An Ehresmann connection on a vector bundle $\pi : E \to X$ is a splitting of the sequence,

$$ 0 \to V \to TE \to \pi^* TX \to 0 $$ which respects the linear structure on $E$ (meaning the section is invariant under the induced automorphism of $T E$ induced by scaling).

In algebraic geometry, I am familiar with the equivalence between connections on a locally free sheaf $\mathcal{E}$ and splitting of the sequence

$$ 0 \to \Omega_X^1 \otimes \mathcal{E} \to J^1(\mathcal{E}) \to \mathcal{E} \to 0 $$ I have heard it said that this is a version of the Ehresmann connection formalism but I am not able to make this precise. If I dualize the top sequence and use the fact that $V \cong \pi^* E$ then I recover,

$$ 0 \to \pi^* T^* X \to T^* E \to \pi^* E^* \to 0 $$ which looks similar to the jet bundle sequence. However, I am not sure how to directly compare these two sequences.

Furthermore, the notion of an Ehresmann connection makes perfect sense in the algebraic category. However, (at least without directly relating it to the jet bundle sequence) I do not see how to show that the data of a splitting recovers an (algebraic) connection.

The usual construction of a connection from an Ehresmann connection goes through algebraically. Call the splitting $v : T E \to \pi^* E$. Then given a section $s : X \to E$ we get $\mathrm{d}{s} : T X \to s^* T E$ and then $s^* v \circ \mathrm{d}{s}$ is a linear map $T X \to E$ defining $X \mapsto \nabla_X s$ thus defining the connection.

However, to reverse this process, it seems that I need to be able to choose, locally, flat sections for a connection $\nabla$ in order to define the kernel of $v$ which is the horizontal subspace. Does this mean the Ehresmann connection is really a transcendental object?

An Ehresmann connection on a vector bundle $\pi : E \to X$ is a splitting of the sequence,

$$ 0 \to V \to TE \to \pi^* TX \to 0 $$ which respects the linear structure on $E$ (meaning the section is invariant under the induced automorphism of $T E$ induced by scaling).

In algebraic geometry, I am familiar with the equivalence between connections on a locally free sheaf $\mathcal{E}$ and splitting of the sequence

$$ 0 \to \Omega_X^1 \otimes \mathcal{E} \to J^1(\mathcal{E}) \to \mathcal{E} \to 0. $$ I have heard it said that this is a version of the Ehresmann connection formalism but I am not able to make this precise. If I dualize the top sequence and use the fact that $V \cong \pi^* E$ then I recover,

$$ 0 \to \pi^* T^* X \to T^* E \to \pi^* E^* \to 0 $$ which looks similar to the jet bundle sequence. However, I am not sure how to directly compare these two sequences.

Furthermore, the notion of an Ehresmann connection makes perfect sense in the algebraic category. However, (at least without directly relating it to the jet bundle sequence) I do not see how to show that the datum of a splitting recovers an (algebraic) connection.

The usual construction of a connection from an Ehresmann connection goes through algebraically. Call the splitting $v : T E \to \pi^* E$. Then given a section $s : X \to E$ we get $\mathrm{d}{s} : T X \to s^* T E$ and then $s^* v \circ \mathrm{d}{s}$ is a linear map $T X \to E$ defining $X \mapsto \nabla_X s$ thus defining the connection.

However, to reverse this process, it seems that I need to be able to choose, locally, flat sections for a connection $\nabla$ in order to define the kernel of $v$ which is the horizontal subspace. Does this mean the Ehresmann connection is really a transcendental object?