The aim of this lecture is to familiarize the students with the basic language of and some fundamental theorems in Riemannian geometry. This course is a continuation of [MTH.B407 : Advanced topics in Geometry C1].
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 in class. |
Class 2 | The definition of geodesics and the equations of geodesics | Details will be provided in class. |
Class 3 | Exponential map | Details will be provided in class. |
Class 4 | Normal coordinate neighborhood, Gauss' lemma | Details will be provided in class. |
Class 5 | Geodesics are locally minimizing curves | Details will be provided in class. |
Class 6 | Einstein equation, Hopf-Rinow theorem | Details will be provided in class. |
Class 7 | Proof of Hopf-Rinow theorem | Details will be provided in class. |
Class 8 | Jacobi field | 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 [Advanced topics in Geometry C1].