The aim of this lecture is to familiarize the students with the basic language of
and some fundamental theorems in Riemannian geometry.
This course will be succeeded by [ZUA.B334 : Advanced courses in Geometry D].
Students are expexted to
・understand the definitions of Riemannian metric, sectional curvature, Ricci curvature, and scalar curvature.
・be familiar with the method of expressing them by using local coordinates of the underlying manifold.
Riemannian metric, connection, covariant derivative, curvature tensor, Levi-Civita connection, sectional curvature, Ricci curvature, scalar curvature, Laplacian
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course
Course schedule | Required learning | |
---|---|---|
Class 1 | the definition and examples of Riemannian metrics | Details will be provided during each class session |
Class 2 | the length of a curve, the distance and the volume form on a Riemannian manifold | Details will be provided during each class session |
Class 3 | connection and covariant derivative, torsion tensor and curvature tensor | Details will be provided during each class session |
Class 4 | Bianchi's identity, extending a covariant derivative to diffenrential of (general) tensor fields | Details will be provided during each class session |
Class 5 | Levi-Civita connection and sectional curvature | Details will be provided during each class session |
Class 6 | relationship between Gaussian curvature and sectional curvature, Ricci curvature and scalar curvature | Details will be provided during each class session |
Class 7 | local expression of covariant derivative | Details will be provided during each class session |
Class 8 | divergence and Green's theoem, Laplacian | Details will be provided during each class session |
None required
M. do Carmo, Riemannian Geometry, Birkhauser
Exams and reports. Details will be provided during class sessions.
Students are expected to have passed [Geometry I], [Geometry II] and [Geometry III].