Let $G$ be a Lie group and $\pi:E_G\rightarrow M $ be a principal $G$-bundle.
I have seen in many places that a connection on $(E_G,M,G)$ is a splitting of the Atiyah sequence
$$ 0\rightarrow \text{ad}(E_G)\rightarrow \text{At}(E_G)\rightarrow T M\rightarrow 0$$
where $\text{ad}(E_G)$ is the adjoint vector bundle for $E_G$ and $\text{At}(E_G)$ is the Atiyah bundle for $E_G$.
Reference given for this is Atiyah's paper. This paper is slightly difficult to read.
Can some one give an outline of this construction or point out some exposition where this is written in detail.
Best Answer
First note that the adjoint bundle $ad(E_G)$ can be canonically identified with the vertical tangent bundle $V E_G / G$: send the pair $(p, \xi)$ consisting of a point $p \in E_G$ and a Lie algebra element $\xi$ to the value $p \cdot \xi$ at $p$ of the fundamental vector field generated by $\xi$. Moreover, the Atiyah bundle $At(E_G)$ is just a fancy way of writing $T E_G / G$.
Thus, a splitting of the Atiyah sequence is nothing else than a diffeomorphism of $T E_G / G$ with the direct sum $V E_G /G \oplus T M$ (taken over $M$). In this way, you recover the definition of a connection as a complement to $V E_G$. The horizontal bundle is the image of $TM$ under the above isomorphism $V E_G \oplus T M \to T E_G / G$. Alternatively, you can view a splitting as a projection onto $V E_G$ or as a $G$-equivariant lift $TM \to TE_G$. These equivalent viewpoints give you the definition of a connection as a connection $1$-form and as a horizontal lift operator, respectively. I've expanded a bit on the different but equivalent viewpoints in an answer to a different question.