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In physics there are phenomenons related to a loop integration in ##M##, such as the Berry's phase

##\gamma = \oint_C A ##

where ##C(t)## is a loop in ##M##, and ##A## is the gauge potential (pull back of connection one-form of ##P## on ##M##).

It seems that if we can find an ##A## that is smooth and single valued on the loop, ##\gamma ## is well defined. Physics literatures always assume this is true. However, we know in general there may not be a smooth and single valued ##A## found over ##M##. Then my question is what determines there is a good ##A## defined on the loop?

My guess is that since ##f:C \to M## is a smooth map, so the topology of the pullback bundle ##{f^ * }P## on ##C## determines whether ##\gamma ## can be well defined on ##M##? i.e. if ##{f^ * }P## is trivial, then a smooth and single valued ##A## can be defined on ##C## in ##M##.

Is this right ? please help.