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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!

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    $\begingroup$ Yes. The relevant keyword is "Ehresmann connection". See en.wikipedia.org/wiki/… $\endgroup$ May 11, 2016 at 2:47
  • $\begingroup$ Many thanks, Yury! I think you should post it as an answer to my question to earn more credit! $\endgroup$ May 11, 2016 at 18:16
  • $\begingroup$ You are welcome, I am glad it helped. I post it as answer. $\endgroup$ May 11, 2016 at 19:50

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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 $e\in E$.

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