# Bundle metric

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In [differential geometry](/source/differential_geometry), the notion of a [metric tensor](/source/metric_tensor) can be extended to an arbitrary [vector bundle](/source/vector_bundle), and to some [principal fiber bundle](/source/principal_fiber_bundle)s. This metric is often called a '''bundle metric''', or '''fibre metric'''.

==Definition==
If ''M'' is a [topological manifold](/source/topological_manifold) and {{pi}} : ''E'' → ''M'' a vector bundle on ''M'', then a metric on ''E'' is a [bundle map](/source/bundle_map) ''k'' : ''E'' &times;<sub>''M''</sub> ''E'' → ''M''&nbsp;&times;&nbsp;'''R''' from the [fiber product](/source/fiber_product) of ''E'' with itself to the [trivial bundle](/source/trivial_bundle) with fiber '''R''' such that the restriction of ''k'' to each fibre over ''M'' is a [nondegenerate](/source/nondegenerate) [bilinear map](/source/bilinear_map) of [vector spaces](/source/vector_spaces).<ref name=jost>{{citation
 | last = Jost | first = Jürgen |author-link=Jürgen Jost
 | doi = 10.1007/978-3-642-21298-7
 | edition = Sixth
 | isbn = 978-3-642-21297-0
 | mr = 2829653
 | page = 46
 | publisher = Springer, Heidelberg
 | series = Universitext
 | title = Riemannian geometry and geometric analysis
 | url = https://books.google.com/books?id=UjzUqF2mRWYC&pg=PA46
 | year = 2011| url-access = subscription
 }}.</ref> Roughly speaking, ''k'' gives a kind of [dot product](/source/dot_product) (not necessarily symmetric or positive definite) on the vector space above each point of ''M'', and these products vary smoothly over ''M''.

==Properties==
Every vector bundle with paracompact base space can be equipped with a bundle metric.<ref name=jost/> For a vector bundle of rank ''n'', this follows from the [bundle charts](/source/local_trivialization) <math>\phi:\pi^{-1}(U)\to U\times\mathbb{R}^n</math>: the bundle metric can be taken as the pullback of the [inner product](/source/inner_product) of a metric on <math>\mathbb{R}^n</math>; for example, the orthonormal charts of Euclidean space. The [structure group](/source/structure_group) of such a metric is the [orthogonal group](/source/orthogonal_group) ''O''(''n'').

==Example: Riemann metric==
If ''M'' is a [Riemannian manifold](/source/Riemannian_manifold), and ''E'' is its [tangent bundle](/source/tangent_bundle) T''M'', then the [Riemannian metric](/source/Riemannian_metric) gives a bundle metric, and vice versa.<ref name=jost/>

==Example: on vertical bundles==
If the bundle {{pi}}:''P'' → ''M'' is a [principal fiber bundle](/source/principal_fiber_bundle) with group ''G'', and ''G'' is a [compact Lie group](/source/compact_Lie_group), then there exists an Ad(''G'')-invariant inner product ''k'' on the fibers, taken from the inner product on the corresponding [compact Lie algebra](/source/compact_Lie_algebra). More precisely, there is a [metric tensor](/source/metric_tensor) ''k'' defined on the [vertical bundle](/source/vertical_bundle) E = V''P'' such that ''k'' is invariant under left-multiplication:

:<math>k(L_{g*}X, L_{g*}Y)=k(X,Y)</math>

for vertical vectors ''X'', ''Y'' and ''L''<sub>''g''</sub> is left-multiplication by ''g'' along the fiber, and ''L''<sub>''g*''</sub> is the [pushforward](/source/pushforward_(differential)).  That is, ''E'' is the vector bundle that consists of the vertical subspace of the tangent of the principal bundle.

More generally, whenever one has a compact group with [Haar measure](/source/Haar_measure) μ, and an arbitrary [inner product](/source/inner_product) ''h(X,Y)'' defined at the tangent space of some point in ''G'', one can define an invariant metric simply by averaging over the entire group, i.e. by defining

:<math>k(X,Y)=\int_G h(L_{g*} X, L_{g*} Y) d\mu_g</math>

as the average.

