The "The Foundations of Relativity" thread was so helpful for me that I was able to learn a lot very quickly and I thank everyone who made the effort and time to help me out - but in an effort to prevent it from being too long I decided to start the next part in the series in a brand new thread. In this thread, I would like to focus on what I consider the next step in learning the mathematics behind relativity, and that is Differential Geometry. Now I know there are a lot of good threads on differential geometry on this website, but somehow, the questions asked are not exactly what I needed to know, or they became too involved too quickly (lethe's thread on this topic comes to mind). I will continue to visit the first part with questions on tensors as I come to them. So lets get this started. The first question obviously is: What is differential geometry? Why is it useful in the study of relativity? And what is a good approach to the topic?