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| In [[mathematics]], more precisely in [[differential geometry]], a '''soldering''' (or sometimes '''solder form''') of a [[fiber bundle]] to a [[smooth manifold]] is a manner of attaching the fibres to the manifold in such a way that they can be regarded as tangent. Intuitively, soldering expresses in abstract terms the idea that a manifold may have a point of [[contact (mathematics)|contact]] with a certain model [[Klein geometry]] at each point. In extrinsic differential geometry, the soldering is simply expressed by the tangency of the model space to the manifold. In intrinsic geometry, other techniques are needed to express it. Soldering was introduced in this general form by [[Charles Ehresmann]] in 1950.<ref>Kobayashi (1957).</ref>
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| ==Soldering of a fibre bundle==
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| Let ''M'' be a smooth manifold, and ''G'' a [[Lie group]], and let ''E'' be a smooth fibre bundle over ''M'' with structure group ''G''. Suppose that ''G'' [[group action|acts transitively]] on the typical fibre ''F'' of ''E'', and that dim ''F'' = dim ''M''. A '''soldering''' of ''E'' to ''M'' consists of the following data:
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| # A distinguished [[section (fiber bundle)|section]] ''o'' : ''M'' → ''E''.
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| # A linear isomorphism of vector bundles θ : T''M'' → ''o''<sup>−1</sup>V''E'' from the [[tangent bundle]] of ''M'' to the [[pullback bundle|pullback]] of the [[vertical bundle]] of ''E'' along the distinguished section.
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| In particular, this latter condition can be interpreted as saying that θ determines a linear isomorphism
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| :<math>\theta_x : T_xM\rightarrow V_{o(x)} E</math>
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| from the tangent space of ''M'' at ''x'' to the (vertical) tangent space of the fibre at the point determined by the distinguished section. The form θ is called the '''solder form''' for the soldering.
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| ==Special cases==
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| ===Affine bundles and vector bundles===
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| Suppose that ''E'' is an affine [[vector bundle]] (a vector bundle without a choice of zero section). Then a soldering on ''E'' specifies first a ''distinguished section'': that is, a choice of zero section ''o'', so that ''E'' may be identified as a vector bundle. The solder form is then a linear isomorphism
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| :<math>\theta\colon TM \to V_oE,</math>
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| However, for a vector bundle there is a canonical isomorphism between the vertical space at the origin and the fibre V<sub>o</sub>''E'' ≈ ''E''. Making this identification, the solder form is specified by a linear isomorphism
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| :<math>TM \to E.</math>
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| In other words, a soldering on an [[affine bundle]] ''E'' is a choice of isomorphism of ''E'' with the tangent bundle of ''M''.
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| Often one speaks of a ''solder form on a vector bundle'', where it is understood ''a priori'' that the distinguished section of the soldering is the zero section of the bundle. In this case, the structure group of the vector bundle is often implicitly enlarged by the [[semidirect product]] of ''GL''(''n'') with the typical fibre of ''E'' (which is a representation of ''GL''(''n'')).<ref>Cf. Kobayashi (1957) section 11 for a discussion of the companion reduction of the structure group.</ref>
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| ====Examples====
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| * As a special case, for instance, the tangent bundle itself carries a canonical solder form, namely the identity.
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| * If ''M'' has a [[Riemannian metric]] (or [[pseudo-Riemannian metric]]), then the [[covariant metric tensor]] gives an isomorphism <math>g\colon TM \to T^*M</math> from the tangent bundle to the [[cotangent bundle]], which is a solder form.
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| * In [[Hamiltonian mechanics]], the solder form is known as the [[tautological one-form]], or alternately as the '''Liouville one-form''', the '''Poincaré one-form''', the '''canonical one-form''', or the '''symplectic potential'''.
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| ====Applications====
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| A solder form on a vector bundle allows one to define the [[torsion tensor]] of a [[connection (vector bundle)|connection]].
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| ===Principal bundles===
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| In the language of principal bundles, a '''solder form''' on a smooth [[principal bundle|principal ''G''-bundle]] ''P'' over a [[smooth manifold]] ''M'' is a horizontal and ''G''-equivariant [[vector-valued differential form|differential 1-form]] on ''P'' with values in a [[linear representation]] ''V'' of ''G'' such that the associated [[bundle map]] from the [[tangent bundle]] ''TM'' to the [[associated bundle]] ''P''×<sub>''G''</sub> ''V'' is a [[bundle map|bundle isomorphism]]. (In particular, ''V'' and ''M'' must have the same dimension.)
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| A motivating example of a solder form is the [[frame bundle#Solder form|tautological or fundamental form]] on the [[frame bundle]] of a manifold.
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| The reason for the name is that a solder form solders (or attaches) the abstract principal bundle to the manifold ''M'' by identifying an associated bundle with the tangent bundle. Solder forms provide a method for studying [[G-structure|''G''-structures]] and are important in the theory of [[Cartan connection]]s. The terminology and approach is particularly popular in the physics literature.
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| == See also ==
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| * [[Tautological one-form]]
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| ==Notes==
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| <references/>
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| ==References==
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| *{{cite journal | first = C. | last = Ehresmann | title = Les connexions infinitésimales dans un espace fibré différentiel | journal = Colloque de Topologie, Bruxelles | year = 1950 | pages = 29–55}}
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| *{{cite journal | first = Shoshichi | last = Kobayashi | title = Theory of Connections | journal = Ann. Mat. Pura Appl. | year = 1957 | volume = 43 | issue = 1 | pages = 119–194 | doi = 10.1007/BF02411907}}
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| *{{cite book | author=Kobayashi, Shoshichi and Nomizu, Katsumi | title = [[Foundations of Differential Geometry]], Vol. 1 & 2 | publisher=[[Wiley Interscience]] | year=1996 (New edition) |isbn = 0-471-15733-3}}
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| [[Category:Differential forms]]
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| [[Category:Fiber bundles]]
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