From: "Chuck Cadman"
Subject: Re: Vector bundles
Date: Fri, 25 Jun 1999 05:56:21 GMT
Newsgroups: sci.math
Tc Hughes1 wrote in message
<19990625001919.08063.00001447@ng-fs1.aol.com>...
>Can someone please explain to me what vector and tensor bundles are? Any help
>would be much appreciated! Thanx in advance!
>-Taylor
The vector bundle on a manifold M is just the collection of vectors on M.
It is endowed with a differentiable structure. In a coordinate neighborhood
U, it just looks like UxR^n, but the global structure might be nontrivial.
To see how the structure arises, first note that the vector bundle in R^n is
R^2n, since there is a canonical isomorphism between all the tangent spaces.
Furthermore, diffeomorphisms give you isomorphisms in the tangent spaces.
So from a coordinate function, you obtain a bijective mapping that sends
vectors in R^n to vectors in a coordinate neighborhood of the manifold.
This is a coordinate function for TM.
Note that M is embedded in TM. The mapping i(p) = 0 (the zero vector at p)
gives you that. Also, a vector field on M is a differentiable mapping
v:M->TM which satisfies P(v(p)) = p, where P is the projection from TM onto
M.
==============================================================================
From: Robin Chapman
Subject: Re: Vector bundles
Date: Fri, 25 Jun 1999 07:52:04 GMT
Newsgroups: sci.math
In article ,
"Chuck Cadman" wrote:
>
> Tc Hughes1 wrote in message
> <19990625001919.08063.00001447@ng-fs1.aol.com>...
> >Can someone please explain to me what vector and tensor bundles are? Any
> help
> >would be much appreciated! Thanx in advance!
> >-Taylor
>
> The vector bundle on a manifold M is just the collection of vectors on M.
You mean the tangent bundle on M.
> It is endowed with a differentiable structure. In a coordinate neighborhood
> U, it just looks like UxR^n, but the global structure might be nontrivial.
> To see how the structure arises, first note that the vector bundle in R^n is
> R^2n, since there is a canonical isomorphism between all the tangent spaces.
> Furthermore, diffeomorphisms give you isomorphisms in the tangent spaces.
> So from a coordinate function, you obtain a bijective mapping that sends
> vectors in R^n to vectors in a coordinate neighborhood of the manifold.
> This is a coordinate function for TM.
>
> Note that M is embedded in TM. The mapping i(p) = 0 (the zero vector at p)
> gives you that. Also, a vector field on M is a differentiable mapping
> v:M->TM which satisfies P(v(p)) = p, where P is the projection from TM onto
> M.
>
The tangent bundle on a smooth manifold is just one of many examples of
vector bundles. Essentially a vector bundle over a topolgical space X
is an assignement of a vector space V_x to each point x of X such
that the V_x "vary continously". What this means is that the disjoint
union V of the V_x is given a topology which satisfies various
conditions which I'm too lazy to list. When one has vector bundles
V and W on a space X one can construct others by vector space
operations. Important cases are the dual V* of V defined by letting
V*_x be the dual space of V_x and the tensor product V (x) W
defined by letting (V (x) W)_x = V_x (x) W_x. [Here (x) denotes the
multiplication sign in a circle].
In manifold theory the dual of the tangent bundle TM is the cotangent
bundle T*M. Taking tensor products of copies of TM and T*M yield
"tensor bundles" which are widely used in physics. The physicists
have developed a gruesome notation for elements of these bundles
with multiple subscripts and superscripts. They also call elements
of the tangent and cotangent bundles contravariant and covariant
vectors, but I cannot remember which is which :-(
Taking alternating powers of the cotangent bundle yields the bundles
of differential forms. These are important in topology as they yield
a cochain complex whose cohomology groups are topological invariants
of the manifold.
--
Robin Chapman
http://www.maths.ex.ac.uk/~rjc/rjc.html
"They did not have proper palms at home in Exeter."
Peter Carey, _Oscar and Lucinda_
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