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  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?

Closed by author request
asked Jul 15 in Theoretical Physics by New Student (165 points) [ revision history ]
closed Jul 16 by author request




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