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?
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.
On your more general question about differential geometry, i.e. why do people study it? There are many answers, some having little to do with each other.
In my opinion differential geometry is perhaps best approached once you have seen the need for formal and computational structure underlying various fairly mundane geometric problems in more than one area. Recall differential geometry got its start with Gauss and his interest in cartography. In principle a good "curves and surfaces in R^3" course should take you through the kinds of considerations Gauss took himself through, deducing connections and the Gauss curvature. You could see this all as motivated by the desire to seek an underlying formalism for the peculiar things that happen when you start measuring on a curved surface, as cartographers do.
Connections are perhaps one of the easiest topics to motivate from your list, as they tie in so closely with the definition of a manifold. In the definition of a manifold, tangent spaces at different points are all isomorphic, but there are many possible isomorphisms (which could be seen as frustrating, lacking a canonical isomorphism). If you use a chart there is a canonical isomorphism, but it is only valid in that chart. On the other hand if your manifold is in Euclidean space there is another way to do it. Provided your two points $p,q \in M$ are nearby, the tangent spaces $T_p M$ and $T_q M$ are a small (essentially canonical) rotation away from each other. If you use this rotation as your identification, you are very close to the connection perspective, as connections are the infinitesimal rotation you get as $p$ and $q$ approach each other. This is why usage of connections always involve differential equations, or integration.
This might be my own bias, but I think differential geometry is a really natural area to study. When we look out at the world around us, we see lots of objects that seem smooth, but are not flat. As such, it is only natural to try to study geometry using techniques from calculus, which is the starting point of differential geometry. In this vein, many of the concepts in differential geometry come from real-world considerations.
Let me try to explain how this works with connections. Most people take it for granted that we have a consistent idea of which direction "East" is. That is to say, if you tell me "the sun rises in the East," we will both know which direction that is, even though we are standing in different locations. However, it turns out that formalizing this idea of consistent directions at different points requires some work, and it's not obvious how to do it.
After a fair amount of thought, it turns out to be equivalent to being able to compute the directional derivative of vector fields, because we can say that two vectors at different base points $p$ and $q$ are "the same" if the directional derivative is identically zero when we move from $p$ to $q$ along the shortest path* from $p$ to $q$ (i.e., a geodesic). And this is exactly what a connection $\nabla$ is: it is a way to compute the covariant derivatives of vector fields.
Now, there is an added complexity in that there are many possible connections and each induce their own notion of parallel transport (and also their own notion of what a geodesic is). However, the basic idea comes from a natural physical idea and the generalization is helpful because it appears in many other mathematical problems.
*Strictly speaking, the correspondence between the curves whose tangents are parallel with respect to $\nabla$ (i.e., $\nabla$-geodesics) and paths which minimize the total length only holds true for metric connections. For general connections, we use the $\nabla$-geodesics rather than the shortest paths to go from $p$ to $q$.
I don’t understand modern abstract differential geometry, but the elementary theory of curves and surfaces in $\mathbb{R}^3$, as expounded by Gauss, Euler, Darboux and others, is very useful in engineering and manufacturing. A few examples:
The real motivation for the Lie derivative is doing differential calculus with vector fields. If we want fo differentiate the vector field $W$ in the direction of the vector field $V$, we take the flow of $V$ through time, use it to pull back $W$, and take the derivative at $t=0$.
To explain better, let $V$ and $W$ be vector fields on a smooth manifold $M$, with $\theta:M\times\mathbb R\supseteq\mathcal D\to M$ the flow of $V$. Because $\theta_t$ is a diffeomorphism from the set $\{p\in M| (t,p)\in \mathcal D\} := M_t$ to the set $M_{-t}$, with inverse $\theta_{-t}$, we can use it to pull back vector fields, as follows: define $(\theta_t^*W)_p$ by $$\theta{_{-t}*} W_{\theta_t(p)}\in T_pM.$$ We can then take the following limit$$\lim_{t\to 0}\frac{(\theta_t^*W)_p -W_p}t := \mathcal (L_VW)_p = \frac d{dt}\big |_{t= 0}(\theta_t^*W)_p$$ which we call the Lie derivative. It allows us to transport a vector field at one point to a vector at another, and find the rate of change of that transformation.
Why do we do this? Because it is a step towards generalizing analysis to an arbitrary smooth manifold. It allows us to take the derivative of a vector field "along" the flow of another.
However, there is a problem. Lie derivatives do not allow us to take directional derivatives along curves, as it leads to problems. This is why we use the notion of a connection.
(My source for the last comment is the following Math.SE question: Link. My main source generally is John Lee's Introduction to Smooth Manifolds.)
To quote Richard Sharpe, Differential Geometry: Cartan's Generalization of Klein's Erlangen Program:
My innocent question, left over from my undergraduate days, was ``Why is differential geometry the study of a connection on a principal bundle?''
and
It became clear to me that Cartan had a subtle and really wonderful idea, which gives a fully satisfying explanation for the study of an Ehresmann connection on a principal bundle.
To find out this explanation, unfortunately, you will need to read Sharpe's book. But Chern, in the forward, gives his answer:
The answer is of course very simple: because Euclidean geometry studies a connection on a principal bundle, and all geometries are in a sense generalizations of Euclidean geometry.
The bundle in Euclidean geometry is the frame bundle of Euclidean space, i.e. the bundle of choices of point and orthonormal basis of tangent vectors at that point, which is a principal bundle over Euclidean space, with gauge group the group of rigid motions fixing a given point, i.e. the orthogonal group. The rigid motions of Euclidean space act in the obvious way on that bundle, becoming precisely the automorphisms of a unique connection on that bundle, the Levi-Civita connection. That connection gives as its parallel transport the obvious translations.
Similarly, taking the group of transformations of affine space which take lines to lines, the affine transformations, we replace that bundle by the bundle of all choices of point and basis of tangent vectors at that point, a principal bundle over affine space. The affine transformations act in the obvious way on that bundle, becoming precisely the automorphisms of a unique connection on that bundle, again the Levi-Civita connection, with the same parallel transport.
We can repeat this idea with other homogeneous spaces, and it succeeds in defining a connection as long as the stabilizer group of a point is reductive. Starting from various different geometries in the sense of Klein's Erlangen programme, we arrive at a description of those geometries as symmetries of a connection. But then we can bend those geometries, by considering all possible connections with given stabilizer group and given representation of that stabilizer group on the Lie algebra of the symmetry group, giving rise to a natural approach to bend Euclidean space into Riemannian geometry, Minkowski space into Lorentzian geometry, and many other examples. You would really need to read Sharpe's book to get a clearer idea of the possibilities.