From: stevens@math.chalmers.se (Jan Stevens)
Subject: Re: integration along the fiber: can Clifford algebra help?
Date: 14 Mar 2000 09:33:51 GMT
Newsgroups: sci.math
Summary: [missing]

In article <8aaruj$apt$1@panix5.panix.com>,
	lrudolph@panix.com (Lee Rudolph) writes:
> Suppose you have a smooth fibration p:E \to B, where the 
> base is a k-manifold-with-boundary, the total space is
> an n-manifold-with-corners, and the fibers p^{-1}(b)
> are compact (n-k)-manifolds-with-boundary.  If Z is a
> differential form on E of degree m >= n-k then there's
> a way to "integrate Z along the fibers" of p to obtain
> a differential form W on B of degree m-(n-k) >= 0; under
> appropriate further conditions, integrals of Z over
> chains in E are then equal to integrals of W over 
> chains in B.  
[...]
>  My questions here are two: can 
> anyone show me (or point me at a good printed explanation 
> of) an effective method of computing examples? and, is 
> there a slick explication of the situation (preferably, 
> but not necessarily, usable in practice) in terms of 
> Clifford algebra?

I leave the Clifford algebra part to Pertti.
It seems overkill to me.

The situation is quite simple. Let an (n+m)-form Z be given.
For every m tangent vectors at a point of the base we want a number.
We choose m vector fields along the fibre, which project onto
the given vectors. Contracting with Z gives an n-form on the fibre,
whose value on n-tuples of vector fields tangent to the fibre is 
independent of the chosen lift. Integrate this n-form over the fibre
to get a number.

As to examples, computing the integral of an n-form over a
n-manifold with boundary can already be complicated, and now you have to
do it depending on parameters. I suppose all depends on the
way your fibration p: E \to B is given to you.

Bye,
Jan. 
--
email: stevens@math.chalmers.se

Matematiska institutionen
G"oteborgs universitet
Chalmers tekniska h"ogskola
SE 412 96 G"oteborg
Sweden