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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?

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2 Answers 2

Kobayashi-Nomizu "Foundations of Differential Geometry Volume 1" is the traditional go to source on connections, it will have what you are looking for.

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OK, thank you, I will check that. – Lao-tzu Nov 1 '13 at 9:06

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.

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Thank you! I will do it. – Lao-tzu Dec 18 '13 at 1:52

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