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- Academic unit or major
- Mathematics
- Instructor(s)
- Yamada Kotaro
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Tue3-4(H115)
- Group
- -
- Course number
- ZUA.B332
- Credits
- 1
- Academic year
- 2017
- Offered quarter
- 2Q
- Syllabus updated
- 2017/3/17
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
-
Course description and aims
As a generalization of theory of surfaces in Euclidean 3-space, differential geometry of surfaces in 3-dimensional non-flat space forms (the sphere/the hyperbolic space) is introduced. In particular, existence of local isometric correspondence between constant mean curvature surfaces in Euclidean 3-space and those in 3-sphere/hyperbolic 3-space (a.k.a. Lawson correpondence) is shown.
Student learning outcomes
Students are expected to know
- the fundamental theorem for surfaces in the 3-sphere and the hyperbolic 3-space,
- local isometric correspondence of surfaces of constant mean curvature in space forms
- and a method to construct constant mean curvature surfaces in space forms.
Keywords
the fundamental theorem for surface theory, constant mean curvature surfaces, a local isometric correspondence for
constant mean curvature surfaces in space forms, Lawson correspondence
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
A standard lecture course.
Homeworks will be assined for each lesson.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Space forms (the sphere, the Euclidean space and the hyperbolic space) | Details will be provided during each class session |
| Class 2 | Surfaces in 3-dimensional space forms | Details will be provided during each class session |
| Class 3 | Totally umbilic surfaces | Details will be provided during each class session |
| Class 4 | Gauss-Weingarten formula | Details will be provided during each class session |
| Class 5 | The fundamental theorem for surfaces in space forms | Details will be provided during each class session |
| Class 6 | Surfaces of constant mean curvature, local isometric correspondence | Details will be provided during each class session |
| Class 7 | Constructing constant mean curvature surfaces | Details will be provided during each class session |
| Class 8 | (Delaunay surfaces and Wente tori) | Details will be provided during each class session |
Textbook(s)
No textbook is set.
Lecture note will be provided.
Reference books, course materials, etc.
Masaaki Umehara and Kotaro Yamada, Differential geometry of curves and surfaces, to be published in 2017.
Assessment criteria and methods
Graded by homeworks
Related courses
- MTH.B211 ： Introduction to Geometry I
- MTH.B212 ： Introduction to Geometry II
- ZUA.B331 ： Advanced courses in Geometry A
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Knowledge on differential geometry of curves and surfaces (as in MTH.B211 "Introduction to Geometry I" and MTH.B212 "Introduction to
Geometry II", or Sections 1 to 10 of the text book "Differential Geometry of Curves and Surfaces" by M. Umehara and K.
Yamada), and knowledge of fundamental notions of space forms (e.g. ZUA.B331) are required.
Contact information (e-mail and phone) Notice : Please replace from "[at]" to "@"(half-width character).
kotaro[at]math.titech.ac.jp
Office hours
N/A. Contact by E-mails, or at the classroom.
Other
For details, visit the web-site of this class http://www.math.titech.ac.jp/~kotaro/class/2017/geom-a
