So, in order to get to some problems with Gauge Theory I have been having over the last few months, let's do this.......

I suppose an n-manifold - for now, I insist on no particular properties, other that it is a manifold.

I suppose that for each point I can find find a vector space each of whose elements are tangent to at that point, and call this vector space as .

I now form the set-theoretic (disjoint) union of all such vector spaces, and call this the "tangent bundle over " - one usually writes for this beast (somewhat confusingly) Ask me why I insist on non-disjunction if you dare!

Recalling that I have placed no restrictions on my manifold, and accepting the fact that therefore I have not excluded the possibility of what is called an "embedding space", I may visualize my tangent vectors as "sticking out" into this embedding space. This is not good (which we may get to later), but first this....

Consider the 1-manifold which is usually called the real circle. Now consider the vector space at each point which is, essentially by definition, the real line - a vector space - , and remember we are not at present excluding the possible existence of an embedding space.

For visualisation, let us take our tangent vector under this (false) scenario to be, not exactly tangent to our circle but perpendicular to it, again a false scenario. Then it is not hard to see intuitively that the bundle is a "cylinder" of sorts. This is of course a manifold of dimension 2. We would be well advised to use this for intuition only, but note this......

The vector bundle over any n-manifold is also a manifold of dimension 2n.

Agreed? Good. So, again without excluding the existence of an embedding space, I want to define a vector field over any manifold as a section of its tangent bundle, where "section" can be taken more-or-less literally, that is to each and every point I may assign a vector, a point without insisting on any sort of uniqueness.

So this is where our problem starts - our embedding space. How do we know whether our section is a set of elements from our embedding space or whether - as we require - it is a set of elements from the bundle itself? Specifically we need to know what is meant by a tangent vector on an n-manifold where we deny the existence of an embedding space

More later (possibly), as this is more than enough for now