Hi, 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 Vp consists of vectors in TpP which are also tangent to the fiber at p, and the horizontal space Hp is a set of vectors such that Vp+Hp=TpP. What I don't understand is why finding Vp doesn't uniquely specify Hp. It should be possible to construct TpP without defining a connection, right? If so, wouldn't Hp just be every element of TpP that is not also in Vp? I don't see how we are free to make this partition ourselves. Where am I going wrong? Thanks for reading!