riemannian Geometry, Birkhauser, 1994.
Assessment criteria and methods
Graded by homeworks
Related courses
- MTH.B211 ： Introduction to Geometry I
- MTH.B212 ： Introduction to Geometry II
- ZUA.B331 ： Advanced courses in Geometry A
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Knowledge on differential geometry of curves and surfaces (as in MTH.B211 "Introduction to Geometry I" and MTH.B212 "Introduction to
Geometry II", or Sections 1 to 10 of the text book "Differential Geometry of Curves and Surfaces" by M. Umehara and K.
Yamada), and knowledge of fundamental notions of space forms (e.g. ZUA.B331) are required.
Contact information (e-mail and phone) Notice : Please replace from "[at]" to "@"(half-width character).
kotaro[at]math.titech.ac.jp
Office hours
N/A. Contact by E-mails, or at the classroom.
Other
For details, visit the web-site of this class http://www.math.titech.ac.jp/~kotaro/class/2019/geom-b
