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?