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.
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.
pseudo Euclidean space, hypersurfaces, sectional curvature, sphere, hyperbolic space.
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
A standard lecture course.
Homeworks will be assined for each lesson.
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 |
No textbook is set.
Lecture note will be provided.
B. O'Neill, Semi-Riemannian Geometry, Academic Press, 1983; ISBN-13: 978-0-12-526740-3
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 differentable manifolds (MTH.301/MTH.302) 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/2017/geom-a