Hi, I have been reading some stuff about differential geometry (with the ultimate goal of trying to understand loop quantum gravity...) and something I have been having trouble with for awhile is the idea of a connection. Basically, I have yet to find a definition that actually enables me to understand what a "connection" is. In general, I think I understand that the motivation of a connection is that when spaces are or can be curved, then the notion (normally taken utterly for granted) that a vector at one coordinate can be compared to a vector at another coordinate becomes false. In order to compare two vectors under these circumstances, we must effectively "transport" the first vector to the location of the second vector. However, since the space may be curved, then the value of the first vector on arrival at the position of the second vector will depend on the path the first vector followed when getting there! Therefore, we must define a notion of "parallel" transport, which will allow us to link up paths between all coordinates in such a way that if we have arbitrarily-positioned vectors A, B, and C, then transporting C to A by our parallel-transport method will have the same effect as moving C first to B and then to A by the same parallel-transport method. Connections provide such a method of "parallel" transport. What I don't get though is: Exactly what is the connection, itself? What is the definition of a connection as a mathematical object? For example if someone makes a statement like "the holonomy of the connection can be identified with a Lie group, the holonomy group" then exactly what is it that this holonomy group is being identified with? When we talk about a particular "connection". Do we mean that the connection is a system of parallel-transport paths, which can be used to pick a parallel-transport path given any two points? Or do we mean simply the connection is the the parallel-transport path between the two points itself? Along similar lines: I recently read a book called "Geometrical Methods of Mathematical Physics" which cleared up a lot of terminology issues for me, but left me still confused about connections. For one thing, they actually did not address connections at all, but only explained something called "affine connections" in an appendix. (Is a connection and an affine connection the same thing?) I ultimately was unable to get anything out of this appendix. Something they did do however in the main part of the book was introduce the idea of a "congruence". They did this because they wanted to introduce the concept of "lie dragging". As I understood it, the point of this was that they needed some unambiguous notion of "direction" on a manifold, in order that that notion of direction could be used to introduce coordinates; the congruence would provide an unambiguous path between any two points on the manifold, and "lie dragging" along the congruence would map the manifold to itself in such a way that each point would be "moved" along its path a certain amount. After defining a sort of basis set of congruences for some manifold, you can then introduce coordinates for that manifold because each of the different ways of "lie dragging" along the basis set of congruences your a manifold would produce one infinitesimal generator of that manifold's lie group. (Does that sound right?) This by itself all made sense, but in some ways it confused me even more, because lie dragging sounded to me exactly like parallel transport! When they first started explaining this, I at first thought that the congruence was going to either be related to, or even the same as, a connection. But apparently they are different. So: What is the difference between a congruence and a connection? What is the difference between lie-dragging along a congruence, and parallel transport along a connection? One last thing I do not understand is why connections are identified with fiber bundles. It seems that this idea of parallel transport makes sense on any manifold with a vector space attached. Or is the idea that if a vector space is "attached" to the manifold in the first place, then this by definition means that we have a fiber bundle? Any help would be appreciated, thanks.