# Why is the interpolation between two connections related via a gauge transformation still a connection?

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I am studying the theory of anomalies in gauge field.

Let $A$ be a gauge field (or a connection for mathematicians). Let $A_{U}$ be an equivalent gauge related via a local gauge transformation
$$A_{U}=U^{-1}dU+U^{-1}AU$$
Then, there is a interpolating between the two
$$A(s)=sA+(1-s)A_{U}$$
where $s\in[0,1]$.

Why is $A(s)$ a connection? If $A$ is a flat connection, then this interpolation $A(s)$ between two flat connections isn't even flat. What are the geometric and physical meaning of such an interpolation?

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