Connections - July 27, 2025
I love geometry, but I don't know much beyond some basic theory of smooth manifolds and the fundamental differential properties. I had always heard of connections or seen them mentioned but never really learned what they are intuitively. So lets do just that.
So consider a smooth manifold along with some vector bundle, for simplicity I will just consider the tangent bundle for now. In a nutshell, a connection on this manifold gives us a way to differntiate a vector field along another vector field. In $\mathbb{R}^2$, the connection is related to the partial differentials of $x$ and $y$. "Flat" spaces in general have simple interpretations of connections. When things get bumpy, we need a better tool. Lets just do the $\mathbb{R}^2$ example first to give a little insight. The standard (Riemannian) metric on $\mathbb{R}^2$ is $ds^2 = dx^2 + dy^2$. I won't get into it now but the connection we usually use for spaces like this is the Levi-Cevita connection. Formally, a connection is an operator on a smooth manifold $M$, that takes in two vector fields and spits out a new vector field that is the covariant derivative. Let $\mathfrak{X}(M)$ be the set of all vector fields on $M$. Then a connection is the operator $$ \nabla : \mathfrak{X}(M) \times \mathfrak{X}(M) \to \mathfrak{X}(M) $$ So in $\mathbb{R}^2$, we take in the usual $X$ and $Y$ vector fields and we get a new vector field $\nabla_X Y$. Formally $$ X \times Y \mapsto \nabla_X Y $$ So suppose we have coordinates $(x_1, \dots , x_n)$ with $X = X^i \partial_i$ and $Y = Y^j \partial_j$. Then we can define the connection as $$ \nabla_X Y = \sum_j ( X \left(Y^j\right) + \sum_i X^i Y^k \Gamma_{ik}^j)\partial_j $$ Here, $\Gamma$ is the Christoffel symbol. Without going into too much detail about that now, just know that it gives some insight into how basis vectors changes as we move along a path on the manifold. We can even boil down this expression a little more and write $$ \nabla_{\partial_i}\partial_j = \sum_k \Gamma_{ij}^k \partial_k $$ In Cartesian coordinates, the Christoffel symbol is just $\Gamma = 0$, so the connection just becomes the usual directional derivative $$ \nabla_X Y = X(Y^j)\partial_j $$ which can be interpreted as "the directional derivative of the vector field $Y$ along $X$ ". So just like we saw in vector calculus, the subscript gives us a "direction" or a thing to differentiate against. As an interesting but simple example, try computing the connection on $\mathbb{R}^2$ in polar coordinates with the Christoffel symbols $$ \begin{align} \Gamma_{\theta r}^\theta &= \Gamma_{r \theta}^\theta = \frac{1}{r}\\ \Gamma_{\theta \theta}^r &= -r \end{align} $$