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2022　Advanced topics in Geometry F

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Yamada Kotaro 
Class Format
Lecture    (Livestream)
Media-enhanced courses
Day/Period(Room No.)
Tue3-4()  
Group
-
Course number
MTH.B502
Credits
1
Academic year
2022
Offered quarter
2Q
Syllabus updated
2022/12/19
Lecture notes updated
-
Language used
English
Access Index

Course description and aims

A local theory of Riemannian manifolds and fundamental theorem of surface theory for surface in Riemannian 3-manifolds of constant scetional curvature are introduced. As an application, a relationship of surfaces of constant mean curvature surface in different spaces,
e.g. minimal surfaces in Euclidean 3-space and constant curvature one surfaces in hyperbolic space, is discussed.

Student learning outcomes

Students will learn: a local theory of Riemannian manifolds, i.e. notions of Riemannian metrics, sectional curvatures; spaces of constant curvature (space forms); an extension of the fundamental theorem of surface theory for surfaces in 3-dimensional space forms.

Keywords

Riemannian metric, curvature, space form, fundamental theorem of surface theory

Competencies that will be developed

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

Class flow

A standard lecture course. Homeworks will be assined for each lesson.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Riemannian metrics and connections Details will be provided during each class session.
Class 2 Curvatures Details will be provided during each class session.
Class 3 Euclidean spaces and Spheres Details will be provided during each class session.
Class 4 Lorentz-Minkowski space Details will be provided during each class session.
Class 5 Hyperbolic spaces Details will be provided during each class session.
Class 6 Fundamental theorem of surface theory revisited Details will be provided during each class session.
Class 7 Constant mean curvature surfaces in 3-dimensional space forms Details will be provided during each class session.

Out-of-Class Study Time (Preparation and Review)

Official Message: 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)

No textbook is set. Lecture note will be provided.

Reference books, course materials, etc.

Masaaki Umehara and Kotaro Yamada, Differential Geometry of Curves and Surfaces, Transl. by Wayne Rossman, World Scientific Publ.,
2017, ISBN 978-9814740234 (hardcover); 978-9814740241 (softcover)

Assessment criteria and methods

Graded by homeworks. Details will be announced through T2SCHOLA

Related courses

  • MTH.B211 ： Introduction to Geometry I
  • MTH.B212 ： Introduction to Geometry II
  • MTH.B501 ： Advanced topics in Geometry E

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

At least, knowledge of undergraduate calculus and linear algebra are required.
Attending the class "Advanced Topics in Geometry E" (MTH.B501) is strongly recommended.

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

Other

Visit http://www.math.titech.ac.jp/~kotaro/class/2022/geom-f for details.

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