2019 Introduction to Geometry I

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Academic unit or major
Undergraduate major in Mathematics
Yamada Kotaro 
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Course description and aims

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.

Student learning outcomes

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.


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

Competencies that will be developed

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

Class flow

Sections 1 to 5 of the textbook are explained according to the flowchart at the beginning of the textbook.
Homework will be assigned in every class session.
The precise contents, speed of lectures will be controlled according to the outcomes of the homework.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Review from Linear Algebra --- Isometries of the Euclidean spaces. Recall the notion of "orthogonal matrices" in Linear Algebra. Details will be provided during the class session.
Class 2 Review from Calculus --- Taylor's theorem, the implicit function theorem, the fundamental theorems for ordinary differential equations. Recall the "Taylor's theorem" in Calculus. Details will be provided during the class session
Class 3 Parametrized plane curves, change of parameters, the arclength. Details will be provided during each class session.
Class 4 The arclength parameter, curvature, Frenet's formula. Details will be provided during each class session.
Class 5 The fundamental theorem for plane curves, geometric meanings of the curvature. Details will be provided during each class session.
Class 6 The total curvature and the rotation index for closed plane curves. Details will be provided during each class session.
Class 7 The curvature and the torsion for space curves, the fundamental theorem for space curves. Details will be provided during each class session.
Class 8 Evaluation of Progress Details will be provided during each class session.


Masaaki Umehara and Kotaro Yamada, Differential geometry of curves and surfaces, World Scienetific, 2017.

Reference books, course materials, etc.

Sebastian Montiel y Antonio Ros, Curvas y superficie, Proyecto Sur, 1998.
Manfredo P. do Carmo, Differenial Geoetry of Curves and Surfaces, Prentice-Hall Inc., 1976.

Assessment criteria and methods

Score of the final exam.
The score of homeworks might be added in the case that the score of the final exam is insifficient, by a formula which will be shown explicitly in the class session.

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 (i.e., required knowledge, skills, courses, etc.)

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).


Office hours

Contact by E-mails, or at the classroom.


Check the web page http://www.math.titech.ac.jp/~kotaro/class/2019/geom-1/index-jp.html and/or OCW, for details.
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

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