The above notion can be extended to the [associated bundle](/source/associated_bundle) <math>P\times_G V</math> where ''V'' is a vector space transforming covariantly under some [representation](/source/Lie_algebra_representation) of ''G''.

==In relation to Kaluza–Klein theory==
If the base space ''M'' is also a [metric space](/source/metric_space), with metric ''g'', and the principal bundle is endowed with a [connection form](/source/connection_form) ω, then {{pi}}<sup>*</sup>g+kω is a metric defined on the entire [tangent bundle](/source/tangent_bundle) ''E'' = T''P''.<ref name=bleecker>David Bleecker, "[https://zulfahmed.files.wordpress.com/2014/05/88623149-bleecker-d-gauge-theory-and-variational-principles-aw-1981-ka-t-201s-pqgf.pdf Gauge Theory and Variational Principles] {{Webarchive|url=https://web.archive.org/web/20210709185749/https://zulfahmed.files.wordpress.com/2014/05/88623149-bleecker-d-gauge-theory-and-variational-principles-aw-1981-ka-t-201s-pqgf.pdf |date=2021-07-09 }}" (1982) D. Reidel Publishing ''(See chapter 9'')</ref>

More precisely, one writes {{pi}}<sup>*</sup>g(''X'',''Y'')&nbsp;=&nbsp;''g''({{pi}}<sub>*</sub>''X'', {{pi}}<sub>*</sub>''Y'') where {{pi}}<sub>*</sub> is the [pushforward](/source/pushforward_(differential)) of the projection {{pi}}, and ''g'' is the [metric tensor](/source/metric_tensor) on the base space ''M''.  The expression ''kω'' should be understood as (''kω'')(''X'',''Y'')&nbsp;=&nbsp;''k''(''ω''(''X''),''ω''(''Y'')), with ''k'' the metric tensor on each fiber. Here, ''X'' and ''Y'' are elements of the [tangent space](/source/tangent_space) T''P''.

Observe that the lift {{pi}}<sup>*</sup>g vanishes on the [vertical subspace](/source/vertical_subspace) T''V'' (since {{pi}}<sub>*</sub> vanishes on vertical vectors), while kω vanishes on the horizontal subspace T''H'' (since the horizontal subspace is defined as that part of the tangent space T''P'' on which the connection ω vanishes). Since the total tangent space of the bundle is a direct sum of the vertical and horizontal subspaces (that is, T''P'' = T''V''&nbsp;⊕&nbsp;T''H''), this metric is well-defined on the entire bundle.

This bundle metric underpins the generalized form of [Kaluza–Klein theory](/source/Kaluza%E2%80%93Klein_theory) due to several interesting properties that it possesses. The [scalar curvature](/source/scalar_curvature) derived from this metric is constant on each fiber,<ref name=bleecker/> this follows from the Ad(''G'') invariance of the fiber metric ''k''.  The scalar curvature on the bundle can be decomposed into three distinct pieces:
:''R''<sub>''E''</sub> = ''R''<sub>M</sub>(''g'') + ''L''(''g'', ω) + ''R''<sub>''G''</sub>(''k'')
where ''R''<sub>''E''</sub> is the scalar curvature on the bundle as a whole (obtained from the metric {{pi}}<sup>*</sup>g+kω above), and ''R''<sub>M</sub>(''g'') is the scalar curvature on the base manifold ''M'' (the [Lagrangian density](/source/Lagrangian_density) of the [Einstein–Hilbert action](/source/Einstein%E2%80%93Hilbert_action)), and ''L''(''g'', ω) is the Lagrangian density for the [Yang–Mills action](/source/Yang%E2%80%93Mills_action), and ''R''<sub>''G''</sub>(''k'') is the scalar curvature on each fibre (obtained from the fiber metric ''k'', and constant, due to the Ad(''G'')-invariance of the metric ''k'').  The arguments denote that ''R''<sub>M</sub>(''g'') only depends on the metric ''g'' on the base manifold, but not ω or ''k'', and likewise, that ''R''<sub>''G''</sub>(''k'') only depends on ''k'', and not on ''g'' or ω, and so-on.

==References==
{{reflist}}

Category:Differential geometry

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Adapted from the Wikipedia article [Bundle metric](https://en.wikipedia.org/wiki/Bundle_metric) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Bundle_metric?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.