We can define a connection 1-form which picks out the vertical part of any vector in the tangent bundle of some principal bundle. This allows us to define parallel transport so that we have a way to connect nearby fibers. As a map, it maps vectors in the tangent bundle to the Lie algebra associated with the Lie group of the principal bundle
In contrast, an affine connection is defined for any manifold and maps the cartesian product of two vector fields to another vector field.
I am confused on the relationship between these two kinds of connections. On one hand they both capture a similar idea of connecting nearby fibers to each other, but as maps they seem completely different with one have values in the space of vector fields and the other having values in a Lie algebra. Furthermore affine connections seem to have nothing to do with horizontal or vertical tangent spaces, which seem to be at the heart of connection 1-forms/Ehresmann connections.
Is there any relationship between these two objects? I took a look at some older posts, such as the one below, but I am still confused.
Best Answer
I will try to indicate how affine connections are related to Ehresmann connections.
Given a vector bundle $\pi:E\to M$ over a manifold, an affine connection is an operator $$\nabla: \mathfrak{X}(M)\times \Gamma(M,E)\to \Gamma(M,E)$$ written as $\nabla(X,s)=\nabla_Xs$ which is $\Bbb{R}$-linear in both factors and satisfies the Leibniz property, i.e. for all $X\in \mathfrak{X}(M)$, $s\in \Gamma(M,E), f\in C^\infty(M)$, $$ \nabla_X(f\cdot s) = X(f)\cdot s+f\nabla_X(s). $$ On the other hand, an Ehresmann connection is defined as a right-invariant horizontal distribution $\mathcal{H}$ on a principal $G$-bundle, $p:P\to M$. In a certain sense, an affine connection is a special case of an Ehresmann connection. Associated to $E$ as above is its bundle of frames $\mathrm{Fr}(E)$, which is a fibre bundle whose fibre over $x\in M$ is $\mathrm{Fr}(E)_x$, the set of bases of $E_x$. Since $E_x$ is a vector space of dimension $r$ for all $x$, one proves that this is a principal $\mathrm{GL}_r(\Bbb{R})$-bundle. Indeed, any pair of bases of a vector space $V$ of dimension $r$ are related by a unique element of $\mathrm{GL}(V)$.
On the other hand, $\nabla$ is locally encoded by a connection $1$-form, which is a differential $1$-form which is valued in $\mathfrak{g} = \mathfrak{gl}(r,\Bbb{R})$. Indeed, choose an open set $U$ such that $E$ admits a local frame $e = \{e_1,\ldots, e_r\}$. Then, there is a unique matrix $(\omega_j^i) = \omega$ of differential $1$-forms such that for all $X$ and $j$ $$ \nabla_X e_j = \sum_i \omega_j^i(X)e_i. $$ Under a change of frame, i.e. a map $g:U\to \mathrm{GL}(r,\Bbb{R})$, we get a new frame $e' = \{ge_1,\ldots, ge_r\}$. Write the connection $1$-form of $e$ as $\omega_e$ and that of $e'$ as $\omega_{e'}$. One has $$ \omega_{e'} = g^{-1}\omega_e g + g^{-1}dg $$ where $dg$ is $d(g)$, where $g$ is regarded as a matrix of smooth functions.
An Ehresmann connection $\mathcal{H}$ as above defines a $\mathfrak{g}$-valued $1$-form on $P$ by the exact sequence: for each $x\in P$, let $\mathcal{V}_x := \ker p_{*,x}$. Then $$ 0\to\mathcal{H}_x\to T_xP\to \mathcal{V}_x \to 0 $$ is exact. However, as $P$ is a principal $G$-bundle, $\mathcal{V}_x\cong \mathfrak{g}$ canonically, and we define a form $\omega:T_xP\to \mathcal{V}_x \cong \mathfrak{g}$ using the arrow in the exact sequence. Such an $\omega$ is $G$-invariant, satisfies $\omega(\underline{X}_x) = X$, where $\underline{X}_x$ is the vector field associated to $X\in \mathfrak{g}$ from differentiating the $G$-action, and smooth. Conversely, one can show that a $\mathfrak{g}$-valued $1$-form with those properties determines an Ehresmann connection, i.e. a right invariant horizontal distribution.
Anyway, an affine connection $\nabla$ defines parallel transportation of vectors in $E$. More precisely, given $\gamma:I=(-\epsilon,\epsilon)\to M$ such that $\gamma(0) = y$, then for a chosen frame around $y$ extending the basis $s_1(y),\ldots, s_r(y)$ of $E_y$, parallel transport gives a lift to a curve $\widetilde{\gamma}:I\to \mathrm{Fr}(E)$ with $\widetilde{\gamma}(0) = (s_1(y),\ldots, s_r(y)) =: s(y)$. There is an associated tangent vector $\widetilde{\gamma}'(0) \in T_{s(y)}\mathrm{Fr}(E)$. The set of all trajectories of such lifts along all $\gamma$ forms a vector subspace of $T_{s(y)}\mathrm{Fr}(E)$. One verifies that this is a right-invariant distribution on $T\mathrm{Fr}(E)$, giving us an Ehresmann connection.
So, an affine connection $\nabla$ on $E$ gives us an Ehresmann connection $\mathcal{H}$ on the associated frame bundle, $\mathrm{Fr}(E)$. This has the following beautiful property (though it is really a tautology once you unwind the definitions).
Note that choosing a frame $e$ on an open set $U$ of $M$ is equivalent to giving a section $e:U\to \mathrm{Fr}(E)|_U$ of the projection $p$.
Here, $\omega_e$ is the matrix of $1$-forms from further up in the post. The upshot of all this is that an affine connection is represented locally by a matrix in a local frame, while an Ehresmann connection is like "working in all frames at once." (so says a friend of mine, anyway!)
I learned most of what I know about this from Tu's book Differential Geometry: Connections, Curvature, and Characteristic Classes, which I highly recommend.