I am teaching myself differential geometry to be able to grasp General relativity. I went through multiple textbook and have a fair amount on my bookshelf ( around 10-15 ).
Connections : I get disappointed most of the time when the topic of connections come up. All of my book (except for one see below ) introduce the principe of connection by listing the properties the one-form fulfill. I much prefer the angle of defining it by the distribution of Horizontal subspaces that makes more intuitive sense for me. Unfortunately one one of my book treat this point of view namely : Fibre bundle The later treatment of covariant derivatives doesn't suit me in this book as it is just a slapping of properties which I do not like ( basically because of the lack of history and intuition ).
Covariant derivatives : The Covariant derivative and the exterior covariant derivative is pretty much the same story. I am quite not at ease when learning definition without much of history or what problem are we trying to solve by introducing such structures.
I would like to know a book or a reference that can fill the gaps in my knowledge/intuition.
- Def of a connection in terms of distribution of horizontal subspaces
- Derive the connection one-form from it
- Derive the Yang mills field by pulling back the one form on the base manifold
- Derive (somehow) the exterior covariant derivative
- Derive the concept of covariant derivative.
To sum up, I would like to get a rigorous treatment of these principles that will build my intuition better than just learning a bunch of properties.
Thanks for any suggestions !