2024 Faculty Courses School of Science Undergraduate major in Mathematics
Geometry II
- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Tamas Kalman
- Class Format
- Lecture/Exercise (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-6 Fri
- Class
- -
- Course Code
- MTH.B302
- Number of credits
- 110
- Course offered
- 2024
- Offered quarter
- 2Q
- Syllabus updated
- Mar 14, 2025
- Language
- Japanese
Syllabus
Course overview and goals
The aim of this course is the same as the one of [MTH. B301 : Geometry I]: it is to familiarize the students with basic notions and properties on differentiable manifolds.
The contents of this course is as follows: differentials of maps, regular values, critical points, inverse function theorem, Sard's theorem, immersions and embeddings, submanifold, partition of unity, vector fileds. Each lecture will be accompanied by a problem solving class. This course is a continuation of [Geometry I] in the first quarter and will be succeeded by [MTH. B331 : Geometry Ⅲ] in the third quater.
Course description and aims
Students are expected to
・understand the definition of defferentials of maps between manifolds.
・know more than 3 examples of submanifolds.
・be able to use ``Partition of unity''.
・understand the definitions of brackets of vector fields and integral curves of vector fields.
Keywords
Differential of a map, regular value, critical point, inverse function theorem, Sard's theorem, immersion and embedding, Whitney's embedding theorem, partition of unity, vector field, bracket, integral curve, 1-parameter group of transformations
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course accompanied by discussion sessions
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | The differential of a map, regular points, critical points | Details will be provided during each class session. |
| Class 2 | Dicussion session | Details will be provided during each class session. |
| Class 3 | Inverse function theorem, the inverse image of a regular value, Sard's theorem | Details will be provided during each class session. |
| Class 4 | Discussion session | Details will be provided during each class session. |
| Class 5 | Immersion, embedding | Details will be provided during each class session. |
| Class 6 | Discussion session | Details will be provided during each class session. |
| Class 7 | Relationship between submanifolds and embeddings | Details will be provided during each class session. |
| Class 8 | Discussion session | Details will be provided during each class session. |
| Class 9 | Whitney's embedding theorem, partition of unity | Details will be provided during each class session. |
| Class 10 | Discussion session | Details will be provided during each class session. |
| Class 11 | Vector field, bracket, integral curves of vector fields | Details will be provided during each class session. |
| Class 12 | Discussion session | Details will be provided during each class session. |
| Class 13 | 1-parameter groups of transformations | Details will be provided during each class session. |
| Class 14 | Discussion session | Details will be provided during each class session. |
Study advice (preparation and review)
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)
None required
Reference books, course materials, etc.
Yozo Matsushima, Differentiable Manifolds (Translated by E.T. Kobayashi), Marcel Dekker, Inc., 1972
Frank W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer-Verlag, 1983
Evaluation methods and criteria
Final exam and discussion session. Details will be provided during class sessions.
Related courses
- MTH.B301 ： Geometry I
- MTH.B331 ： Geometry III
Prerequisites
Students are expected to have passed [Geometry I].