2022 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Geometry F
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Kotaro Yamada
- Class Format
- Lecture (Livestream)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Tue
- Class
- -
- Course Code
- MTH.B502
- Number of credits
- 100
- Course offered
- 2022
- Offered quarter
- 2Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
A local theory of Riemannian manifolds and fundamental theorem of surface theory for surface in Riemannian 3-manifolds of constant scetional curvature are introduced. As an application, a relationship of surfaces of constant mean curvature surface in different spaces,
e.g. minimal surfaces in Euclidean 3-space and constant curvature one surfaces in hyperbolic space, is discussed.
Course description and aims
Students will learn: a local theory of Riemannian manifolds, i.e. notions of Riemannian metrics, sectional curvatures; spaces of constant curvature (space forms); an extension of the fundamental theorem of surface theory for surfaces in 3-dimensional space forms.
Keywords
Riemannian metric, curvature, space form, fundamental theorem of surface theory
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 | Riemannian metrics and connections |
Details will be provided during each class |
| Class 2 | Curvatures |
Details will be provided during each class |
| Class 3 | Euclidean spaces and Spheres |
Details will be provided during each class |
| Class 4 | Lorentz-Minkowski space |
Details will be provided during each class |
| Class 5 | Hyperbolic spaces |
Details will be provided during each class |
| Class 6 | Fundamental theorem of surface theory revisited |
Details will be provided during each class |
| Class 7 | Constant mean curvature surfaces in 3-dimensional space forms |
Details will be provided during each class |
Study advice (preparation and review)
Official 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)
Evaluation methods and criteria
Graded by homeworks. Details will be announced through T2SCHOLA
Related courses
- MTH.B211 ： Introduction to Geometry I
- MTH.B212 ： Introduction to Geometry II
- MTH.B501 ： Advanced topics in Geometry E
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
Visit http://www.math.titech.ac.jp/~kotaro/class/2022/geom-f for details.