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I'm trying to understand the fiber bundle formulation of gauge theory at the moment, and I'm stuck on the connection. Every reference I've found introduces the idea of a connection on a principle bundle as a kind of partitioning of the tangent space at all points in the total space into a "vertical space" and a "horizontal space". The vertical space V_{p}consists of vectors in T_{p}P which are also tangent to the fiber at p, and the horizontal space H_{p}is a set of vectors such that V_{p}+H_{p}=T_{p}P.

What I don't understand is why finding V_{p}doesn't uniquely specify H_{p}. It should be possible to construct T_{p}P without defining a connection, right? If so, wouldn't H_{p}just be every element of T_{p}P that is not also in V_{p}? I don't see how we are free to make this partition ourselves. Where am I going wrong?

Thanks for reading!

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# The connection as a choice of horizontal subspace?

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