A strange enough name for constructions that have been long known to mathematicians, and relatively recently found applications in physics - by relatively I mean quite a long time ago.

I start with a simple, and possibly familiar example.

Suppose a manifold - if it helps you can think of the sphere, a 2-manifold - but it is better to think more generally.

Now, at any point I may define a vector space denoted as to signify the fact that these define a set of vectors tangent to at each and every point. A problem arises, however.......

Taking the naive view of a tangent, we must assume that our tangent vectors "protrude" into some "surrounding" space in which our manifold is embedded. We cannot allow this - the inventors of differential geometry originally called it "intrinsic geometry". So we need a different definition of a tangent vector. Let us nonetheless be a little naive, and say a vector has direction and magnitude (not the best definition).

So one defines the gadget as being a tangent vector, which we can very loosely think of as the "tendency" of our point to move in the direction on . It's a definition! Magnitude comes from applying a real scalar to this gadget, and, assuming for simplicity , the real 2-sphere, we write where

Obviously this extends to any n-manifold.

So the set-theoretic union (actually it's strictly a disjoint union) of all for all is called the "tangent bundle over ". One writes somewhat confusingly, for this bundle

So a section of this bundle - in the usual usage of the word section - defines a vector field on , that is it "selects" a single vector and assigns it to each and every point . It does this as follows........

First one defines a projection , so that for any point , which is quite obviously a vector, is vector at . Like all projections known to man, beast or bacterium, this mapping is highly surjective - given a point , the pre-image set at that very point.

For reasons that escape me, this is called the "fibre at ". We are of course entitled to regard all such fibres at all points in our manifold as equivalent, so hence the term - the disjoint union of all such fibres at all such points is called a "fibre bundle".

Let's return to our section of this bundle, and write it as . Then quite obviously if any of the above is to make any sense - and it may not! - we must have, if our section is to be field, that .

This may seem a very complicated way of explaining something rather simple, and indeed it is, But I have my reasons. There are also some complications around what are called local vs. global triviality, which, if you want to follow the direction I am travelling, I need to explain.

But for now, I am out of puff, and you, no doubt, are out of patience