Solder form

In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fibre 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 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. [Kobayashi (1957).]

oldering of a fibre bundle

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" 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:
# A distinguished section "o" : "M" → "E".
# A linear isomorphism of vector bundles θ : T"M" → "o"^-1V"E" from the tangent bundle of "M" to the pullback of the vertical bundle of "E" along the distinguished section.In particular, this latter condition can be interpreted as saying that θ determines a linear isomorphism:$heta\_x\; :\; T\_xM\; ightarrow\; V\_\{o(x)\}\; E$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.

pecial cases

Affine bundles and vector bundles

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:$hetacolon\; TM\; o\; V\_oE,$However, for a vector bundle there is a canonical isomorphism between the vertical space at the origin and the fibre V_o"E" ≈ "E". Making this identification, the solder form is specified by a linear isomorphism:$TM\; o\; E.$

In other words, a soldering on an affine bundle "E" is a choice of isomorphism of "E" with the tangent bundle of "M".

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")). [Cf. Kobayashi (1957) section 11 for a discussion of the companion reduction of the structure group.]

Examples

* As a special case, for instance, the tangent bundle itself carries a canonical solder form, namely the identity.
* If "M" has a Riemannian metric (or pseudo-Riemannian metric), then the covariant metric tensor gives an isomorphism $gcolon\; TM\; o\; T^*M$ from the tangent bundle to the cotangent bundle, which is a solder form.

Applications

A solder form on a vector bundle allows one to define the torsion tensor of a connection.

Principal bundles

In the language of principal bundles, a solder form on a smooth principal "G"-bundle "P" over a smooth manifold "M" is a horizontal and "G"-equivariant 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"×_"G" "V" is a bundle isomorphism. (In particular, "V" and "M" must have the same dimension.)

A motivating example of a solder form is the tautological or fundamental form on the frame bundle of a manifold.

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"-structures and are important in the theory of Cartan connections. The terminology and approach is particularly popular in the physics literature.

Notes

See also

* vielbein

References

*cite journal | first = C. | last = Ehresmann | title = Les connexions infinitésimals dans un espace fibré differential | journal = Colloque de Topologie, Bruxelles | year = 1950 | pages = 29–55
*cite journal | first = Shoshichi | last = Kobayashi | title = Theory of Connections | journal = Ann. Mat. Pura Appl. | year = 1957 | volume = 43 | pages = 119–194 | doi = 10.1007/BF02411907
*cite book | author=Kobayashi, Shoshichi and Nomizu, Katsumi | title = Foundations of Differential Geometry, Vol. 1 & 2 | publisher=Wiley-Interscience | year=1996 (New edition) |id = ISBN 0471157333