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_ Sent via Deja.com http://www.deja.com/ Share what you know. Learn what you don't.