Talk:Jet bundle

Holonomic coordinates

I think a holonomic bundle can be used to define a holonomic coordinate system on the underlying manifold, but I'm not quite sure, can someone verify and link as appropriate? linas 01:14, 24 Jan 2005 (UTC)

Article is a bit confused

I have placed a cleanup tag in this article. Iterated jets are not generally regarded as being the same things as jets themselves. Of course there are different types of jets, for instance: holonomic, semiholonomic, and non-holonomic. For example, the following inequality of bundle-valued functors is readily verified: $J^{1}\circ J^{1}\not =J^{2}$. I am currently working on an article Jet_(mathematics). Hopefully this article will inspire the required related work on this article. Thanks, 151.204.6.171

You will find that many (most?) articles on WP, especially those dealing with less well-known topics, to be a random assortment of facts, often poorly stated, sometimes misleading, and occasionally wrong. The well-written articles will be those that are either on a popular topic, and have been combed over by many editors, or the work of a lone, dedicated individual. linas 05:09, 5 November 2005 (UTC)

In that case, I hope you have no objection to me taking over this article once I've finished up on jet (mathematics). Silly rabbit 11:01, 5 November 2005 (UTC)

Anyway, it is clear that the article should agree with the references provided. It may seem like a conflict of interests that I was the one who provided the references. However, the Ehresmann paper is widely regarded as the original paper on jets and jet bundles, and the KMS book is an extensive expository work on the defintion and application of jets to problems in differential geometry. (As of this writing, KMS is freely available on the internet.) Best, Silly rabbit 12:34, 5 November 2005 (UTC)

I just have a question on the part "vector field" Can we always decompose a vector field on a bundle in a horizontal and vertical part or do we need a (Ehresman) connection? — Preceding unsigned comment added by 131.169.87.137 (talk) 15:31, 9 April 2013 (UTC)

Jet Bundles Notes

I am currently writing up some notes on jet bundles, by defining jets of local sections of a fiber bundle. Current headings are

Jets

Jet Manifolds

Jet Bundles

Contact Forms

Vector Fields on the Jet Bundle

Jet Prolongation

Partial Differential Equations

I hope to post when they are ready. Twalton 11.21, 07 Dec 2005

Very nice. I was hoping to do just such a thing, but it turned out to be too significant a project for myself alone. Please see jet_(mathematics) for some seed material. (Actually, I was just developing what I needed for the symbol of a differential operator as it pertains to an elliptic complex. But it would be very good to have a thorough treatment of the subject.) Thanks, Silly rabbit 23:08, 7 December 2005 (UTC)

Ok, so here are the posted notes. Feel free to rip them apart or tell me if there are objects that aren't defined very well. Hope this entry is a slight improvement on before. Twalton 12.09, 09 Dec 2005

Basic Defn

Surely the Jet of a bundle is defined only for smooth vector bundles, not for general fiber bundles? We should make sure that's stated correctly.

Also, the first section really needs to demonstrate that the definition of jet is independent of the choice of co-ordinates.

It's great that the article is here - let me know what I can do to help clean it up. Ewjw 08:32, 19 December 2005 (UTC)

I believe that the jet bundle is defined in the generic case of a fiber bundle and doesn't need the extra requirements that a vector bundle adds. But I would be interested to see the arguments of your claim. Twalton 15:45, 19 December 2005 (UTC)

At a minimum to define the jets on a vector bundle you need to be able to take (formal) partial derivatives of sections. This isn't going to be the case unless the fibres are objects on which this makes sense, which means that they are locally like euclidean space. In other words, the fibre bundle must be a vector bundle. Further, to take the r-jet of a section then one needs the section to be at least a C^r map from the manifold M to the bundle E. So, we should be clear about what category we're working in: I would prefer to put the definitions in as smooth vector bundles over smooth manifolds so that all partial derivatives are well-defined.

