Talk:Cotangent bundle

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Mathematics rating: Start Class Mid Priority  Field: Geometry

Apparently, nobody wants to define the cotangent bundle. Silly rabbit 08:04, 10 November 2005 (UTC)

Would anyone be able add a slightly less high level, maybe more qualitative introduction? Halio 6 Dec 2005

Maybe we should go on about co-vector fields being dual to vector fields and do it components before we lose most of our readers by mentioning a sheaf? Billlion 05:26, 19 May 2006 (UTC)

I agree with what you guys have said. The definition given is very high level, and I found it rather impossible to understand in half an hour. I'll have to look it up, but if the definition naturally paralles the definition of the tangent bundle, then we really should state it in elementary form:

T^*(M) = \coprod_{x\in M}T^*_x(M) = \left\{(v,x) \colon x \in M \mbox{ and } v \in T^*_x M\right\}

That is, ordered pairs of a point in the manifold with cotangent vectors at that point. Very simple. --Chris Foster 05:55, 30 July 2006 (UTC)

Well yes as a set, and a vector space at each point, but this misses out the topology let alone the differntiable structure.Billlion 20:44, 30 July 2006 (UTC)
You're quite right, but perhaps we can attach the topological and smooth structure to the simple definition, as is done in the article on the tangent bundle? It seems to me that the balance between redundency and explanatory power lies in having a simple definition at the top of the article. Of course, the mathematically sophisticated version is always nice if you can understand without reading twenty other pages of definitions. It can be moved further down the page for those who understand the deeper structure. --Chris Foster 12:06, 7 August 2006 (UTC)

Can someone please clarify the difference between the examples of tangent and cotangent bundles as given in the Tangent bundle article: "Another simple example is the unit circle, S1. The tangent bundle of the circle is also trivial and isomorphic to S1 × R. Geometrically, this is a cylinder of infinite height." and the one given in this Cotangent bundle article: " The entire state space looks like a cylinder. The cylinder is the cotangent bundle of the circle." I could understand it if the tangent bundle example was a plane of tangents in the plane of the circle (with a hole in it) - but as written they seem to suggest the same image for both tangent and cotangent bundles, or am I missing something? Thanks in anticipation. PaulGEllis <pellis@london.edu>

Well for any (finite dimensional) manifold the tangent bundle and cotangent bundle are isomorphic as bundles, so of course the total spaces are diffeomorphic. However there is not a natural isomorphism between them, unless you have some additional structure such as Riemannian metric.Billlion 11:55, 30 December 2006 (UTC)