Inmathematics, more precisely indifferential geometry, asoldering (or sometimessolder form) of afiber bundle to asmooth manifold is a manner of attaching the fibers 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 ofcontact with a certain modelKlein 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 byCharles Ehresmann in 1950.^[1]

LetM be a smooth manifold, andG aLie group, and letE be a smooth fibre bundle overM with structure groupG. Suppose thatGacts transitively on the typical fibreF ofE, and that dimF = dimM. Asoldering ofE toM consists of the following data:

1. A distinguishedsectiono :M →E.
2. A linear isomorphism of vector bundles θ : TM →o^*VE from thetangent bundle ofM to thepullback of thevertical bundle ofE along the distinguished section.

In particular, this latter condition can be interpreted as saying that θ determines a linear isomorphism

${\displaystyle {\theta }_x:T_{x}M\to V_{o(x)}E}$

from the tangent space ofM atx to the (vertical) tangent space of the fibre at the point determined by the distinguished section. The form θ is called thesolder form for the soldering.

By convention, whenever the choice of soldering is unique or canonically determined, the solder form is called the canonical form, or the tautological form.

Suppose thatE is an affinevector bundle (a vector bundle without a choice of zero section). Then a soldering onE specifies first adistinguished section: that is, a choice of zero sectiono, so thatE may be identified as a vector bundle. The solder form is then a linear isomorphism

${\displaystyle \theta :TM\to V_{o}E,}$

However, for a vector bundle there is a canonical isomorphism between the vertical space at the origin and the fibre V_oE ≈E. Making this identification, the solder form is specified by a linear isomorphism

${\displaystyle TM\to E.}$

In other words, a soldering on anaffine bundleE is a choice of isomorphism ofE with the tangent bundle ofM.

Often one speaks of asolder form on a vector bundle, where it is understooda 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 thesemidirect product ofGL(n) with the typical fibre ofE (which is a representation ofGL(n)).^[2]

• As a special case, for instance, the tangent bundle itself carries a canonical solder form, namely the identity.
• IfM has aRiemannian metric (orpseudo-Riemannian metric), then thecovariant metric tensor gives an isomorphism${\displaystyle g:TM\to T^{\ast }M}$ from the tangent bundle to thecotangent bundle, which is a solder form.
• InHamiltonian mechanics, the solder form is known as thetautological one-form, or alternately as theLiouville one-form, thePoincaré one-form, thecanonical one-form, or thesymplectic potential.
• Consider the Mobius strip as a fiber bundle over the circle. The vertical bundleo^*VE is still a Mobius strip, while the tangent bundle TM is the cylinder, so there is no solder form for this.

• A solder form on a vector bundle allows one to define thetorsion andcontorsion tensors of aconnection.

• Solder forms occur in thesigma model, where they glue together the tangent space of a spacetime manifold to the tangent space of the field manifold.

• Vierbeins, ortetrads in general relativity, look like solder forms, in that they glue together coordinate charts on the spacetime manifold, to the preferred, usually orthonormal basis on the tangent space, where calculations can be considerably simplified. That is, the coordinate charts are the${\displaystyle TM}$ in the definitions above, and the frame field is the vertical bundle${\displaystyle VE}$. In the sigma model, the vierbeins are explicitly the solder forms.

In the language of principal bundles, asolder form on a smoothprincipalG-bundleP over asmooth manifoldM is a horizontal andG-equivariantdifferential 1-form onP with values in alinear representationV ofG such that the associatedbundle map from thetangent bundleTM to theassociated bundleP×_GV is abundle isomorphism. (In particular,V andM must have the same dimension.)

A motivating example of a solder form is thetautological or fundamental form on theframe bundle of a manifold.

The reason for the name is that a solder form solders (or attaches) the abstract principal bundle to the manifoldM by identifying an associated bundle with the tangent bundle. Solder forms provide a method for studyingG-structures and are important in the theory ofCartan connections. The terminology and approach is particularly popular in the physics literature.

• Ehresmann, C. (1950). "Les connexions infinitésimales dans un espace fibré différentiel".Colloque de Topologie, Bruxelles:29–55.
• Kobayashi, Shoshichi (1957)."Theory of Connections".Ann. Mat. Pura Appl.43 (1):119–194.doi:10.1007/BF02411907.
• Kobayashi, Shoshichi & Nomizu, Katsumi (1996).Foundations of Differential Geometry, Vol. 1 & 2 (New ed.).Wiley Interscience.ISBN 0-471-15733-3.

• Differential forms
• Fiber bundles

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