Christoffel Symbols, Curvature, and Riemann Tensor - Lecture 8

Delve into an advanced graduate-level lecture on differential geometry and general relativity, part of the Warsaw4PhD and GeoPlanet PhD schools' Autumn/Winter 2023-2024 semester. Explore key concepts including Christoffel symbols, curvature, and the Riemann tensor with lecturer Mikołaj Korzyński from the Center for Theoretical Physics, Polish Academy of Sciences. Begin with problem sheet discussions, then progress through topics such as geodesics in Newtonian approximation, properties of the Riemann tensor, Bianchi identities, and tensor contractions. Conclude with proofs on commuting covariant derivatives and the tensor nature of the Riemann, as well as an examination of the Riemann tensor for a round 2-sphere.

00:00- Problem sheets 3 and 2
30:25- Christoffels, geodesics in the Newtonian approximation
50:35- Break
50:50- Curvature, Riemann tensor
01:07:22- Riemann tensor in locally flat coordinates
01:10:35- Algebraic properties of the Riemann
01:14:16- Commuting covariant derivatives gives the Riemann
01:17:56- Bianchi identities
01:21:21- Contractions of the Riemann
01:24:38- Commuting covariant derivatives gives the Riemann - proof
01:33:02- Riemann is a tensor - proof
01:39:02- Riemann tensor of a round 2-sphere