I am currently reading through differential geometry as a mathematics graduate.


Can somebody give me a brief explainer on the purpose of connections?


I could also use explainers on differential forms. Lie derivatives, and the tangent bundle generally.


I understand that you can have these structures, I just don't understand why we want to. What information can we learn from these structures that we couldn't learn about the manifolds otherwise?



The purpose of connections in differential geometry
If you are interested in local-to-global results, i.e., collecting local info about the manifold and then patch it together to get a global info then you need tools for the patching part of the process. These often come under the guise of some form of integration.


Here is a simple example. If a compact surface admits a metric with negative curvature (local info) then it has nontrivial fundamental group (global info). This is a special case of Gauss-Bonnet that tells you more.


On the other hand, a theorem of Myers shows that if the Ricci curvature of a Riemann manifold is bounded below by a positive constant (local info) then the manifold is compact and its fundamental group is finite (global info).


This is a more sophisticated result and its proof uses the concept of connection.


The revolution in low dimensional topology brought by gauge theory relies in an essential fashion on the concept of connection. The instantons are simply connections with minimal "energy".


As far as Lie derivatives are concerned, I recommend Arnold's "*Mathematical Methods of Classical Mechanics*" to see a geometric interepration and appreciate some of their uses.


