### 2019　Introduction to Geometry II

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Academic unit or major
Undergraduate major in Mathematics
Instructor(s)
Course component(s)
Lecture
Day/Period(Room No.)
Thr3-4(H136)
Group
-
Course number
MTH.B212
Credits
1
2019
Offered quarter
4Q
Syllabus updated
2019/3/18
Lecture notes updated
2020/1/31
Language used
Japanese
Access Index

### Course description and aims

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.
This course is a succession of " Introduction to Geometry I" in 3Q.

### Student learning outcomes

The students will learn the basic matters of differential geometry of surface 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 the relationship between the shape of surfaces and curvatures.
(3) To know examples of global properties and local properties of surfaces.
(4) To confirm the theories by calculations on concrete examples.

### Keywords

Differential Geometry, Surfaces, Gaussian cruvature, Mean curvature, the Gauss-Bonnet theorem.

### Competencies that will be developed

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

### Class flow

Sections 6 to 10 of the textbook are treated 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 Parametrization for regular surfaces Details will be provided during each class session.
Class 2 The first fundamental form, the length, the angle and the area Details will be provided during each class session.
Class 3 The second fundamental form, the principal curvatures, the Gaussian and the mean curvatures. Details will be provided during each class session.
Class 4 The normal curvature, the principal and the asymptotic directions. Details will be provided during each class session.
Class 5 The Gauss and the Weingarten formula, and Theorema Egregium. Details will be provided during each class session.
Class 6 Geodesics and the Gauss-Bonnet theorem Details will be provided during each class session.
Class 7 A brief introduction to the fundamental theorem for surfaces. Details will be provided during each class session.
Class 8 Evaluation of Progress Details will be provided during each class session.

### Textbook(s)

Masaaki Umehara and Kotaro Yamada, DIfferential Geometry of curves and surfaces, World Scientific, 2017

### Reference books, course materials, etc.

Sebastian Montiel y Antonio Ros, Curvas y superficie, Proyecto Sur, 1998.Details will be provided during each class session.

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 teh final exam is insufficient, by aformula which will be shown explicitly in the class session.

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

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).

kotaro[at]math.titech.ac.jp

### Office hours

N/A.
Contact by E-mails, or at the classroom.

### Other

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