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Concept
Solder form
Summary
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 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 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.
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 : TM → o*VE 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
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.
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 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
However, for a vector bundle there is a canonical isomorphism between the vertical space at the origin and the fibre VoE ≈ E.
Official source
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