I have been reading through Wald's book on General Relativity, often using other sources to help gain a deeper understanding of the mathematics. Through this supplemental learning, I encountered something I'm finding very interesting.
In his book, Wald discusses the Riemann curvature tensor, showing that it can be thought of as a multilinear map from dual vectors to type $(0,3)$ tensors. Hence, by the isomorphism $\operatorname{Hom}(V,W) \cong V^*\otimes W$, there is a tensor of type $(1,3)$ which represents the mapping. In trying to read more about this I stumbled upon Geroch's notes on differential geometry. Here Geroch uses the difference of a Lie derivative/bracket(?) in the base manifold with a Lie derivative/bracket(?) in the total space to define the curvature tensor. And in his explanation of its geometric interpretation he says that the difference is a vertical vector in the total space.
My questions are:
(1) Is there any reference(s) that treats GR in terms of fiber bundles, using language along the same lines as Geroch and other differential geometry texts (Ehresmann connections, etc.)? I'd like to read/learn about GR in this language.
(2) Does this vector characterize the curvature (or holonomy(?)) of the space? If so, how?
Thanks in advance for any thoughts and comments. And please let me know if anything I wrote is incoherent and/or wrong (which is probably the case :)).
I’m not quite entirely sure what you mean by (2), perhaps I’m not reading your question carefully enough. For (1) however, you should look at “Formulations of General Relativity” by Kiril Krasnov. He gives a brief overview of principal bundles and principal connections, then discusses various equivalent formulations of the einstein Hilbert actions in terms of a soldering form (the canonical one form on the frame bundle of the tangent bundle), and a connection one form. He also discusses the equivalent of the Riemann curvature tensor, as well as other curvature’s (ricci snd ricci scalr) in terms of this formalism. It may be what you are looking for.