Also, I think in the definition of this article we want to have r-th partial derivates, rather than these rogue alphas. What do you think? Ewjw 01:00, 25 December 2005 (UTC)

You don't need a vector bundle structure to define the r-jet of a section. In fact, you don't even need local triviality. So long as the projection map from E to M is surjective and a submersion then you're OK. The fibres are locally like Euclidean space because they are submanifolds of E (consequence of the submersion property) so it makes sense compare derivatives. It is, though, convenient to use smooth manifolds and a smooth projection.83.104.131.53 12:21, 5 February 2006 (UTC)

the projection can only be a submersion if the required differentiable structure is present. And I see nothing in fiber bundle which suggests that topological fiber bundles are meaningless. I think you've used manifold as a synonym of differentiable manifold, but it is better to be explicit. -MarSch 11:29, 16 March 2006 (UTC)

On the other hand, the formulation is such that it allows for other types of manifold which have a notion of differentiation. -MarSch 11:32, 16 March 2006 (UTC)

No, you don't need a differentiable structure to define a submersion. You just need the existence of split charts on the total space, or (equivalently) the existence of local sections. The surjectivity of the tangent map is a convenient test when you do have a differentiable structure, but the property can be defined on a map between topological manifolds. See, for instance, Lang's "Differential manifolds". But you do, of course, need differentiable structures of order k or more to see whether local sections are k-equivalent. —Preceding unsigned comment added by 83.104.131.53 (talk) 08:04, 27 February 2008 (UTC)

fiber projection for the fiber bundle

I really dislike the practice of saying

"$(T,\pi ,B)$ is a fiber bundle. $j\pi$ is the jet bundle of the fiber bundle π."

Instead I much prefer

"$F=(T,\pi ,B)$ is a fiber bundle. $jF$ is the jet bundle of the fiber bundle F."

-MarSch 11:40, 16 March 2006 (UTC)

Multiindex should be allowed to have length zero?

I'm not an expert on jet bundles, but it seems that one should allow the multiindex $I$ in the section "Jets" to have length 0 as well, i.e. the two jets are equivalent if and only if the equality holds for $0\leq |I|\leq r$. Otherwise, the target projection $\pi _{r,0}$ is not well defined. 194.95.184.201 13:47, 3 January 2007 (UTC)

I changed this.--Udoh (talk) 13:23, 16 August 2010 (UTC)

Merge with Jet (Mathematics)?

I don't see a point in having this page and also the page Jet (Mathematics). In general I find this page to be better written and to contain more information (though it needs to be cleaned up and made shorter), so i propose to delte Jet (Mathematics) and replace it with this one, keeping as main title Jet (Mathematics). —Preceding unsigned comment added by 134.157.61.172 (talk) 16:46, 27 May 2010 (UTC)

Other readers have found Jet (mathematics) to be useful as well. There does seem to be sufficient scope for two articles. See my comments at Talk:Jet (mathematics). Sławomir Biały (talk) 12:31, 29 March 2011 (UTC)

Let p: Y -> M be a smoot locally trivial fiber bundle. The the r-th jet bundle is only a closed submanifold of the jet space $J^r(M,Y)$. Hence jet space is a different, more general framework and the article should be stay separated. — Preceding unsigned comment added by 92.225.85.129 (talk) 14:52, 16 September 2011 (UTC)

yeah, dont' merge, the two are dealing are written at different levels of accessibility, and are on related but not identical topics. (p.s. I thought there s different merge proposal for jet group somewhere??) linas (talk) 05:35, 22 December 2011 (UTC)

coordinate free approach?

Could please, please someone explain the coordinate free approach? This article is a mess. --131.220.201.124 (talk) 20:09, 5 May 2012 (UTC)

Jets

The section Jet bundle#Jets has and undefined quantity, α. I guess it runs from 1 to the the dimension of the fiber. YohanN7 (talk) 15:01, 5 October 2012 (UTC)