2019 Advanced courses in Geometry B

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Academic unit or major
Mathematics
Instructor(s)
Yamada Kotaro 
Course component(s)
Lecture
Day/Period(Room No.)
Tue3-4(H104)  
Group
-
Course number
ZUA.B332
Credits
1
Academic year
2019
Offered quarter
2Q
Syllabus updated
2019/3/18
Lecture notes updated
-
Language used
English
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Course description and aims

Definition and meanings of the "curvature" of Riemannian manifolds, especially those obtained as submanifolds of (pseudo) Euclidean space, are introduced.

Student learning outcomes

Students are expected to know
- the integrability condition of linear system of partial differential equations,
- the sectional curvature of a Riemannian manifolds,
- the curvature as an integrability condition,
- and the local uniqueness of Riemannian manifolds of constant sectional curvature.

Keywords

Riemannian manifolds, curvature, integrability conditions

Competencies that will be developed

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

Class flow

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

Course schedule/Required learning

  Course schedule Required learning
Class 1 Linear ordinary differential equations Details will be provided during each class session
Class 2 The fundamental theorem for linear ordinary differential equations Details will be provided during each class session
Class 3 The second fundamental forms of hypersurfaces and the sectional curvature Details will be provided during each class session
Class 4 Spheres and hyperbolic spaces Details will be provided during each class session
Class 5 The curvature tensor and the sectional curvature Details will be provided during each class session
Class 6 Local uniqueness of Riemannian manifolds of constant sectoinal curvature Details will be provided during each class session
Class 7 Models of hyperbolic spaces Details will be provided during each class session

Textbook(s)

No textbook is set.
Lecture note will be provided.

Reference books, course materials, etc.

S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, Third Edition, Springer-Verlag, 2013.
M. P. do Carmo (transl. F. Flaherty), Riemannian Geometry, Birkhauser, 1994.

Assessment criteria and methods

Graded by homeworks

Related courses

  • MTH.B211 : Introduction to Geometry I
  • MTH.B212 : Introduction to Geometry II
  • ZUA.B331 : Advanced courses in Geometry A

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

Knowledge on differential geometry of curves and surfaces (as in MTH.B211 "Introduction to Geometry I" and MTH.B212 "Introduction to
Geometry II", or Sections 1 to 10 of the text book "Differential Geometry of Curves and Surfaces" by M. Umehara and K.
Yamada), and knowledge of fundamental notions of space forms (e.g. ZUA.B331) are required.

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

For details, visit the web-site of this class http://www.math.titech.ac.jp/~kotaro/class/2019/geom-b

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