Introduction to Geometry I

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.B211

Number of credits

100

Course offered

2026

Offered quarter

3Q

Syllabus updated

Mar 5, 2026

Language

Japanese

Syllabus

Course overview and goals

After brief reviews of Linear Algebra and Calculus, the following items about curves in the Euclidean spaces are introduced:
parametrized plane curves, the arc length, the curvature, Frenet's formula and the fundamental theorem of plane curves; parametrized space curves, the curvature, torsion and the fundamental theorem of space curves.
Through the basic matters in the differential geometry of plane/space curves, the students will observe the scenes of applications of Linear Algebra and Calculus, and get a notion of "transformations" and "invariants" which are fundamental concept of the modern geometry. This course is succeeded by " Introduction to Geometry II" in 4Q.

Course description and aims

The students will learn the basic matters of differential geometry of plane curves and space curves. In particular
(1) To understand that the curvature and the torsion of curves as invariants under isometries and parameter changes, and that they determine a curve, that is the fundamental theorem for curves.
(2) To know the difference between "local" notions and "global" notions through the relationship between the topological property for closed curves and curvature.
(3) To confirm the theories by calculations on concrete examples.

Keywords

Differential Geometry, Plane Curves, Space Curves, Curvature, Torsion, Isometries

Competencies

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

Class flow

Standard lecture course.

Course schedule/Objectives

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
Manfredo P. do Carmo, Differenial Geoetry of Curves and Surfaces,　Prentice-Hall Inc., 1976.

Evaluation methods and criteria

Details will be provided on the first class.

Related courses

• MTH.B212 ： Introduction to Geometry II
• LAS.M102 ： Linear Algebra I / Recitation
• LAS.M106 ： Linear Algebra II
• LAS.M101 ： Calculus I / Recitation
• LAS.M105 ： Calculus II

Prerequisites

The contents of Linear Algebra I/II, and Calculus I/II are assumed, but not formal prerequisite.

Other

In addition to the subjects in "Related Courses“， the following courses are related to this subject:
Differential Equations I/II; Introduction to Topology I/II/III/IV; Geometry I/II/III