### 2023　Geometry II

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Academic unit or major
Undergraduate major in Mathematics
Instructor(s)
Honda Nobuhiro
Class Format
Lecture / Exercise    (Face-to-face)
Media-enhanced courses
Day/Period(Room No.)
Fri3-6(M-107(H113))
Group
-
Course number
MTH.B302
Credits
2
2023
Offered quarter
2Q
Syllabus updated
2023/3/20
Lecture notes updated
-
Language used
Japanese
Access Index

### Course description and aims

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.

### Student learning outcomes

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 that will be developed

 ✔ 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/Required learning

Course schedule Required learning
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.

### Out-of-Class Study Time (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.

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

### Assessment criteria and methods

Final exam and discussion session. Details will be provided during class sessions.

### Related courses

• MTH.B301 ： Geometry I
• MTH.B331 ： Geometry III

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students are expected to have passed [Geometry I].