Levi-Civita Connection, Christoffel Symbols, and Geodesics - Lecture 7

Explore the fundamental concepts of differential geometry in this graduate-level lecture from the Warsaw4PhD and GeoPlanet PhD schools. Delve into the intricacies of covariant derivatives, connection coefficients, and the Levi-Civita connection. Examine Christoffel symbols and their significance in curved spaces. Investigate locally flat coordinates and the properties of covariant derivatives. Understand parallel transport and its applications. Study geodesics and their variational principles. Apply these concepts to practical examples, including Christoffel symbols on a 2-sphere and in the Newtonian approximation. Gain a comprehensive understanding of these advanced mathematical tools essential for theoretical physics and cosmology.

00:00- Covariant derivative and connection
04:30- Transformation law for connection coefficients
16:40- Levi-Civita connection, Christoffel symbols
28:23- Locally flat coordinates
33:29- Properties of the covariant derivative
37:29- Parallel transport
42:29- Geodesics
52:28- Break
52:40- Variational principle for geodesics
01:09:09- Christoffels on a 2-sphere
01:28:12- Christoffels in the Newtonian approximation