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  Jets and vertical differential

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For a vector bundle (E,π,M) let ϕ:ME be a section of π, xM and u=ϕ(x). The vertical differential of the section ϕ at point uE is the map: dvuϕ:TuEVuπ

In coordinates on E (xi,uα) we write; dvuϕ=(duαϕαxidxi)uα
Apparently it is obvious from this that dvuϕ depends only on the first order jet space j1xϕ.

What is Vuπ in this case? It is clearly related to the jet manifold J1π whose total space is the product TMEVπ . But I don't really understand what an associated vector bundle is!

References:

  1. C.M. Campos, Geometric Methods in Classical Field Theory and Continuous Media, pages 24-25.


This post imported from StackExchange Mathematics at 2015-08-12 17:47 (UTC), posted by SE-user Janet the Physicist

asked Apr 14, 2015 in Mathematics by Janet the Physicist (15 points) [ revision history ]
edited Aug 12, 2015 by Dilaton
Janet the Physicist. Looks like your bundle is endowed with the connection, i.e., family of "horizontal" subspaces, while the vertical differential is the projection of TuE to vertical fibers Vπ of the bundle. How else you can define projection dV?

This post imported from StackExchange Mathematics at 2015-08-12 17:47 (UTC), posted by SE-user user2612
i.e., Vuπ is the vector space tangent to the fiber of the bundle π:EM. In the book local coordinates (x,u) provide TuE with the splitting Vuπ+HuE, where (horizontal) subspace is identified with TuM. so that taking vertical differential of a section equals the projection of the ordinary differential dϕ to VuE along this Hu. Jet manifold J1π has projections on E and M which makes it bundle, it is called associated since its structure group is defined by the structure group of the initial bundle π, see (3.5).

This post imported from StackExchange Mathematics at 2015-08-12 17:47 (UTC), posted by SE-user user2612
@user2612 So the tangent space to a point uE, TuE is the sum of vertical and horizontal spaces Vuπ+HuE. The horizontal space in this case are HuE which is also (?) identified with TuM. Taking the vertical differential of a section of an element in TuE maps to the vertical space of the tangent space? Thank you so much for your help :)

This post imported from StackExchange Mathematics at 2015-08-12 17:47 (UTC), posted by SE-user Janet the Physicist
Yes, you are welcome! I will put this in more details below as an answer.

This post imported from StackExchange Mathematics at 2015-08-12 17:47 (UTC), posted by SE-user valeri

1 Answer

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Vuπ is the vector space tangent to the fiber π1(x) of the bundle π:EM going through u. In the book local coordinates (x,u) provide TuE with the splitting Vuπ+HuE, where (horizontal) subspace is identified with TuM. We may think of the horizontal space as the tangent to locally constant sections ϕ:ME going through the point u, i.e., ϕ(x)u. Now, taking vertical differential of a section ϕ equals the projection of the ordinary differential dϕ to VuE along this Hu. Jet manifold J1π has projections on both E and M which makes it bundle, it is called associated since its structure group is defined by the structure group of the initial bundle π, see (3.5).

This post imported from StackExchange Mathematics at 2015-08-12 17:47 (UTC), posted by SE-user valeri
answered Apr 16, 2015 by valeri (10 points) [ no revision ]
thank you for your answer, is Vuπ the same as VuE in this case?

This post imported from StackExchange Mathematics at 2015-08-12 17:47 (UTC), posted by SE-user Janet the Physicist
yes, probably, I should write the splitting above as TuE=VuE+HuE.

This post imported from StackExchange Mathematics at 2015-08-12 17:47 (UTC), posted by SE-user valeri
or. stack to authors notations - i.e., use Vuπ everywhere, sorry for confusion!

This post imported from StackExchange Mathematics at 2015-08-12 17:47 (UTC), posted by SE-user valeri
I can't begin to thank you enough, I have been here three days without a clue! :D

This post imported from StackExchange Mathematics at 2015-08-12 17:47 (UTC), posted by SE-user Janet the Physicist
Janet the Physicist welcome!

This post imported from StackExchange Mathematics at 2015-08-12 17:47 (UTC), posted by SE-user valeri

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