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 [MTH.B408 : Advanced topics 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 in class. |
Class 2 | The length of a curve, the distance and the volume form on a Riemannian manifold | Details will be provided in class. |
Class 3 | Connection and covariant derivative, torsion tensor and curvature tensor | Details will be provided in class. |
Class 4 | Bianchi's identity, extending a covariant derivative to diffenrential of (general) tensor fields | Details will be provided in class. |
Class 5 | Levi-Civita connection and sectional curvature | Details will be provided in class. |
Class 6 | Relationship between Gaussian curvature and sectional curvature, Ricci curvature and scalar curvature | Details will be provided in class . |
Class 7 | Local expression of covariant derivative | Details will be provided in class. |
Class 8 | Divergence and Green's theoem, Laplacian | Details will be provided in class. |
None required
M.do Carmo, Riemannian Geometry, Birkhauser
Exams and reports. Details will be provided in class.
Students are expected to have passed [Geometry I], [Geometry II] and [Geometry III].