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:
- A distinguished section o : M → E.
- 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
Famous quotes containing the words fibre and/or bundle:
“In the man whose childhood has known caresses and kindness, there is always a fibre of memory that can be touched by gentle issues.”
—George Eliot [Mary Ann (or Marian)
“We styled ourselves the Knights of the Umbrella and the Bundle; for, wherever we went ... the umbrella and the bundle went with us; for we wished to be ready to digress at any moment. We made it our home nowhere in particular, but everywhere where our umbrella and bundle were.”
—Henry David Thoreau (1817–1862)