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- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Yamada Kotaro
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Tue3-4(H115)
- Group
- -
- Course number
- MTH.B405
- Credits
- 1
- Academic year
- 2017
- Offered quarter
- 1Q
- Syllabus updated
- 2017/3/17
- Lecture notes updated
- 2017/5/30
- Language used
- Japanese
- Access Index
-
Course description and aims
As a generalization of theory of surfaces in the Euclidean 3-space, differential geometry of hypersurfaces in (pseudo) Euclidean spaces,
with examples including spheres and hyperbolic spaces as complete Riemannian manifolds with constant sectional curvature, is
introduced.
Student learning outcomes
Students are expected to learn
- definition of geometric invariatns of hypersurfaces in (pseudo) Euclidean spaces,
- the fact that the sectional curvature for hypersurfaces is intrinsic invariant,
- and the fact that the spheres and the hyperbolic spaces are complete Riemannian manifolds of constant sectional curvatuure.
Keywords
pseudo Euclidean space, hypersurfaces, sectional curvature, sphere, hyperbolic space.
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 | Indefinite inner products | Details will be provided during each class session |
| Class 2 | Pseudo Euclidean space | Details will be provided during each class session |
| Class 3 | Hypersurfaces, induced metrics | Details will be provided during each class session |
| Class 4 | Non degenerate hypersurfaces | Details will be provided during each class session |
| Class 5 | The second fundamental form and the sectional curvature | Details will be provided during each class session |
| Class 6 | The spheres and the hyperbolic spaces | Details will be provided during each class session |
| Class 7 | Geodesics and completeness | Details will be provided during each class session |
| Class 8 | (de Sitter space and anti de Sitter space) | Details will be provided during each class session |
Textbook(s)
No textbook is set.
Lecture note will be provided.
Reference books, course materials, etc.
B. O'Neill, Semi-Riemannian Geometry, Academic Press, 1983; ISBN-13: 978-0-12-526740-3
Assessment criteria and methods
Graded by homeworks
Related courses
- MTH.B211 ： Introduction to Geometry I
- MTH.B212 ： Introduction to Geometry II
- MTH.B211 ： Introduction to Geometry I
- MTH.B212 ： Introduction to Geometry II
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 differentable manifolds (MTH.301/MTH.302) are required.
Contact information (e-mail and phone) Notice : Please replace from "[at]" to "@"(half-width character).
kotaro[at]math.titechac.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/2017/geom-a
