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- Common courses
- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Yamada Kotaro
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Fri5-6(H119A)
- Group
- -
- Course number
- MTH.B501
- Credits
- 1
- Academic year
- 2016
- Offered quarter
- 1Q
- Syllabus updated
- 2016/12/14
- Lecture notes updated
- 2016/5/27
- Language used
- Japanese
- Access Index
-
Course description and aims
After reviewing the theory of surfaces, students will be given proof for the fundamental theorem of surface theory. The instructor will introduce, as an application of it, the relationship between surfaces of constant negative curvature and the sine-Gordon equation, Hilbert's theorem, and Bäcklund transformations.
Students will learn the relationship between fundamental equations and appropriate coordinate systems through the class theory of a certain class of surfaces with specified properties, not general theory of surfaces. Students are given specific examples of methods for approaching problems in differential geometry.
Student learning outcomes
As a continuation of a basic course of differential geometry of curves and surfaces (e.g. MTH.B211 Introduction to Geometry I/MTH.B212 Introductioｎ to Geometry II), the fundamental theorem of surface theory is introduced, and as its application, theory of surfaces of constant negative curvature is treated. This course will be continued to MTH.B502 Advanced Topics in Geometry F.
In particular students will experience
(1) to know a method to translate a differential geometric problem to a problem on partial differential equation, and
(2) to know an explicit example of a relationship between differential geometric objects and so called an "integrable system".
Keywords
The fundamental theorem of surface theory, Gauss and Codazzi equations, surface of constant negative curvature, sine-Gordon equation, Bäcklund transformation.
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
Standard lecture. Homeworks will be assigned in each class.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Preliminaries: A review of classical surface theory | Details will be provided during each class session |
| Class 2 | Preliminaries: The Gauss and Weingarten equations | Details will be provided during each class session |
| Class 3 | Preliminaries: The Gauss and Codazzi Equations | Details will be provided during each class session |
| Class 4 | Preliminaries: The fundamental theorem of the surface theory | Details will be provided during each class session |
| Class 5 | Application: Surfaces of constant negative curvature 1: Asymptotic Chebyshev nets | Details will be provided during each class session |
| Class 6 | Application: Surfaces of constant negative curvature 2: sine Gordon equation | Details will be provided during each class session |
| Class 7 | Application: Surfaces of constant negative curvature 3: Hilbert's theorem | Details will be provided during each class session |
| Class 8 | Application: Surfaces of constant negative curvature 4: Bäcklund transformations | Details will be provided during each class session |
Textbook(s)
None required
Reference books, course materials, etc.
Masaaki Umehara and Kotaro Yamada, Differential geometry of curves and surfaces, to be published in 2016.
C. Rogers and W. K. Schief, Bäcklund and Darboux transformations, Cambridge Texts in Applied Mathematics, 2002
Course materials are provided during class，and also found in OCW/OCW-i.
Assessment criteria and methods
Graded by homeworks
Related courses
- MTH.B211 ： Introduction to Geometry I
- MTH.B212 ： Introduction to Geometry II
- MTH.B502 ： Advanced topics in Geometry F
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) is 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.
