The aim of this lecture is to familiarize the students with the basic l
anguage of and some fundamental theorems in Riemannian geometry.
This course is a continuation of [ZUA.B333 : Advanced courses in Geometry C].
Students are expected to
・understand the definition of geodesic and the theorem on completeness.
・understand that Einstein equation is a second order non-linear partial differential equation for Riemannian metrics.
Parallel translation, geodesic, exponential map, normal cocordinate neighborhood, Einstein equation, Hopf-Rinow Theorem
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course
Course schedule | Required learning | |
---|---|---|
Class 1 | Parallel translation | Details will be provided during each class session |
Class 2 | The definitions of geodesics and the equations of geodesics | Details will be provided during each class session |
Class 3 | Exponential map | Details will be provided during each class session |
Class 4 | Normal coordinate neighborhood, Gauss' lemma | Details will be provided during each class session |
Class 5 | Geodesics are locally minimizing curves | Details will be provided during each class session |
Class 6 | Einstein equation and Hopf-Rinow theorem | Details will be provided during each class session |
Class 7 | Proof of Hopf-Rinow theorem | Details will be provided during each class session |
Class 8 | Jacobi field | Details will be provided during each class session |
Non required
M.do Carmo, Riemannian Geometry, Birkhauser
S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, Springer
Exams and reports. Details will be provided during class sessions.
Students are expected to have passed [Geometry I], [Geometry II] , [Geometry III] and [Advanced courses in Geometry C].