Solder Form - Soldering of A Fibre Bundle

Soldering 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:

  1. A distinguished section o : M → E.
  2. A linear isomorphism of vector bundles θ : TM → o−1VE 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.

Read more about this topic:  Solder Form

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