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!