Vertical and horizontal bundles

<h2 id="formal-definition">Formal definition</h2>
Let π:E→B be a smooth fiber bundle over a <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifold</a> B. The vertical bundle is the <a href="/facts/Kernel_(linear_algebra)/adhGUYa0">kernel</a> VE := ker(dπ) of the <a href="/facts/Tangent_map/xLsoLwVH">tangent map</a> dπ : TE → TB.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
Since dπe is surjective at each point e, it yields a regular subbundle of TE. Furthermore, the vertical bundle VE is also <a href="/facts/Integrable/Crgzjw6G">integrable</a>.
An <a href="/facts/Ehresmann_connection/qIpSLmFz">Ehresmann connection</a> on E is a choice of a complementary subbundle HE to VE in TE, called the horizontal bundle of the connection. At each point e in E, the two subspaces form a <a href="/facts/Direct_sum/x3ljVsP6">direct sum</a>, such that 
TeE = VeE ⊕ HeE.

<h2 id="example">Example</h2>

The <a href="/facts/M%25C3%25B6bius_strip/35kzqqFL">Möbius strip</a> is a <a href="/facts/Line_bundle/CupAl1ft">line bundle</a> over the circle, and the circle can be pictured as the middle ring of the strip. At each point 
 
 
 
 e
 
 
 {\displaystyle e}
 
 on the strip, the projection map projects it towards the middle ring, and the fiber is perpendicular to the middle ring. The vertical bundle at this point 
 
 
 
 
 V
 
 e
 
 
 E
 
 
 {\displaystyle V_{e}E}
 
 is the tangent space to the fiber.
A simple example of a smooth fiber bundle is a <a href="/facts/Cartesian_product/BfqBWzdL">Cartesian product</a> of two <a href="/facts/Manifold/rXPERJtu">manifolds</a>. Consider the bundle B1 := (M × N, pr1) with bundle projection pr1 : M × N → M : (x, y) → x. Applying the definition in the paragraph above to find the vertical bundle, we consider first a point (m,n) in M × N. Then the image of this point under pr1 is m. The preimage of m under this same pr1 is {m} × N, so that T(m,n) ({m} × N) = {m} × TN. The vertical bundle is then VB1 = M × TN, which is a subbundle of T(M ×N). If we take the other projection pr2 : M × N → N : (x, y) → y to define the fiber bundle B2 := (M × N, pr2) then the vertical bundle will be VB2 = TM × N.
In both cases, the product structure gives a natural choice of horizontal bundle, and hence an Ehresmann connection: the horizontal bundle of B1 is the vertical bundle of B2 and vice versa.

<h2 id="properties">Properties</h2>
Various important <a href="/facts/Tensor/pNjFo5tV">tensors</a> and <a href="/facts/Differential_form/T9gyHtMv">differential forms</a> from <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a> take on specific properties on the vertical and horizontal bundles, or even can be defined in terms of them. Some of these are:

<ul><li>A vertical vector field is a <a href="/facts/Vector_field/ccFNuPPp">vector field</a> that is in the vertical bundle. That is, for each point e of E, one chooses a vector 
 
 
 
 
 v
 
 e
 
 
 ∈
 
 V
 
 e
 
 
 E
 
 
 {\displaystyle v_{e}\in V_{e}E}
 
 where 
 
 
 
 
 V
 
 e
 
 
 E
 ⊂
 
 T
 
 e
 
 
 E
 =
 
 T
 
 e
 
 
 (
 
 E
 
 π
 (
 e
 )
 
 
 )
 
 
 {\displaystyle V_{e}E\subset T_{e}E=T_{e}(E_{\pi (e)})}
 
 is the vertical vector space at e.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></li>
<li>A differentiable <a href="/facts/Differential_form/T9gyHtMv">r-form</a> 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 on E is said to be a horizontal form if 
 
 
 
 α
 (
 
 v
 
 1
 
 
 ,
 .
 .
 .
 ,
 
 v
 
 r
 
 
 )
 =
 0
 
 
 {\displaystyle \alpha (v_{1},...,v_{r})=0}
 
 whenever at least one of the vectors 
 
 
 
 
 v
 
 1
 
 
 ,
 .
 .
 .
 ,
 
 v
 
 r
 
 
 
