2025 (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: 2025
Offered quarter: 4Q
Syllabus updated: Mar 19, 2025
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, manifold

Competencies

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

Class flow

Each lecture will be constructed under the hormworks assigned on the previous week, including problems related to the lecture and questions related to the material.

Course schedule/Objectives

	Course schedule	Objectives
Class 1	Parametrizatin, lengths and areas	Details will be provided during each class session.
Class 2	The first and second fundamental forms and some curvature I	Details will be provided during each class session.
Class 3	The first and second fundamental forms and some curvature II	Details will be provided during each class session.
Class 4	The fundamental theorem for surface theory (Theorema Egregium)	Details will be provided during each class session.
Class 5	Geodesic lines	Details will be provided during each class session.
Class 6	Tangent bundle and differential form	Details will be provided during each class session.
Class 7	Survey of manifolds	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 (including assignments) for each class.
They should do so by referring to textbooks and other course material.

Textbook(s)

Details will be provided during each class session.

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.

Contact information (e-mail and phone) Notice : Please replace from ”[at]” to ”@”(half-width character).

naoe[at]math.titech.ac.jp

Office hours

Contact by E-mails, or by chats of the online course.

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