I'm trying to self-study general relativity, but I encounter a problem. I can easily understand the ideas and the results, but there are some things related to 4-d geometry that I can't find, like what are Christoffel symbols and how to solve the equation of a geodesic line. I searched for those on the internet, but I found only complicated materials, not the introductory stuff, like definitions and explanations with applications. So can anyone suggest some FREE introductory materials on general relativity and 4-d geometry? I'm looking for something with complete proofs and applications.