2024 Faculty Courses School of Science Undergraduate major in Mathematics
Geometry I
- 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.B301
- Number of credits
- 110
- Course offered
- 2024
- Offered quarter
- 1Q
- Syllabus updated
- Mar 14, 2025
- Language
- Japanese
Syllabus
Course overview and goals
The aim of this course is to familiarize the students with basic notions and properties on diferentiable manifolds, which are important not only in mahematics but in related areas such as theoretical physics. It is not easy for beginners to comprehend these abstract notions without suitable training. We supply many concrete examples in each lecture.
The contents of this course is as follows: definition and examples of manifolds, smooth functions on manifolds, smooth maps between manifolds, constructing manifolds by using the inverse images of regular values, definition of tangent vectors and tangent spaces. Each lecture will be accompanied by a problem solving class. This course will be succeeded by [MTH. B302 : Geometry II] in the second quater.
Course description and aims
Students are expected to
・understand the definition of manifolds.
・know more than 5 examples of manifolds.
・understand the definitions of smooth functions on manifolds, and smooth maps between manifolds.
・be familiar with the method of constructing manifolds by using the inverse images of regular values.
・understand the definition of tangent vectors and tangent spaces.
Keywords
Manifolds, differentiable structures, smooth function, smooth map, regular value, projective space, tangent vector, tangent space
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 definition of manifolds, examples of manifolds (spheres) | Details will be provided during each class session. |
| Class 2 | Discussion session | Details will be provided during each class session. |
| Class 3 | Examples of manifolds (examples which are not spheres), differentiable structures | Details will be provided during each class session. |
| Class 4 | Discussion session | Details will be provided during each class session. |
| Class 5 | Smooth functions and maps, construction of manifolds as the inverse image of a regular value of a map | Details will be provided during each class session. |
| Class 6 | Discussion session | Details will be provided during each class session. |
| Class 7 | Proof of the fact that the inverse image of a regular value is a manifold | Details will be provided during each class session. |
| Class 8 | Discussion session | Details will be provided during each class session. |
| Class 9 | Real projective spaces, curves on real projective spaces | Details will be provided during each class session. |
| Class 10 | Discussion session | Details will be provided during each class session. |
| Class 11 | Complex projective spaces, the definition of tangent vectors | Details will be provided during each class session. |
| Class 12 | Discussion sesssion | Details will be provided during each class session. |
| Class 13 | The definition of tangent spaces, vector space structure on tangent spaces | 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.B302 ： Geometry II
- MTH.B203 ： Introduction to Topology III
- MTH.B204 ： Introduction to Topology IV
- MTH.C201 ： Introduction to Analysis I
- MTH.C202 ： Introduction to Analysis II
Prerequisites
Students are expected to have passed [Introduction to Topology Ⅲ, Ⅳ], and [Introduction to Analysis Ⅰ, Ⅱ].