I am currently reading through differential geometry as a mathematics graduate, but I have some questions regarding the subject in general.
The problem I'm running into is that most textbooks and resources I have read just throw definitions and theorems at you without really explaining why we bother to work with the objects in question at all. Are we studying the manifolds themselves? Are we studying structures on the manifold and the manifold is just the setting? What general questions do we hope to answer?
Can somebody give me a brief explainer on the purpose of the following objects of studyconnections?
- Differential forms (I understand that you can only integrate forms on manifolds, I just don't understand why we want to)
- Connections
- Lie derivatives
As an additional question: I understand the general idea that "linear objects are easier to study than non linear ones" as an explainer for understandingcould also use explainers on differential forms. Lie derivatives, and the purpose tangent bundle, but what does this look like in practice generally.
I understand that you can have these structures, what realI just don't understand why we want to. What information can we learn from the tangent bundlethese structures that we couldn't learn about the manifolds otherwise?