 
 {\displaystyle v_{1},...,v_{r}}
 
 is vertical.</li>
<li>The <a href="/facts/Connection_form/Hnn6FJFN">connection form</a> vanishes on the horizontal bundle, and is non-zero only on the vertical bundle. In this way, the connection form can be used to define the horizontal bundle: The horizontal bundle is the kernel of the connection form.</li>
<li>The <a href="/facts/Solder_form/1zNxoKvT">solder form</a> or <a href="/facts/Tautological_one-form/FH8WKyxV">tautological one-form</a> vanishes on the vertical bundle and is non-zero only on the horizontal bundle. By definition, the solder form takes its values entirely in the horizontal bundle.</li>
<li>For the case of a <a href="/facts/Frame_bundle/Gugt3qyL">frame bundle</a>, the <a href="/facts/Torsion_form/LVfDhPnW">torsion form</a> vanishes on the vertical bundle, and can be used to define exactly that part that needs to be added to an arbitrary connection to turn it into a <a href="/facts/Levi-Civita_connection/16KGSQUY">Levi-Civita connection</a>, i.e. to make a connection be torsionless. Indeed, if one writes θ for the solder form, then the torsion tensor Θ is given by Θ = D θ (with D the <a href="/facts/Exterior_covariant_derivative/eKHqDq6Q">exterior covariant derivative</a>). For any given connection ω, there is a unique one-form σ on TE, called the <a href="/facts/Contorsion_tensor/09XxGsu3">contorsion tensor</a>, that is vanishing in the vertical bundle, and is such that ω+σ is another connection 1-form that is torsion-free. The resulting one-form ω+σ is nothing other than the Levi-Civita connection. One can take this as a definition: since the torsion is given by 
 
 
 
 Θ
 =
 D
 θ
 =
 d
 θ
 +
 ω
 ∧
 θ
 
 
 {\displaystyle \Theta =D\theta =d\theta +\omega \wedge \theta }
 
, the vanishing of the torsion is equivalent to having 
 
 
 
 d
 θ
 =
 −
 (
 ω
 +
 σ
 )
 ∧
 θ
 
 
 {\displaystyle d\theta =-(\omega +\sigma )\wedge \theta }
 
, and it is not hard to show that σ must vanish on the vertical bundle, and that σ must be G-invariant on each fibre (more precisely, that σ transforms in the <a href="/facts/Adjoint_representation/bWG4c2su">adjoint representation</a> of G). Note that this defines the Levi-Civita connection without making any explicit reference to any metric tensor (although the metric tensor can be understood to be a special case of a solder form, as it establishes a mapping between the tangent and cotangent bundles of the base space, i.e. between the horizontal and vertical subspaces of the frame bundle).</li>
<li>In the case where E is a principal bundle, then the <a href="/facts/Fundamental_vector_field/NM7NG4Q9">fundamental vector field</a> must necessarily live in the vertical bundle, and vanish in any horizontal bundle.</li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Yvonne_Choquet-Bruhat/wd3xA1SJ">Choquet-Bruhat, Yvonne</a>; DeWitt-Morette, Cécile (1977), <a href="https://archive.org/details/analysismanifold0000choq">Analysis, Manifolds and Physics</a>, Amsterdam: Elsevier, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-7204-0494-4</li>
<li>Kobayashi, Shoshichi; Nomizu, Katsumi (1996). <a href="/facts/Foundations_of_Differential_Geometry/lgGBjhhd">Foundations of Differential Geometry, Vol. 1</a> (New ed.). <a href="/facts/Wiley_Interscience/53g3LqJc">Wiley Interscience</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-15733-3.</li>
<li>Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), <a href="https://www.mat.univie.ac.at/~michor/kmsbookh.pdf">Natural Operations in Differential Geometry</a> (PDF), Springer-Verlag</li>
<li>Krupka, Demeter; Janyška, Josef (1990), Lectures on differential invariants, Univerzita J. E. Purkyně V Brně, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 80-210-0165-8</li>
<li>Saunders, D.J. (1989), <a href="https://archive.org/details/geometryofjetbun0000saun">The geometry of jet bundles</a>, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-36948-7</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">David Bleecker, Gauge Theory and Variational Principles (1981) Addison-Wesely Publishing Company ISBN 0-201-10096-7 (See theorem 1.2.4) <a href="https://zulfahmed.files.wordpress.com/2014/05/88623149-bleecker-d-gauge-theory-and-variational-principles-aw-1981-ka-t-201s-pqgf.pdf" target="_blank">https://zulfahmed.files.wordpress.com/2014/05/88623149-bleecker-d-gauge-theory-and-variational-principles-aw-1981-ka-t-201s-pqgf.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural Operations in Differential Geometry (PDF), Springer-Verlag (page 77) <a href="http://www.emis.de/monographs/KSM/kmsbookh.pdf" target="_blank">http://www.emis.de/monographs/KSM/kmsbookh.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural Operations in Differential Geometry (PDF), Springer-Verlag (page 77) <a href="http://www.emis.de/monographs/KSM/kmsbookh.pdf" target="_blank">http://www.emis.de/monographs/KSM/kmsbookh.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Vertical and horizontal bundles open-in-new

Vertical and horizontal bundles