There is the following result: If $D$ is a connection on a vector bundle $E$ over $N$ and $φ$ is a smooth map from $M$ to $N$, then there is a pullback connection on $φ^*E$ over M, determined uniquely by the condition that $(φ^*D)_X(φ^*s)=φ^*(D_{dφ(X)}s)$.
I want to know how to show the existence and uniqueness of the pullback connection. Can you show it to me or give some reference which discuss this point?
On
Locally, the connection is determined by the connection 1 forms. The pull back of the connection 1 forms over a coordinate neighborhood in N are the connection 1 forms of the induced connection on the induced vector bundle.
The only insight here is that a local maximal linearlly independent set of sections of the pull back bundle over M can always be obtained by pulling back local sections on N. These pullbacks together with the pullbacks of the connection 1 forms determine the connection on the pullback bundle.
Kobayashi-Nomizu "Foundations of Differential Geometry Volume 1" is the traditional go to source on connections, it will have what you are looking for.