2026 (Current Year) Faculty Courses School of Science Undergraduate major in Mathematics
Introduction to Geometry II
- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Hironobu Naoe
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Class
- -
- Course Code
- MTH.B212
- Number of credits
- 100
- Course offered
- 2026
- Offered quarter
- 4Q
- Syllabus updated
- Mar 5, 2026
- Language
- Japanese
Syllabus
Course overview and goals
As a continuation of "Introduction to Geometry I" MTH.B211, the following items about surfaces in the Euclidean 3-space are introduced:
parametrized surface, the first fundamental form; the length, the angle, and the area, the second fundamental form, the principal curvatures, the Gaussian and mean curvatures, geodesics, the Gauss-Bonnet theorem, the fundamental theorem of surface theory.
The goal is an understanding fudamental materials of classical differential geometry of surfaces, and a preparation of modern differential geometry.
Course description and aims
The students will learn the basic matters of differential geometry of surfaces in the Euclidean 3-space. In particular
(1) To understand that the parametrization of surfaces and a notion of quantities which do not depend on parameters.
(2) To know examples of global properties and local properties of surfaces.
(3) To confirm the theories by calculations on concrete examples.
Keywords
Differential Geometry, Surfaces, Cruvature, Gauss-Bonnet theorem
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Parametrizatin, tangent spaces and areas |
Details will be provided during each class session. |
| Class 2 | The first fundamental forms |
Details will be provided during each class session. |
| Class 3 | The second fundamental forms |
Details will be provided during each class session. |
| Class 4 | Curvatures |
Details will be provided during each class session. |
| Class 5 | Geodesic lines |
Details will be provided during each class session. |
| Class 6 | Theorema Egregium and geodesic triangles |
Details will be provided during each class session. |
| Class 7 | Topology of sufaces and the Gauss-Bonnet theorem |
Details will be provided during each class session. |
Study advice (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 for each class. They should do so by referring to textbooks and other course material.
Textbook(s)
In the first class, I will introduce some textbooks for this class.
Reference books, course materials, etc.
Masaaki Umehara and Kotaro Yamada, DIfferential Geometry of curves and surfaces, World Scientific, 2017
Shoshichi Kobayahi, DIfferential Geometry of curves and surfaces, Shoukabou
Yukio Matsumoto, Basics of manifolds
Evaluation methods and criteria
Details will be explained in the course.
Related courses
- MTH.B211 ： Introduction to Geometry I
- LAS.M102 ： Linear Algebra I / Recitation
- LAS.M106 ： Linear Algebra II
- LAS.M101 ： Calculus I / Recitation
- LAS.M105 ： Calculus II
Prerequisites
Students is required to take the class MTH.B211 "Introduction to Geometry I", or to study the contents of the class.
Other
In addition to the subjects in "Related Courses“， the following cources are related to this subject:
Differential Equations I/II; Introduction to Topology I/II/III/IV; Geometry I/II/III