For a principal bundle $\pi: P \to M$ we have the following Atiyah sequence that can be used to define a connection on it
\begin{equation}
 0 \to V{P} \to T{P} \to \pi^\*{T{M}} \to 0
\end{equation}
A connection is defined as a $G$-equivariant splitting of the above short exact sequence.


My question is, is there a similar definition (i.e. as a splitting of some short exact sequence) of connections for vector bundles?


Thank you very much for your attention!



Atiyah-sequence-like definition of connection on vector bundles?
Yes, there is a general notion of a connection, which makes sense for any fiber bundle, namely, Ehresmann connection, see <https://en.wikipedia.org/wiki/Ehresmann_connection>


In the case of vector bundle $E\to M$, you have to require additionally that for $x\in M$, $e\in E\_x$ the splitting of
$$
0 \to Vert\_{x,e}\,E\to T\_{x,e}E\to \pi^\* T\_xM\to 0 
$$
 is linear in $e\in E$.


