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- Common courses
- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Yamada Kotaro
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Tue3-4(H104)
- Group
- -
- Course number
- MTH.B406
- Credits
- 1
- Academic year
- 2019
- Offered quarter
- 2Q
- Syllabus updated
- 2019/3/18
- Lecture notes updated
- 2019/7/23
- 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
Homeworks will be assined for each lesson.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | The fundamental theorem for linear ordinary differential equations | Details will be provided during each class session |
| Class 2 | The integrability condition of liner systems of partial differential equatons | 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
- MTH.B406 ： Advanced topics in Geometry B1
Prerequisites (i.e., required knowledge, skills, courses, etc.)
MTH.B211 幾何学概論第一, MTH.B212 幾何学概論第二に相当する知識 (梅原・山田著「曲線と曲面」(改訂版) の§1から§10 程度の内容),およ
び3次元空間形の基礎的な事項（ZUA.B331 幾何学特別講義) を前提とする．
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. MTH.B406) 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.
