Definition and meanings of the "curvature" of Riemannian manifolds, especially those obtained as submanifolds of (pseudo) Euclidean space, are introduced.
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.
Riemannian manifolds, curvature, integrability conditions
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
A standard lecture course.
Homeworks will be assigned for each lesson.
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 |
No textbook is set.
Lecture note will be provided.
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.
Graded by homeworks
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.
kotaro[at]math.titech.ac.jp
N/A. Contact by E-mails, or at the classroom.
For details, visit the web-site of this class http://www.math.titech.ac.jp/~kotaro/class/2019/geom-b