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But because the Riemann tensor is a genuine tensor, if it vanishes in one. This introduction to the foundatations of General Relativity by Chris Blake is also very good. the flat space Christoffel symbols in polar coordinates (1.10).

There are also some decent lectures introducting GR on Youtube if you prefer this (a 5min overview: General Relativity Explained in 7 Levels of Difficulty and this great full course by Alex Flournoy). For a slightly longer and more detailed introduction to General Relativity I recommend A No-Nonsense Introduction to General Relativity by Sean Carroll. There notes are just to give you the very basics of the General Relativity that is needed to do cosmology. 1.9 Newtonian limit of General Relativity.1.8 Minimal coupling and the Geodesic equation In a flat space, and only in a rectangular coordinate system is each coordinate independent of the other three.