2025 (Current Year) Faculty Courses School of Science Undergraduate major in Mathematics
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
- 2025
- Offered quarter
- 3Q
- Syllabus updated
- Mar 19, 2025
- 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
Each lecture consists of "comments on homeworks of the previous class", "a lecture on new topics", and "a presentation of homeworks".
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Curves in the Euclidean space | Details will be provided during each class session. |
| Class 2 | The fundamental theorem for plane curves | Details will be provided during each class session. |
| Class 3 | Closed curves | Details will be provided during each class session. |
| Class 4 | Osculating circles | Details will be provided during each class session. |
| Class 5 | The Frenet-Serret formula | Details will be provided during each class session. |
| Class 6 | The fundamental theorem for space curves | Details will be provided during each class session. |
| Class 7 | The implicit function theorem |
Study advice (preparation and review)
As an average, it will be take 100 minutes for homeworks.
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.
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-mail.
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