2023 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Geometry F1
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Kotaro Yamada
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Tue (M-143B(H119B))
- Class
- -
- Course Code
- MTH.B506
- Number of credits
- 100
- Course offered
- 2023
- Offered quarter
- 2Q
- Syllabus updated
- Jul 8, 2025
- Language
- English
Syllabus
Course overview and goals
Definition and meanings of the "curvature" of Riemannian manifolds, especially those obtained as submanifolds of (pseudo) Euclidean space, are introduced.
Course description and aims
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
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
A standard lecture course.
Homeworks will be assined for each lesson.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Linear ordinary differential equations | Details will be provided during each class session. |
| Class 2 | The integrability condition | Details will be provided during each class session. |
| Class 3 | The fundamental theorem for hypersurfaces | Details will be provided during each class session. |
| Class 4 | The sectional curvature | Details will be provided during each class session. |
| Class 5 | The curvature tensor | Details will be provided during each class session. |
| Class 6 | Riemannian manifolds of constant sectional curvature | Details will be provided during each class session. |
| Class 7 | Realization | Details will be provided during each class session. |
Study advice (preparation and review)
Formal Message: To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and
another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
No textbook is set. Lecture note will be provided.
Reference books, course materials, etc.
Masaaki Umehara and Kotaro Yamada, Differential Geometry of Curves and Surfaces, Transl. by Wayne Rossman, World Scientific Publ.,
2017, ISBN 978-9814740234 (hardcover); 978-9814740241 (softcover)
Loring W. Tu, Differential Geometry,Graduate Texts in Mathematics, Springer-Verlag, 2017, ISBN 978-3-319-55082-4, 978-3-319-55084-8 (eBook)
Evaluation methods and criteria
Graded by homeworks. Details will be announced through T2SCHOLA
Related courses
- MTH.B505 ： Advanced topics in Geometry E1
Prerequisites
At least, knowledge of undergraduate calculus and linear algebra are required.
Attending the class "Advanced Topics in Geometry E" (MTH.B501) is strongly recommended.
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
Other
Web page:
http://www.math.titech.ac.jp/~kotaro/class/2023/geom-f1
http://www.official.kotaroy.com/class/2023/geom-f1