### 2021　Topics in Geometry

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Graduate major in Mathematical and Computing Science
Instructor(s)
Umehara Masaaki  Nishibata Shinya  Miura Hideyuki  Murofushi Toshiaki  Suzuki Sakie
Course component(s)
Lecture    (ZOOM)
Day/Period(Room No.)
Mon5-6()  Thr5-6()
Group
-
Course number
MCS.T504
Credits
2
2021
Offered quarter
2Q
Syllabus updated
2021/3/19
Lecture notes updated
-
Language used
Japanese
Access Index

### Course description and aims

When we recognize planar curves and surfaces as wave fronts, and we can consider their time evolutions. Singular points appear frequently. In this course, we review differential geometry of curves and surfaces, and also give an introduction to singularities on curves and surfaces. We introduce criteria for important singularities as well as their fundamental properties. Students attending this course will have a better familiarity with curves, surfaces and the concept of manifolds. The course itself is almost fully self-contained. So it is possible to join this course without prior knowledge of these materials.

### Student learning outcomes

[Theme] The fundamental properties of curves and surfaces are explained from the viewpoint of differential geometry. In particular, we explain several types of curvatures on curves and surfaces. We also explain topological properties, criteria and geometric properties of singularities appearing in curves and surfaces. In each class, we try to explain the material by showing examples, sometimes using computers.
[Goal] The students are expected to understand the fundamentals of curves and surfaces for handling geometric structures appearing in mathematical
and computing science, and also to be able to apply them to practical problems.

### Keywords

curves, surfaces, singular points, Gaussian curvature, wave fronts

### Competencies that will be developed

 ✔ Specialist skills ✔ Intercultural skills Communication skills Critical thinking skills ✔ Practical and/or problem-solving skills

### Class flow

The course provides the fundamentals of curves, surfaces and singularities.

### Course schedule/Required learning

Course schedule Required learning
Class 1 planar curves (singular points, regular points, curvature) Understand the contents covered by the lecture.
Class 2 planar curves (four vertex theorem, rotation index) Understand the contents covered by the lecture.
Class 3 evolute, cusps as singularities Understand the contents covered by the lecture.
Class 4 wave fronts as planar curves Understand the contents covered by the lecture.
Class 5 behaviour of curvature functions near singular points Understand the contents covered by the lecture.
Class 6 a criterion for cusps and its applications Understand the contents covered by the lecture.
Class 7 fundamentals of surface theory 1 (the first and second fundamental forms) Understand the contents covered by the lecture.
Class 8 fundamentals of surface theory 2 (Gaussian curvature, mean curvature, principal curvature) Understand the contents covered by the lecture.
Class 9 the Gauss Bonnet theorem Understand the contents covered by the lecture.
Class 10 Gaussian curvature and mean curvature of parallel surfaces Understand the contents covered by the lecture.
Class 11 wave fronts as surfaces Understand the contents covered by the lecture.
Class 12 important singularities appearing in surfaces Understand the contents covered by the lecture.
Class 13 a proof of the criterion for cusps Understand the contents covered by the lecture.
Class 14 a proof of the criterion for cross caps Understand the contents covered by the lecture.

### 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.

Masaaki Umehara Differential Geometry of curves and surfaces with singularities, Keio Universities Suuri-Kagakuka lecture note No. 38 (2009) .
Masaaki Umehara and Kotaro Yamada, Curves and surfaces revised edition, Shokabo (2015) .

### Assessment criteria and methods

Final report and class attendance

### Related courses

• MCS.T331 ： Discrete Mathematics

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

The student has better to have a knolwedge of Topology　and vector analysis