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.
[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.
curves, surfaces, singular points, Gaussian curvature, wave fronts
✔ Specialist skills | ✔ Intercultural skills | Communication skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
The course provides the fundamentals of curves, surfaces and singularities.
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. |
Class 15 | applications of criteria for singularities | Understand the contents covered by the lecture. |
None required
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) .
Final report and class attendance
The student has better to have a knolwedge of Topology and vector analysis